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Casciaro1B.Casciaro, M.Francaviglia, V.Tapia,
On the Variational Characterisation of Generalized Jacobi Equations,
Differential Geometry and Applications, Proc. Conf., Aug. 28 - Sept. 1, 1995, Brno, Czech Republic (Masaryk University, Brno, 1996), pp. 353-372;

Casciaro2B. Casciaro, M. Francaviglia,
Covariant second variation for first order Lagrangians on fibered manifolds. I: Generalized Jacobi fields,
Rend. Matem. Univ. Roma, VII (16) (1996), pp. 233-264

Casciaro3O. Amici, B. Casciaro, M. Francaviglia,
The Perturbation Functor in the Calculus of Variations,
Rend. Mat. Roma yy, xx-xx (2004)

Note  Claudio Gorodski,
An introduction to Riemannian geometry,
(to appear)

Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory,
Lambert Academic Publishing, 2013. ISBN 978-3-659-37815-7; arXiv: 0908.1886

HES.W.Hawking,  G.F.R. Ellis,
The Large Scale Structure of Space-Time,
Cambridge University Press, 1973

The Geometry of Jet Bundles,
Cambridge University Press (1989)

AndersonIan M. Anderson,
Introduction to the Variational Bicomplex,
in Mathematical Aspects of Classical Field Theory (ed. byM. Gotay, J.Marsden, V.Moncrief), Comptemporary Mathematics Vol 132, 1992.

Book1L.Fatibene, M.Francaviglia,
Natural and Gauge-Natural theories,
Kluwer, Dordrecht, (2003)

Relativistic theories, gravitational theories and General Relativity,
in preparation, draft version 1.0.2.

Palese1M.Francaviglia, M.Palese,
Generalized Jacobi morphisms in variational sequences,
in: Jan Slovák and Martin Čadek (eds.):
Proceedings of the 21st Winter School "Geometry and Physics".
Circolo Matematico di Palermo, Palermo, 2002.
Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento No. 69. pp. [195]—208.)

SternbergH.Goldschmidt, S. Sternberg,
The Hamilton–Cartan Formalism in the Calculus of Variations,
Ann. Inst. Fourier, Grenoble 23 (1) (1973) 203–267.

Palese2M. Francaviglia, M. Palese, R. Vitolo,
The Hessian and Jacobi Morphisms for Higher Order Calculus of Variations,
Differential Geometry and its Applications, vol. 22 (1) (2005), p. 105-120,
ISSN: 0926-2245

Palese3M. Palese, E. Winterroth,
The relation beetween the Jacobi Morphism and the Hessian in gauge-natural field theories,
Theoretical and Mathematical Physics, vol. 152(2) (2007), p. 377-389,
ISSN: 0040-5779

Accornero1L. Accornero, M. Palese,
Symmetry transformations of extremals and higher conserved quantities: Invariant Yang–Mills connections,
Journal of Mathematical Physics 62, 043504 (2021)

Accornero2L. Accornero, M. Palese,
The Jacobi morphism and the Hessian in higher order field theory; with applications to a Yang–Mills theory on a Minkowskian background,
Int. J. Geom. Meth. Mod. Phys., Vol. 17, No. 08, 2050114 (2020)

A variational framework for higher order perturbations

F.Chiaffredo^1, L.Fatibene^1,2, M.Ferraris^1, E.Ricossa^1, D.Usseglio^3,4

$^1$ Dipartimento di Matematica - University of Torino
Via C. Alberto 10, 10123, Torino, Italy

$^2$ INFN - University of Torino
Via P.Giuria, 1, 10125, Torino, Italy

$^3$ Scuola Superiore Meridionale,
Largo San Marcellino 10, 80138, Naples, Italy

$^4$ INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo
Via Cintia Edificio 6, 80126, Naples, Italy

a covariant, global, variational framework for perturbations in field theories is presented.
Perturbations are obtained as vertical vector fields on the configuration bundle and they drag, exactly, solution into solutions.

The flow of a perturbation drags solutions into solutions and the dragged perturbed solutions can be expanded in a series
with respect to the flow parameter,
hence it contains perturbations at any order.

Mechanics is included as a special case.
As a simple application, we recover the well-known discussion about stability of geodesics on a sphere $S_2$.

We present a geometric framework which extends variational calculus to
(higher order) perturbations,
aiming to  deal with perturbations and stability both in mechanics and field theory; see <ref>, <ref>, <ref>.

We shall provide a variational derivation of generalized Jacobi equations,
which is valid for mechanics and field theories.
Solutions of Jacobi equations are vertical vector fields on the configuration bundle, the prolongations of which are tangent to the field equations of the original Lagrangian $L$.
Accordingly, they exactly drag solutions into solutions, at first order they reproduce first order perturbations, although they account for perturbations at any order.

Usually, first order perturbations of a solution $\bar y^i(x)$ are obtained by considering a deformation $y^i(x)\simeq  \bar y^i(x) + \de y^i(x)$ and using field equations to determine $\de y^i$
so that $y^i(x)$ are still approximate solutions at first order.
That is more or less how one defines a vector $\de y^i$ tangent to a manifold $M$ at the point $\bar y^i$
so that the curve $\ga:t\mapsto \bar y^i + t\de y^i$ lies on the manifold at first order.
In this sense, a first order perturbation is analogous to a tangent vector to the space of solutions at a given reference solution $\bar y^i(x)$.

Jacobi fields instead are analogous to a vector field tangent to the space of solutions. Their flows preserve the space of solutions and define integral curves made of exact solutions.
This framework is then expected to provide a geometric setting to discuss, for example, gravitational waves or propagation of acoustic baryonic oscillations in cosmology.

At the same time, having a Lagrangian setting for Jacobi fields allows us to iterate the procedure and look for the analogous of second order (or higher) variations;
see <ref>, <ref>, <ref>, <ref>, <ref>, <ref> for a different, somehow equivalent, framework.
Second order variations are useful to discuss stability of solutions, as second derivatives are useful to discuss critical points.

The framework is based on calculus of variations on bundles to keep under control global properties and transformation laws of the objects involved.
Here, for the sake of simplicity, we shall consider first order Lagrangians even though most results extend to higher order.

The paper is organized as follows.
In Section 2, we shall briefly recall standard notation for variational calculus on fiber bundles.
In Section 3, we shall introduce composite bundles which are a slight generalization and extend variational calculus on them.

In Section 4, we introduce the Jacobi Lagrangian which determines Jacobi fields, i.e. perturbations of solutions, of a first order Lagrangian.
Jacobi Lagrangian provides also a class of examples of variational principles on composite bundles.
We shall show in general that the  prolongation (of a suitable order) of Jacobi fields are tangent to the original field equations thought as a sub-manifold
of a jet prolongation of the configuration bundle, in this case on $J^2B$. That also proves that a Jacobi flow drags, exactly, a solution into solutions.

In Section 5, we discuss the iteration of the process defining second variations into the variations of the variations of the original theory.
That is just as the second order tangent bundle $T^2M$ can be defined as a sub-bundle of $T(TM)$, or as the second jet prolongation $J^2B$
can be defined as a sub-bundle of $J^1(J^1B)$.
The key point is that linearization to obtain Jacobi equations can be recognized as a functor, just as $T(\cdot)$ and $J^1(\cdot)$ and, consequently, it can be iterated at will.
With second variations we can define stability and discuss it in a completely classical setting.

In Section 6, we shall discuss as an example how one can directly discuss stability of geodesics on the  sphere $S_2$; see <ref>.
That is a relatively well-known and classical result in differential geometry, although here we discuss it by a methods which closely recalls the discussion about propagation of acoustic baryonic density perturbations.

Finally, we add an Appendix to show directly that the variational framework for Jacobi equations is well suited also for second order Lagrangians.

Standard setting for variational calculus

We consider a fiber bundle $[B\arrLabel{\pi} M]$, called the configuration bundle. A configuration is a (global) section $\si:M\arr B$ of the configuration bundle.
The space $\Ga(\pi)$ of all (global) sections of the configuration bundle is called the configuration functional space.

For mechanics, we can set $M=\R$. Any bundle onto a contractible base manifold, as $\R$, is trivial, thus we necessarily have $B\simeq \R\times Q$.
In this case, sections of the configuration bundle $\pi: \R\times Q \arr \R$ are in one-to-one correspondence with parameterized curves $\ga:\R\arr Q$.

More generally, we can consider $M$ to be the spacetime, so that sections are locally represented as $\si: x^\mu\mapsto (x^\mu, y^i(x))$
and $y^i(x)$ are fields on spacetime. Transition maps of the configuration bundle $[B\arr M]$ account for transformation laws of fields and one can specialize to scalar, vector, tensor fields
or any other geometric object.

In other words, the formalism can be specialized to both mechanics and field theories.

In order to capture the geometric essence of variational calculus, we need  a bundle which accounts for partial derivatives of sections, just as in mechanics the tangent bundles account for the derivatives
of parameterized curves in terms of a tangent vector.
For a general introduction to jet bundles see <ref>, <ref>, <ref>, <ref>, <ref>.
This bundle is called the (first) jet prolongation and it is denoted by $[J^1B\arrLabel{\pi^1_0} B\arrLabel{\pi} M]$.
If $B$ has fiber coordinates $(x^\mu, y^i)$, then its first jet prolongation $J^1B$ has fibered coordinates $(x^\mu, y^i, y^i_\mu)$, where
$y^i_\mu$ account for first partial derivatives of fields with respect to base points.

Given transition maps on $B$ in the form

x'^\mu = x'^\mu(x)
\qquad
y'^i = y'^i(x, y)
\fn$$

they induce transition maps on $J^1B$ as

y'^i_\mu = \bar J_\mu^\nu \( J^i_\nu + J^i_k y^k_\nu \)
\qquad\qquad\(\hbox{where we set $\bar J_\mu^\nu=\frac[\del x^\nu/\del x'^\mu]$, $J^i_\nu=\frac[\del y'^i/\del x^\nu]$, $J^i_k=\frac[\del y'^i/\del y^k]$} \)
\fn$$

That simply means one knows how first derivatives of fields transform once one knows how fields transform.

Jet prolongations can be considered at higher orders, e.g. $[J^2B\arrLabel{\pi^2_1} J^1B\arrLabel{\pi^1_0} B\arrLabel{\pi} M]$ is the second jet prolongation,
$J^2B$ has fiber “coordinates” $(x^\mu, y^i, y^i_\mu, y^i_{\mu\nu})$, where $y^i_{\mu\nu}$ account for the second order partial derivatives of fields,
they are hence understood to be symmetric in the lower indices, and they transform as second derivatives.

Jet bundles provide a framework in which systems of partial differential equations (PDE) of order $k$ become constraints and they are intrinsically
represented as sub-manifolds $\E\subset J^{2k}B$
In the special case of mechanics, one has $J^{2k}B\simeq \R\times T^{2k}Q$ and sub-manifolds $\E\subset \R\times T^{2k}Q$ represent systems of ODE.

Let us remark that $J^k(\cdot)$ are covariant functors from bundles to bundles. In fact, $J^kB$ is a bundle, and any bundle map $\Phi:B\arr B'$
(which projects onto a diffeomorphism $\phi:M\arr M'$) is prolonged to a bundle morphism $J^k\Phi:J^kB\arr J^kB'$.

Accordingly, one has for free that any section $\si:M\arr B$ can be prolonged to $J^k\si:M\arr J^kB$, any fibered flow $\Phi_s$ can be prolonged to a flow $J^k\Phi_s$, any projectable vector field $\Xi$
can be prolonged to a vector field $J^k\Xi$.
That directly generalises the situation one has in mechanics with tangent bundles, tangent maps, the tangent prolongation of a curve, the prolongation of flows,
and the tangent lift of vector fields.

In variational calculus, the dynamics can be given by a Lagrangian, namely a global horizontal form on the jet bundle $J^kB$.
Locally, a Lagrangian is

\Lag=L(J^ky) d\si
=L(x^\mu, y^i, y^i_\mu, \dots,  y^i_{\mu_1\dots \mu_k}) d\si
\fn$$

where $d\si=dx^1\land\dots \land dx^m = d^mx$ is the local basis of $m$-forms on $M$ induced by coordinates,
hence the coefficient $L(J^ky)$ is a scalar density (iff $\Lag$ is a global form).

Since $\Lag$ is a form on $J^kB$ and $J^k\si: M\arr J^kB$, we have a form

(J^k\si)^\ast \Lag = L(x^\mu, y^i(x), \del_\mu y^i(x), \dots,  \del_{\mu_1\dots \mu_k} y^i(x)) d\si
\fn$$
on $M$
that can be integrated on a compact domain $D$ (with a compact boundary $\del D$)  to define the action functional

A_D[\si] = \int_D (J^k\si)^\ast \Lag
\fn$$

which is guaranteed to be finite as long as fields are regular.
The action functional is a functional on the functional space  $\Ga(\pi)$ of
global sections of the configuration bundle.

A deformation on $D$ is a vertical vector field $X$ on the configuration bundle, which vanishes on the boundary $\del D$ (together with derivatives up to order $k-1$
for a Lagrangian of order $k$).
A deformation defines a vertical flow $\Phi_s:B\arr B$ on the configuration bundle, which can be prolonged to a vertical flow on the jet bundle $J^k\Phi_s:J^kB\arr J^kB$
or used to produce a 1-parameter family of configurations $\si_s=\Phi_s\circ \si$ that deforms a reference section $\si$.

We define a critical configuration $\si$ to be a section such that for all compact domains $D$ and for all deformations on $D$, the action $A_D[\si]$ is left unchanged at first
order, i.e. such that

\de_X A_D[\si] = \int_D \Frac[d/ds] (J^k\si_s)^\ast \Lag \Big|_{s=0}
=\int_D(J^k\si)^\ast  \Frac[d/ds]  \(J^k\Phi_s\)^\ast  \Lag \Big|_{s=0} =0
\fn$$

There is a beautiful geometric way of expressing the stationarity of the action, namely, since $X$ is vertical, we obtain

\de_X\Lag := \Frac[d/ds] \(J^k\Phi_s\)^\ast  \Lag \Big|_{s=0} = \Lie_{J^kX} \Lag= i_{J^kX}  d\Lag + d\(i_{J^kX} \Lag\)= i_{J^kX}  d\Lag =0
\fn$$

which is particularly interesting because it can be written in terms of forms on a finite dimensional manifold, with no reference whatsoever to the functional space.
The expression we found for $\de_X \Lag=  i_{J^kX}  d\Lag $ is particularly interesting since it is a global horizontal $m$-form on $J^k V(B)$ (as a bundle over $M$) which is intrinsically given.

By a suitable series of integrations by parts, this can be recast as

\de_X\Lag = \E(X) + d \(\F(J^{k-1}X)\)
\fn$$

which is called the first variation formula and it encodes everything about variational calculus in an intrinsic way.

The global horizontal $m$-form $\E(X)=\E_i X^i d\si$ is on $J^{2k}B$, it is called the Euler-Lagrange form.
The global horizontal $(m-1)$-form $\F(J^{k-1}X)$ is on $J^{2k-1}B$, it is called the Liouville form.
In the action, these forms are evaluated along the prolongation of a configuration and then integrated on $D$. In particular, one can use Stokes theorem and integrate $\F(J^{k-1}X)$ on
the boundary where the deformations $J^{k-1}X$ vanish, hence the Liouville form does not contribute to the variation of the action.
Therefore,  field equations are $\E_i=0$, while the Liouville form encodes conservation laws, which will not be discussed  here.

Any global Lagrangian $\Lag$ (of order $k$) produces global equations $\E_i = 0$.

For a first order Lagrangian, we can set $p_i=\frac[\del L/\del y^i]$ and $p_i^\mu=\frac[\del L/\del y^i_\mu]$ and one can easily show that, in coordinates, one has

\de_X\Lag = i_{J^1X}  d\Lag = \( p_iX^i +p_i^\mu X^i_\mu\) d\si
=  \( p_i -d_\mu p_i^\mu\) X^i d\si + d\( p_i^\mu X^id\si_\mu \)
\fn$$

Then we have field equations in the form  $\E_i =p_i -d_\mu p_i^\mu= 0$.
We could directly show that $\E_i'= \bar J \E_k \bar J^k_i$, hence the Euler-Lagrange form is a global form.

Composite bundles

Before proceeding, we wish to generalize variational principles on a bundle to composite bundles.
A composite bundle of height $k$ is a ladder $[B_k\arr \dots \arr B_2\arr B_1\arr M]$ of bundles.
We have projections $\pi^{n+1}_n:B_{n+1}\arr B_n$ at each level, and compositions $\pi^{n+h}_n: B_{n+h}\arr B_n$.

A composite morphism from $[B_2\arr B_1\arr M]$ to $[B'_2\arr B'_1\arr M']$ is a triple of maps with $\Phi_2:B_2\arr B'_2$, $\Phi_1:B_1\arr B'_1$, $\Phi_0:M\arr M'$
such that they define a fibered morphism at each level (hence even at any multilevel block).

That means that, locally, composite morphisms (as well as transition maps) are stratified, i.e. they are in the form

x'^\mu = \Phi^\mu(x)
\qquad\qquad
y'^i = \Phi^i(x, y)
\qquad\qquad
z'^a = \Phi^a(x, y, z)
\fn$$

Let us remark that a general fibered morphism, e.g. for $[B_2\arr M]$, is not a composite morphism, just as it is not stratified.
We really restrict morphisms, hence composite bundles have additional structure to be preserved with respect to ordinary bundles.

A composite section of a composite bundle $[B_2\arrLabel{\pi^2_1} B_1\arrLabel{\pi^1_0} M]$ is a pair of sections $\si_1:M\arr B_1$ and $\si_2:B_1\arr B_2$.

Locally, a composite section is given by $(y^i(x), z^a(x, y))$, while a section $\si: M\arr B_2$ is given by $(y^i(x), z^a(x))$.
Accordingly, when we define jet prolongations, we need to account for first derivatives $(y^i_\mu, z^a_\mu, z^a_i)$, as well as the second derivatives
$(y^i_{\mu\nu}, z^a_{\mu\nu}, z^a_{k\mu}, z^a_{ik})$.

For first derivatives, the construction is simple.
If we consider the first prolongation $J^1(\pi^1_0)$ of the bundle $[B\arr M]$, we obtain a bundle $[\pi^1_B:J^1(\pi^1_0)\arr B]$ with coordinates $(x^\mu, y^i; y^i_\mu)$.

to 9cm[1]

(190, 180)[P1]$\P_1$
(190, 240)[P2]$\P_2$

(130, 240)[JC]$\vdots$
(130, 200)[J2C]$J^2C$
(130, 140)[J1C]$J^1C$
(190, 100)[C]$C$
(190, 50)[B]$B$
(250, 110)[J1B]$J^1B$
(250, 170)[J2B]$J^2B$
(250, 210)[JB]$\vdots$

(190, 0)[M]$M$
(220, 95)[L]$\pi^1_B$

\mor{P1}{J1B}{$$}
\mor{P2}{P1}{$\pi^4_3$}
\mor{P2}{J2C}{$$}
\mor{P2}{J2B}{$$}
\mor{B}{M}{$\pi^1_0$}
\mor{C}{B}{$\pi^2_1$}
\mor{J1B}{B}{$$}[\atright, \solidarrow]%%% un po' più in alto?
\mor{J1C}{C}{$\pi^1_C$}[\atleft, \solidarrow]%%% un po' più in alto?
\mor{J2B}{J1B}{$\pi^2_{B1}$}
\mor{JB}{J2B}{$$}
\mor{J2C}{J1C}{$\pi^2_{C1}$}
\mor{JC}{J2C}{$$}
\cmor((200,2)(208,12)(211,25)(208,45)(200,50)) \pleft(215,40){$\si$}
\cmor((200,-3)(230,35)(248,93)) \pup(250,45){$J^1\si$}
\cmor((180,52)(172,60)(169,76)(172,92)(180,100)) \pright(165,85){$\rho$}
\cmor((180,50)(160,70)(130,130)) \pup(130,100){$J^1\rho$}
%\cmor((245,95)(270,105)(278,123)) \pup(280,105){$\hat \xi$}
\enddc

\SetMargin{12cm}
\vskip-9.3cm
On the other hand, we consider the first prolongation $J^1(\pi^2_1)$ of the bundle $[C\arr B]$, i.e.~a bundle $[\pi^1_C:J^1(\pi^2_1)\arr C]$ with coordinates $(x^\mu, y^i, z^a; z^a_\mu, z^a_i)$.
We  need to smash these two spaces together, which could be done by fiber product if they were bundles on the same base manifold.
Then we can define
\P_1 := J^1(\pi^2_1)\times_C (\pi^2_1)^\ast \( J^1(\pi^1_0)\)
\fn$$
which, in fact, has coordinates $(x^\mu, y^i, z^a; y^i_\mu, z^a_\mu, z^a_i)$. The fiber product comes with a projection $\pi^3_2:\P_1\arr C$.

For the second jet prolongation, we consider the second prolongation $J^2(\pi^1_0)$ of the bundle $[B\arr M]$, we obtain a bundle $[\pi^2_B:J^2(\pi^1_0)\arr B]$ with coordinates $(x^\mu, y^i; y^i_\mu,  y^i_{\mu\nu})$,
as well as the second prolongation $J^2(\pi^2_1)$ of the bundle $[C\arr B]$, i.e.~a bundle $[\pi^2_C:J^2(\pi^2_1)\arr C]$ with coordinates
$(x^\mu, y^i, z^a; z^a_\mu, z^a_i, z^a_{\mu\nu}, z^a_{i\mu}, z^a_{ij})$.

Again, we can smash these bundles together by
\P_2 := J^2(\pi^2_1)\times_C (\pi^2_1)^\ast \( J^2(\pi^1_0)\)
\fn$$
which, in fact, has coordinates

(x^\mu, y^i, z^a; y^i_\mu, z^a_\mu, z^a_i,y^i_{\mu\nu},z^a_{\mu\nu}, z^a_{i\mu}, z^a_{ij}  )
We can define a projection $\pi^4_3:\P_2\arr \P_1$.
Similarly, at higher orders.

\Note
Let us define the {\it total derivatives}
$d_\mu y^i=y^i_\mu$ and $d_\mu z^a =  z^a_\mu + y^k_\mu z^a_k$,
as well as at second order
$d_{\mu\nu} y^i=y^i_{\mu\nu}$ and $d_{\mu\nu} z^a =  z^a_{\mu\nu} + 2z^a_{k(\mu} y^k_{\nu)} + y^k_\mu y^h_\nu z^a_{kh}+ y^k_{\mu\nu} z^a_k$.
\EndNote

Given a composite section $(\si:M\arr B, \rho:B\arr C)$
we can prolong $\si$ to $J^1\si: M\arr J^1B$ and $\rho$ to $J^1\rho: C\arr J^1C$, as in the diagram alongside.
We cannot define a section from $C$ to $\P_1$ (or from $B$ to $\P_1$).
As in the standard case, we can define, however, a section $J^1(\si,\rho): M\arr \P_1$ such that $\pi^3_2\circ J^1(\si, \rho)= \rho\circ \si$
and $\pi^3_1\circ J^1(\si, \rho)= \si$.
One starts from $x\in M$ and mark $\si(x)\in B$ and $\rho\circ\si(x)\in C$, as well as $J^1\si(x)\in J^1B$ and $J^1\rho( \si(x))\in J^1C$.

By construction, we have $\pi^1_B\(J^1\si(x)\) = \si(x) = \pi^2_1\( \rho\circ \si(x)\)$.
Now check in the definition of pull-back bundles, when one has a point $J^1\si(x)\in J^1B$ and $ \rho\circ \si(x)\in C$ such that $\pi^1_B\(J^1\si(x)\) = \pi^2_1\( \rho\circ \si(x)\)$,
that defines a point $(\rho\circ \si(x), J^1\si(x))\in (\pi^2_1)^\ast \( J^1(\pi^1_0)\)$.
The projection on $C$ is defined as $\pi^\ast: (\pi^2_1)^\ast \( J^1(\pi^1_0)\)\arr C: (\rho\circ \si(x), J^1\si(x))\mapsto \rho\circ \si(x)$.
Analogously, as one can check in the definition of product, when one has  a point $(\rho\circ \si(x), J^1\si(x))\in (\pi^2_1)^\ast \( J^1(\pi^1_0)\)$
and a point $J^1\rho\(\si(x)\)\in J^1C$
such that $\pi^\ast(\rho\circ \si(x), J^1\si(x)) = \rho\circ\si(x) = \pi^1_C(J^1\rho\(\si(x)\)) \in C$, that defines a point $\(J^1\rho\(\si(x)\), (\rho\circ \si(x), J^1\si(x))\)\in J^1C\times_C  (\pi^2_1)^\ast \( J^1(\pi^1_0)\) =\P_1$.

Then we can define a map
J^1(\si,\rho): M\arr \P_1: x\mapsto \(J^1\rho\(\si(x)\), (\rho\circ \si(x), J^1\si(x))\)
\fn$$
and check it is a section by computing $\pi^3_1\circ  J^1(\si,\rho)= \si $ and
$\pi^3_2 \circ J^1(\si,\rho)=  \rho\circ \si$.
Finally, one has $\pi^3_0 \circ J^1(\si,\rho) = \pi^1_0 \circ \si$.
In other words, $J^1(\si,\rho): M\arr \P_1$ is a section on $\rho\circ \si$ and hence  on $\si$.
Similarly, one can extend the construction to higher order prolongations.

\Note
Locally, if the composite section is expressed as $\si:M\arr B: x\mapsto (x, y(x))$ and $\rho:B\arr C: (x, y)\mapsto (x, y; z(x, y))$ (and, accordingly,
$\rho\circ \si:M\arr C: x\mapsto (x, y(x), z(x, y(x)))$), then its prolongation to $\P_1$ is
J^1(\si,\rho): M\arr \P_1:x\mapsto \(x^\mu, y^i(x), z^a(x, y(x)), \del_\mu y^i(x), \del_\mu z^a(x, y(x)), \del_i z^a(x, y(x)) \)
\fn$$
\EndNote

\NewSection{Linearized equations}

Let us now consider an ordinary variational principle, defined on a configuration bundle $[B\arr M]$ by a first order Lagrangian $\Lag$.
As a matter of fact, the variation of the Lagrangian $\de \Lag:J^1B \arr V^\ast(J^1 B)\otimes A_m(M)$ induces an auxiliary first order Lagrangian
$J(\Lag)= J^1X \ip d\Lag$
which is called the {\it Jacobi Lagrangian}, on the
composite bundle $[V(B)\arr B\arr M]$.
Let us stress that $J(\Lag)$ is a horizontal $m$-form in $J^1V(B)\simeq V(J^1B)$, i.e.~in the form
J(\Lag)= L_J(x^\mu, y^i, y^i_\mu, X^i, X^i_\mu)\> d\si
= (p_i X^i + p_i^\mu d_\mu X^i)d\si
\fn$$
Notice that the Jacobi Lagrangian $J(\Lag)$ happens to be linear in the vertical vector fields $X$.

Then we can consider a deformation $\Xi = \de y^i\frac[\del/\del y^i] + \de X^i \frac[\del/\del X^i] $ on $V(B)$ and the variation of the Lagrangian $J(\Lag)$.

\Note

We have
\eqalign{
J^1\Xi \ip d(J(\Lag))=& (p_i \de X^i + p_i^\mu d_\mu \de X^i  )
+  (\del_k p_i \de y^k + \del_k^\al p_i d_\al \de y^k ) X^i
+ (\del_k p_i^\mu \de y^k  + \del_k^\al  p_i^\mu d_\al \de y^k   ) d_\mu X^i =\cr
=& \(p_i -d_\mu p_i^\mu  \)  \de X^i+ \(  \del_k p_i  X^i + \del_k p_i^\mu  d_\mu X^i
- d_\al\( \del_k^\al p_i  X^i + \del_k^\al  p_i^\mu  d_\mu X^i   \)  \) \de y^k +\cr
&+d_\mu\(   p_i^\mu   \de X^i +  \( \del_k^\mu p_i  X^i + \del_k^\mu  p_i^\la  d_\la X^i   \)  \de y^k  \) \cr
%=& \de^2 L
\fl{JacobiEqs}$$
We obtain as a first field equation  $\E_i= p_i -d_\mu p_i^\mu =0$, i.e.~the same field equations as for the original Lagrangian $\Lag$.
For the second field equation, we can write it as
X^i \del_i \E_k  +  d_\mu X^i \del_i^\mu \E_k + d_{\mu\nu} X^i \del_i^{\mu\nu} \E_k    =0
\qquad\then
J^2X (\E_i) =0
\fl{Feq}$$
which means the jet prolongation of the  vector field $X$ (vertical on the original configuration bundle $[B\arr M]$) is tangent to the original field equations
thought as a sub-manifold into $J^2B$.
\EndNote

\ni
As a consequence, the flow of $X$ drags solutions of $\Lag$ into solutions.
Because of that, the second field equation is called the {\it linearized equations}
(or the {\it Jacobi equation}) of the original field equations $\E_k= p_k-d_\mu p_k^\mu =0$.
A solution $X$ of linearized equations is called a {\it perturbation} (or a {\it Jacobi field}).

Let us stress that the  linearized equations  come with a variational origin,
they are generated by the Jacobi Lagrangian $J(\Lag)$.

\Note
Usually, in first order perturbation theory, one  expands the original field equation at first order about a solution $\bar y^i(x)$.
That means that one considers $y^i(x)\simeq \bar y^i(x) + \ep X^i(x)$, replaces it into the field equations $\E_k=0$, keeps first order terms in $\ep$,
and interprets it as equations of $X^i$.
When these equations are satisfied, $y^i(x)$ is still a solution {\it at first order}.

Jacobi equations are more complicated, however, the configurations generated by a flow along a Jacobi field of a solution are {\it exact} solutions.
Thus they can be expanded at any order providing us with the whole series of perturbations at any order.

To check that the two prescriptions produce the same linearized equations, we restrict, for the sake of simplicity, to a first order Lagrangian $\Lag=L(x, y, dy)d\si$
which produces equations $E_k=p_k-d_\mu p_k^\mu=0$.
The zero order term is identically satisfied since we assumed $\bar y^i$ to be a solution of the original equations.
The first order term is
X^i  \del_i p_k +  X^i_\mu\del_i^\mu p_k -d_\mu (X^i\del_i p_k^\mu    + X^i_\ep \del_i^\ep p_k^\mu) =0
\fn$$

On the other hand, when we specialize the Jacobi equations \ShowLabel{JacobiEqs} and we expand them at first order, we obtain
\eqalign{
&X^i  \del_i p_k +  X^i_\mu   \del_i^\mu p_k - X^i  d_\mu \del_i p_k^\mu - X^i_\mu  \del_i^\mu d_\al p_k^\al -  X^i_{\mu\nu} \del_i^\nu  p_k^\mu =\cr
%& \qquad = X^i  \del_i p_k +  X^i_\mu   \del_i^\mu p_k - d_\mu (X^i  \del_i p_k^\mu) + \red{X^i_\mu  \del_i p_k^\mu}- `X^i_\mu d_\al \del_i^\mu  p_k^\al  - \red{X^i_\mu  \del_i  p_k^\mu}-  X^i_{\mu\nu} \del_i^\nu  p_k^\mu   =\cr
& \qquad = X^i  \del_i p_k +  X^i_\mu   \del_i^\mu p_k - d_\mu (X^i  \del_i p_k^\mu+ X^i_\nu  \del_i^\mu  p_k^\nu)
\fn$$
Thus we see we obtain the same equations for first order perturbations, as expected.
\EndNote

Jacobi field equations \ShowLabel{Feq} are linear in $X$, therefore Jacobi fields form a real vector space $\calJ$.
Of course, that does not mean it is easy to find Jacobi fields.

\NewSection{Second variations and stability}

%\rightskip -2cm
\vbox to 7cm{\begindc{\commdiag}[1]
%\obj(260, 90)[hxi]{$\hat \xi$}
\obj(200, 130)[JVB]{$J^1V(B)$}
\obj(220, 143)[L']{$L_J$}
\obj(230, 90)[VB]{$V(B)$}
\obj(270, 130)[VVB]{$V^2(B)$}
%\obj(220, 50)[xi]{$\xi$}
\obj(190, 50)[B]{$B$}
\obj(150, 90)[JB]{$J^1B$}
\obj(165, 100)[L]{$L$}
\obj(190, 0)[M]{$M$}
\mor{B}{M}{$\pi$}
\mor{VB}{B}{$\pi_V$}
\mor{JVB}{VB}{$\pi^1_V$}
\mor{JB}{B}{$\pi^1_0$}
\mor{VVB}{VB}{}
\cmor((200,0)(208,10)(211,25)(208,45)(200,50)) \pleft(220,25){$\si$}
\cmor((200,55)(230,65)(238,83)) \pup(240,65){$X$}
\cmor((245,95)(270,105)(278,123)) \pup(280,105){$\hat X$}
\enddc

\vskip -7.3cm
\SetMargin{11cm}

For linearized equations, we started on $B$, we considered a deformation $X$ on $B$, then we considered a new Lagrangian $J(\Lag)$ on $V(B)$, a new deformation $\Xi$ on $V(B)$ and consider the variation of $J(\Lag)$
with respect to the deformation $\Xi$. Let us remark that there is no direct relation between $X$ and $\Xi$.

Now, for second order variation, we act differently. We consider a theory on $B$, consider a deformation $X= X^i\del_i$ (i.e.~a vertical vector field on $B$).
By the flow $\Phi_s$ of $X$, we can drag a section $\si$ as $\si_s = \Phi_s \circ \si$, which is locally represented by $y^i_s(x)$ and expand it as

y^i_s(x)= y^i(x) + s \de y^i(x) +\frac[1/2] s^2 \de^2 y^i(x) + O(s^3)
\fn$$
where we set $\de y^i(x):= X^i(x, y(x))$ and $\de^2 y^i(x): = \del_j X^i(x, y(x))  \de y^j(x)$.

\Note
We start from a first order Lagrangian $L(x, y_s(x), d y_s(x))$. For $s=0$, we get $L$.
The first derivative is
\eqalign{
&\Frac[d L/ ds]
=p_i (x, y_s, d y_s) X^i(x, y_s)+ p_i^\mu  (x, y_s, d y_s) d_\mu X^i(x, y_s)
\fn$$
which then specializes, for $s=0$, to $\de \Lag=J(\Lag)$.
The second derivative is then
\eqalign{
&\Frac[d^2 L/ ds^2] \Big|_{s=0}\kern-10pt
= \( p_i  \>\de^2 y^i + p_i^\mu \> d_\mu  \de^2 y^i \)
+ \(\del_k p_i \de y^k+ \del_k^\al p_i d_\al \de y^k\) \de y^i
+ \(\del_k p_i^\mu \de y^k+ \del_k^\al p_i^\mu  d_\al \de y^k\) d_\mu \de y^i \cr
\fn$$

Notice how that corresponds to set $X^i=\de y^i$ (and consequently $\de X^i= \de^2 y^i$) in the variation $\de\(J(\Lag)\)$ of the Jacobi Lagrangian.

\EndNote

The Lagrangian  can be expanded at second order.
\eqalign{
&L(x, y_s, dy^i_{s}) = L + s \(p_i \de y^i + p_i^\mu d_\mu \de y^i\) +\frac[s^2/2]  \Big(\( p_i  \>\de^2 y^i + p_i^\mu \> d_\mu  \de^2 y^i \) +\cr
+ \(\del_k p_i\> \de y^i  \de y^k + 2\del_i p_k^\al \> \de y^i d_\al \de y^k + \del_k^\al p_i^\mu \> d_\mu \de y^i  d_\al \de y^k \) \Big) +O(s^3)
\fn$$

To summarize, we have
L_s =L+ s \de L +\frac[1/2] s^2  \de^2 L +O(s^3)
\fn$$
where we set
\eqalign{
& \de^2 \Lag  :=  \(\( p_i  \>\de^2 y^i + p_i^\mu \> d_\mu  \de^2 y^i \)
+ \(\del_k p_i\> \de y^i  \de y^k + 2\del_i p_k^\al \> \de y^i d_\al \de y^k + \del_k^\al p_i^\mu \> d_\mu \de y^i  d_\al \de y^k \) \) d\si
\fn$$
Let us check that the second variation $\de^2 \Lag$ of a global first order Lagrangian $\Lag$ is a geometric object.

\Note
We consider a global Lagrangian on $B$, i.e.
\Lag= L(x^\mu, y^i, y^i_\mu) d\si
\fn$$

For globality, we have $JL'(x', y'^i, y'^i_\mu)= L(x^\mu, y^i, y^i_\mu)$ and consequently
p_k^\al = J \bar J^\al_\mu p'^\mu_i J^i_k
\qquad
p_k=J p'_i J^i_k  + J \bar J^\mu_\al p'^\al_i\(J^i_{\mu k} + J^i_{jk} y^j_\mu \)=
J \(p'_i J^i_k  + \bar J^\la_\mu p'^\mu_i (d_\la J^i_{k})\)
\fn$$

For second derivatives, we have
\del_i^\mu p_k^\al =J \bar J^\al_\be \bar J^\mu_\nu \del'^\nu_jp'^\be_h J^h_k  J^j_i
\fn$$
%$ \del_J^\mu= \bar J^\mu_\nu J_J^i \del'^\nu_i$
\del_i p_k^\al = J \bar J^\al_\mu ( \del'_l  p'^\mu_j  J^j_k J_i^l  +   \bar J_\rho^\si  \del'^\rho_ l p'^\mu_j J^j_k (d_\si J^l_i )    +   p'^\mu_j J^j_{ik})
= \del_k^\al p_i
\fn$$
%$ \del_i= J_i^l \del'_l  +  \bar J_\mu^\nu (d_\nu J^l_i ) \del'^\mu_ l $

Finally, we have
\eqalign{
\del_j p_k=&J \(\del'_l p'_i J_J^l  J^i_k+ 2 \bar J_\rho^\si  \del'^\rho_ l p'_i  J^i_k   (d_\si J^l_j )+  \bar J^\la_\mu \bar J_\rho^\si\del'^\rho_ l p'^\mu_i   (d_\si J^l_j ) (d_\la J^i_{k})  + \bar J^\la_\mu p'^\mu_i (d_\la J^i_{jk})
+ p'_i J^i_{jk} \)
\fn$$
%$ \del_j= J_J^l \del'_l  +  \bar J_\rho^\si (d_\si J^l_j ) \del'^\rho_ l $

We see that, although $p_k^\al$ and $\del_i^\mu p_k^\al$ are well-defined geometrical objects, the other momenta are not.
However, there are a number of geometric objects defined out of them which should be pin pointed, since we expect to provide a good intrinsic formalism for variational calculus.

One can then show directly that $\de^2\Lag=\de^2 \Lag'$.
\EndNote

We can also expand the deformed action at second order, i.e.~as
A_D[\si_s]  = \int_D (J^1 \si )^\ast \( L   +  s\>  \de L + \frac[1/2] s^2\>  \de^2 L + O(s^3) \)
\fn$$

Suppose $\si$ is a critical section for $L$. We say that it is {\it stable} if it is a minimum of the action functional, {\it unstable} otherwise.

To decide that, we have to check how the action changes when we deform a solution $\si$. For that we need to consider the second variation.
\eqalign{
A_D[\si_s] & - A_D[\si]=  \Frac[s^2/2] \int_D  \de^2 L    + O(s^3)= \Frac[s^2/2]\( \int_{ D}  (p_i  -d_\mu p_i^\mu  ) \de^2 y^i  d\si  + \int_{\del D}   p_i^\mu   \de^2 y^i  d\si_\mu \)+ \cr
+ \Frac[s^2/2] \int_D   \(\del_k p_i \de y^i  \de y^k + 2\del_i p_k^\al  \de y^i d_\al \de y^k + \del_k^\al p_i^\mu d_\mu \de y^i  d_\al \de y^k \) d\si +O(s^3)
\cr
\fn$$

Practically speaking, if the bilinear form
\H =\(\matrix{\de y^i &  \de y^i_\mu }\)\(\matrix{
\del_k p_i 		&	\del_i p_k^\al	\cr
\del_k p_i^\mu	&	\del_k^\al p_i^\mu	\cr
\de y^k \cr
\de y^k_\al
\fn$$
is definite-positive, then the critical section is in a {\it minimum} of the action functional and it is called a {\it stable} solution.
Let us remark, however, that the bilinear form $\H$ is applied to a vector $(\de y^k , \de y^k_\al)$ which is holonomic, not generic.
Thus, when the form is not definite positive, we need to discuss the sign of the integral.
We shall show that in an example, in the next Section.

\NewSection{Example: unstable geodesics on a  sphere}

Let us consider, as an example, the Lagrangian for geodesics on $[B:=\R\times Q \arr \R]$ given by
\Lag= \frac[1/2] g_{\mu\nu} u^\mu u^\nu ds
\fn$$
By variation, we obtain
\de L= g_{\mu\nu} u^\mu \frac[d/dt]\de x^\nu +\frac[1/2] \del_\la g_{\mu\nu}u^\mu u^\nu \de x^\la
%= u^\mu  g_{\mu\ep}\frac[d/dt]\de x^\ep +\frac[1/2] \( -\red{\del_\mu g_{\nu\la}}+ \red{\del_\nu g_{\la\mu}}+ \del_\la g_{\mu\nu} \) u^\mu u^\nu \de x^\la
=u^\mu  g_{\mu\ep}\(  \frac[d/dt]\de x^\ep + \{g\}^\ep_{\nu\la}  u^\nu \de x^\la \)
\fn$$
Hence, the Jacobi Lagrangian on $[\R\times TQ\arr \R\times Q \arr \R]$ is
J(\Lag):= \de Lds
= u^\mu g_{\mu\al}  \( \frac[d X^\al /ds]  + \{g\}^\al_{\la \nu}X^\la u^\nu \)   ds
\fn$$
which is regarded as a Lagrangian on $V(B)\simeq\R\times TQ$, which happens to be linear in $X= X^\mu(t, x)\del_\mu$
which is assumed to be a vertical vector on $\R\times Q$, or simply a vector field $X= X^\mu(x)\del_\mu$ on $Q$.
The variation of the  auxiliary Lagrangian $J(\Lag)$ defines Jacobi equation for $X$.

\Note
By variation with respect to $\de x^\mu$ and $\de X^\mu$, we obtain
\eqalign{
\de (J(L))=&
\(  R_{\ep\mu \nu\si }X^\si   u^\mu  u^\nu
-   g_{\ep\be}\(  \frac[d/ds] \(\dot X^\be   + \{g\}^\be_{\si\nu}X^\si u^\nu \)
+ \{g\}^\be_{\al\rho}  \(\dot X^\al   + \{g\}^\al_{\si\nu}X^\si u^\nu \) u^\rho \) \)   \de x^\ep +\cr
& -    \(  \dot  u^\be     +  \{g\}^\be_{\mu\nu}   u^\mu  u^\nu   \) \( g_{\al\be} \{g\}^\al_{\si\ep}  X^\si \de x^\ep +  g_{\ep\al} \de  X^\ep \)  +\cr
+\frac[d /ds] \(  g_{\ep\al}  \( \dot X^\al   +  \{g\}^\al_{\si\nu}   X^\si u^\nu \) \de x^\ep
+ u^\mu g_{\mu\al}  \(  \de  X^\ep + \{g\}^\al_{\si\ep}  X^\si \de x^\ep    \)  \)
\cr
\fn$$

Since a composite section is given by $(x(t), X(t, x))$ we have
\eqalign{
\dot X^\al  =& \del X^\al + u^\be d_\be X^\al
\qquad\qquad\then\quad
\dot X^\al  + \{g\}^\al_{\si\nu}X^\si u^\nu= \del X^\al + u^\be \na_\be X^\al\cr
\ddot X^\al
%=&  \del^2 X^\al + \dot u^\be d_\be X^\al +  2u^\be d_\be \del X^\al  + u^\mu u^\nu  ( \na_{\mu\nu} X^\al  -\{g\}^\al_{\rho\mu} \na_\nu X^\rho + \{g\}^\rho_{\mu\nu}\na_\rho X^\al)- u^\mu u^\nu  d_\mu  \{g\}^\al_{\si\nu} X^\si- u^\mu u^\nu     \{g\}^\al_{\si\nu} (\na_\mu X^\si -\{g\}^\si_{\rho\mu} X^\rho)=\cr
=& \del^2 X^\al +  2u^\be \na_\be \del X^\al
+ (\dot u^\be  + \{g\}^\be_{\mu\nu} u^\mu u^\nu) (\na_\be X^\al -\{g\}^\al_{\si\be} X^\si)-  2u^\be \{g\}^\al_{\si\be} \del X^\si  +\cr
&+ u^\mu u^\nu  ( \na_{\mu\nu} X^\al  -2\{g\}^\al_{\si\mu} \na_\nu X^\si
%+ \{g\}^\rho_{\mu\nu}\na_\rho X^\al - \{g\}^\be_{\mu\nu} \na_\be X^\al
- d_\mu  \{g\}^\al_{\si\nu} X^\si
+  \{g\}^\al_{\be\nu} \{g\}^\be_{\si\mu} X^\si
+ \{g\}^\be_{\mu\nu}\{g\}^\al_{\si\be} X^\si)
%u^\mu u^\nu ( - \{g\}^\be_{\mu\nu} \na_\be X^\al + \{g\}^\be_{\mu\nu}\{g\}^\al_{\si\be} X^\si )
\cr
\fn$$
where $\del$ denotes partial derivatives with respect to $t$.
We also obtain along geodesics
\eqalign{
\frac[d/ds] \(\dot X^\be   + \{g\}^\be_{\si\nu}X^\si u^\nu \)
=  \del^2 X^\al +  2u^\be \na_\be \del X^\al
-  u^\be \{g\}^\al_{\si\be} \del X^\si
+ u^\mu u^\nu  \( \na_{\mu\nu} X^\al  -\{g\}^\al_{\si\mu} \na_\nu X^\si \)
\cr
\fn$$
\EndNote

Then, in view of the original equation, the Jacobi equation becomes
\del^2 X^\ep +  2u^\be \na_\be \del X^\ep
+ u^\mu u^\nu   \na_{\mu\nu} X^\ep
=  u^\mu  u^\nu  R^\ep{}_{(\mu \nu)\si }X^\si
\fn$$

\Note
This is the Jacobi equation for a time-dependent Jacobi field $X$.
In the literature, often one finds the Jacobi equations for vector fields on the manifold $Q$, i.e.~with $\del X^\al=0$, which in fact is
\na_{\mu\nu} X^\ep   = R^\ep{}_{\mu \nu\si }X^\si
\fn$$
\EndNote

Therefore,  Euler-Lagrange equations for $J(\Lag)$, i.e.~linearized equations for $\Lag$, are
\dot u^\la + \{g\}^\la_{\al\be}u^\al u^\be=0
\qquad\qquad
\na_{\mu\nu}  X^\al  = R^\al{}_{(\nu\mu)\la}X^\la
\fn$$
and the solution vector fields $X$ are called {\it Jacobi fieds} on $(M, g)$ and their flow sends (affinely parameterised) geodesics into  (affinely parameterised) geodesics.

The variation $\de J(\Lag)$ is something on $V(V(B))$ which can be restricted on $V^2(B)$ to get the second variation of the original Lagrangian $L$,
evaluated along a critical configuration, and neglecting boundary term that will vanish for boundary conditions, that is
\eqalign{
\de^2 L=&  %- \(u^\mu u^\nu g_{\ep\al}   \na_{\mu} \na_{\nu}  X^\al - u^\mu u^\nu R_{\ep\nu\mu\la}X^\la   \)X^\ep=
\(   g_{\ep\al}  \hat  X^\al \hat  X^\ep + u^\mu u^\nu R_{\ep\nu\mu\la}X^\la X^\ep   \) -  \frac[d/ds] \( u^\nu g_{\ep\al}   \na_{\nu}  X^\al X^\ep\)
\cr
\fn$$
where we set  $\hat X^\al := \dot X^\al + u^\mu\{g\}^\al_{\mu\nu} X^\nu = u^\mu \na_\mu X^\al$.

The bilinear form $\H$ for the geodesic Lagrangian specializes to
\H= \(\matrix{X^\al &  \hat X^\al }\)\(\matrix{
R_{\al\mu\nu\be} u^\mu u^\nu	&	0	\cr
0	&	g_{\al\be}	\cr
\hat X^\be
\fn$$

The curvature, top-left block, on the equator of the sphere in stereographic coordinates, reads as
\(\matrix{X^1 &  X^2}\)\(\matrix{
-\cos^2(t)	&	-\cos(t)\sin(t)	\cr
-\cos(t)\sin(t)	&	-\sin^2(t)	\cr
\fn$$
which has eigenvalues $\la=0, -1$.
Since the metric, bottom-right block $(g_{\al\be})$ is definite positive, if we hope to get a negative value to prove instability, we need to focus on the negative eigenvector which, in this case, is $X= \cos(t)\del_x + \sin(t)\del_y$,
i.e.~it is orthogonal to the equator as one could geometrically guess.

Of course, the components $\hat X^\al$ are not independent of the component $X^\al$ (otherwise we could simply look for negative eigenvectors).
Hence, we select a vector $\xi= f(t)\( \cos(t)\del_x + \sin(t)\del_y\)$ along the equator in the direction of a negative eigenvector and $\hat \xi = \dot f(t)\( \cos(t)\del_x + \sin(t)\del_y\)$.
Then we can compute the bilinear form along $(\xi, \hat \xi)$ to be
\H=  \dot f^2 - f^2
\fl{FTH}$$
We have to determine $f$ so that $\int \H \> dt<0$, which proves instability of the equator.
That depends on the boundary points; suppose we integrate between $t=0$ and $t=t_0$.
We do not need $\xi$ to be a Jacobi field, we still need it to be fixed at the boundary, $f(0)= f(t_0)=0$.

If $t_0= a \pi$, we can set
f(t)= \sin\( \frac[t/a]\)
\fn$$
In fact, we have $f(0)= f(a \pi)=0$, and
\int_0^{t_0} \H dt=  \frac[1/2a](1-a^2)
\fn$$
If $a<1$ (i.e.~$t_0<\pi$) that is positive and action functional in that direction is growing, which is compatible with the configuration being stable.
When $a>1$   (i.e.~$t_0>\pi$) that is negative and we found one deformation along which the action functional is not increasing which is enough to prove that the solution is unstable.

This technique is similar to consider the Fourier transform of equation \ShowLabel{FTH} and look for modes that exponentially grow.
These modes give the directions in functional space along which the action functional is changing.

\NewSection{Conclusions and perspectives}

We showed that, given any Lagrangian field theory (including, as a special case, any holonomic mechanical system)  on a bundle $[B\arr M]$,
one can define the Jacobi Lagrangian $J(\Lag) = J^kX\ip d\Lag$ on the composite bundle $[V(B)\arr B\arr M]$.
A composite configuration for the Jacobi Lagrangian is a field $y^i(x)$ and a field $X^\al(x, y)$.
That is already quite odd since usually variational principles have as configurations a number of fields on the same manifold, while composite variational principles allow to have configurations with fields defined on different manifolds.

For the Jacobi Lagrangian, $\si$ still obeys the same field equations as in the original system.
The other field $X:B\arr V(B)$ is a vertical vector field on the configuration bundle and it obeys field equations which force it to drag solutions into solutions.
Since the Jacobi Lagrangian is a global form, we know its equations are global or one can check it directly by changing coordinates on $[B\arr M]$,
inducing a change of coordinates on $[V(B)\arr B\arr M]$, then on $\P_2$, computing transformation laws of momenta, and from them transformation laws of field equations, which eventually transforms as tensor densities, hence they are satisfied in one chart iff they are satisfied in any chart.

This framework applies to a variety of situations in physics.
Since the Jacobi field $X$ drags solution into solutions they can be called {\it (exact) perturbations}.
The framework allows to discuss gravitational waves without resorting to the usual splitting $g_{\mu\nu}=\eta_{\mu\nu}+ h_{\mu\nu}$ which is nonsensical for a number of reasons, firstly because metrics are not a vector (or affine) space and linear combinations do not make sense. Moreover, the linear combination depends on the coordinates as well as on the choice of the unperturbed state $\eta$.
Secondly, when first order Einstein equations are found, solved for a first order perturbation $h$, the corresponding $g$ is not a solution of Einstein equations, it is just a solution at first order.

The framework we presented here, instead, is the natural setting to discuss the issue in a covariant way.

The same argument holds for computing the evolution of acoustic perturbations in a cosmological setting, which is routinely used to compute the power spectrum of the cosmic microwave background radiation.

One can also consider applications to classical mathematical GR.
Many issues with gravity directly stem from the fact that the space of fields are not linear, one for all, there is no superposition principle in GR.
Well, one can try and parameterize configurations in terms of $X$ which are (vertical) vector fields!

\Note
Proceeding backwards, one can consider a family of exact solutions, e.g.~Schwarzschild solutions which are parameterized by a parameter $m$, and
we can define a Jacobi field (along the family) by taking the derivative with respect to the family parameter, obtaining in this way precisely a vertical vector field $X$
(along the family)  on the bundle $[\Lor(M)\arr M]$.
Moreover, the flow of such a Jacobi field is known, by construction, to be the family of Schwarzschild metrics themselves.
In other words,  $X$ drags Minkowski spacetime into the whole family of Schwarzschild metrics.
\EndNote

Once we have two Jacobi fields $X_1, X_2$, we can safely sum them being linear solutions of linear equations.
Of course, there is no simple relation among the flows of $X_1, X_2, X_1+X_2$.
The operation of taking the flow is not linear itself.
However, $ X_1+X_2$ drags Minkowski into solutions which are not the original metrics.
For example, if we consider the fields that generate the Schwarzschild family at two different points, then
$X_1+X_2$ is not spherically symmetric, it is rather what we should call the field generated by two masses.
Analytic expressions for it are probably hopeless in most cases, that does not mean one cannot
infer properties of solutions from the fact that the geometry is produced by the flow of a particular field.
Further investigations will be devoted in this direction.

The approach in terms of flows is not new (see \ref{HE}).
The new part of the framework is the variational setting to compute the Jacobi equations.

Finally, let us mention that Jacobi setting also defines second variation of the original Lagrangian and define a global intrinsic notion of stability.
For example, by applying this framework one is able to prove
Breitenlohner-Freedman bound for the mass of Klein-Gordon fields which makes the vacuum state stable in curved different backgrounds.
We shall show that in a forthcoming paper.

Thanks to the functoriality of linearization, one can iterate the variation to any order.
Indeed, for example, we have
\eqalign{
\de y^i =& X^i(x, y_s(x))\Big|_{s=0}\cr
\de^2 y^i =& \del_k  X^i(x, y_s(x)) X^k(x, y_s(x)) \Big|_{s=0} \kern-10pt=  X^i_k \de y^k\cr
\de^3 y^i =&  \del_{kn}  X^i(x, y_s(x)) X^k(x, y_s(x))X^n(x, y_s(x))  \Big|_{s=0} \kern-10pt+\del_k  X^i_k \de^2 y^k=\cr
=& X^i_k\> \de^2 y^k + X^i_{kn}\> \de y^k \de y^n \cr
\dots
\fn$$
at third order.

\NewAppendix{A}{Linearized equations for a general second order Lagrangians}

Let us now consider an ordinary variational principle, defined on a configuration bundle $[B\arr M]$ and described by a second order Lagrangian $\Lag=L(J^2y)d\si$.
As a matter of fact, the variation of the Lagrangian $\de \Lag:J^2B \arr V^\ast(J^2 B)\otimes A_m(M)$ is a global form and it induces an auxiliary global second order Jacobi Lagrangian
$J(\Lag)= J^2X \ip d\Lag $
on the
composite bundle $[V(B)\arr B\arr M]$.
Let us stress that $J(\Lag)$ is a horizontal $m$-form in $J^2V(B)\simeq V(J^2B)$, i.e.
J(\Lag)= L_J(x, y, X, dy, dX, d^2y, d^2X)\> d\si= (p_i X^i + p_i^\mu d_\mu X^i+ p_i^{\mu\nu} d_{\mu\nu} X^i )d\si
\fn$$

Then we can consider a deformation $\Xi = \de y^i\frac[\del/\del y^i] + \de X^i \frac[\del/\del X^i] $ on $V(B)$ and the variation of the Lagrangian $J(\Lag)$.

\Note

We have
\eqalign{
J^2\Xi& \ip d(J(\Lag))= \(p_i -d_\mu p_i^\mu + d_{\mu\nu} p_i^{\mu\nu} \)  \de X^i
+ \Big(  \del_k p_i  X^i + \del_k p_i^\mu  d_\mu X^i + \del_k p_i^{\mu\nu}  d_{\mu\nu} X^i +\cr
&- d_\al\( \del_k^\al p_i  X^i + \del_k^\al  p_i^\mu  d_\mu X^i   + \del_k^\al  p_i^{\mu\nu} d_{\mu\nu} X^i \)
+d_{\al\be} \( \del_k^{\al\be} p_i  X^i + \del_k^{\al\be}  p_i^\mu  d_\mu X^i + \del_k^{\al\be}  p_i^{\mu\nu}  d_{\mu\nu} X^i  \)  \Big) \de y^k +\cr
&+d_\mu\Big(   \(p_i^\mu   -  d_{\nu} p_i^{\mu\nu} \) \de X^i + p_i^{\mu\nu} d_{\nu} \de X^i
+  \( \del_k^\mu p_i  X^i + \del_k^\mu  p_i^\la  d_\la X^i   + \del_k^\mu  p_i^{\la\nu} d_{\la\nu} X^i \)  \de y^k+\cr
&-d_\al \( \del_k^{\al\mu} p_i  X^i + \del_k^{\al\mu}  p_i^\la  d_\la X^i + \del_k^{\al\mu}  p_i^{\la\nu}  d_{\la\nu} X^i  \) \de y^k
+ \( \del_k^{\mu\be} p_i  X^i + \del_k^{\mu\be}  p_i^\la  d_\la X^i + \del_k^{\mu\be}  p_i^{\la\nu}  d_{\la\nu} X^i  \)  d_{\be} \de y^k
\Big) \cr
%=& \de^2 L
\fn$$
We obtain as a first field equation  $\E_i= p_i -d_\mu p_i^\mu + d_{\mu\nu} p_i^{\mu\nu}=0$, i.e.~the same field equations as for the original Lagrangian $\Lag$.
For the second field equation we can write it as
X^i \del_i \E_k  +  d_\mu X^i \del_i^\mu \E_k + d_{\mu\nu} X^i \del_i^{\mu\nu} \E_k
+  d_{\mu\nu\al} X^i  \(  -  \del_i^{\mu\nu}  p_k^\al    +  2   \del_i^{\mu\nu} d_\be p_k^{\al\be}  -   \del_i^{\mu} p_k^{\al\nu}    \)
+  d_{\mu\nu\al \be} X^i \del_i^{\mu\nu}  p_k^{\al\be}
\fn$$
where we used that $[d_\al, \del_k] =0$, $[d_\al, \del_k^\mu] = -  \de^\mu_\al \del_k$,  $[d_\al, \del_k^{\mu\nu}]  = -  \de^{(\nu}_\al \del_k^{\mu)}$,
and $-[d_\al, \del_k^{\mu\nu\ep}]=   \del_k^{\mu\nu\ep} d_\al = \de^{(\ep}_\al \del_k^{\mu\nu)}$.
In order to further simplify the second field equation, let is compute on a side
\eqalign{
d_{\al\be\ep\la}X^i \del_i^{\al\be\ep\la} \E_k=& d_{\al\be\ep\la}X^i \del_i^{\al\be\ep\la} (p_k -d_\mu p_k^\mu + d_{\mu\nu}p_k^{\mu\nu})
= d_{\al\be\ep\la}X^i \del_i^{\al\be\ep\la}  d_{\mu\nu}p_k^{\mu\nu}= d_{\al\be\ep\la}X^i   \del_i^{\al\be} p_k^{\ep\la}\cr
d_{\al\be\ep}X^i \del_i^{\al\be\ep} \E_k=& d_{\al\be\ep}X^i \del_i^{\al\be\ep} (p_k -d_\mu p_k^\mu + d_{\mu\nu}p_k^{\mu\nu})
= d_{\al\be\ep}X^i  \( -\del_i^{\be\ep}  p_k^\al  - \del_i^{\al}   p_k^{\be\ep}   + 2\del_i^{\al\be} d_\mu p_k^{\mu\ep}  \)
\fn$$
where we used the identities
d_{\al\be\ep}X^i  \del_i^{\al\be\ep} d_{\mu\nu} p_k^{\mu\nu}=  d_{\al\be\ep}X^i (  - \del_i^{\al}   p_k^{\be\ep}   + 2\del_i^{\al\be} d_\mu p_k^{\mu\ep})
\fn$$
d_{\al\be\ep\la}X^i  \del_i^{\al\be\ep\la}  d_{\mu\nu}p_k^{\mu\nu}=d_{\al\be\ep\la}X^i   \del_i^{\al\be} p_k^{\ep\la}
\fn$$

In view of these results, the second field equation is recasted as $J^4X (\E_i) =0$
which means the jet prolongation of the  vector field $X$ (vertical on the original configuration bundle $[B\arr M]$) is tangent to the original field equations
thought as a sub-manifold into $J^4B$.

\Acknowledgements
We also acknowledge the contribution of INFN (Iniziativa Specifica QGSKY), the local research project {\it  Metodi Geometrici in Fisica Matematica e Applicazioni (2022)} of Dipartimento di Matematica of University of Torino (Italy). This paper is also supported by INdAM-GNFM.

\ShowBiblio
\ClosePreMode

\end


# Vibration-based communication for deafblind people

David C. Kutner1 and Suncica Hadzidedic1 1Department of Computer Science,
Durham University, UK<EMAIL_ADDRESS>
###### Abstract

Deafblind people have both hearing and visual impairments, which makes
communication with other people often dependent on expensive technologies
e.g., Braille displays, or on caregivers acting as interpreters. This paper
presents Morse I/O (MIO), a vibrotactile interface for Android, evaluated
through experiments and interviews with deafblind participants. MIO was shown
to enable consistent text entry and recognition after only a few hours of
practice. The participants were willing to continue using the interface,
although there were perceived difficulties in learning to use it. Overall, MIO
is a cost-effective, portable interface for deafblind people without access to
Braille displays or similar.

##  I Introduction

Deafblind people have both vision and hearing impairments resulting in
significant communication challenges in their day-to-day lives. The World
Federation of the Deafblind (WFDB) has estimated that 0.21% of all individuals
aged five and over have severe deafblindness, while noting that there is no
agreement on “best practice” definitions or measurements of deafblindness [1].

Medical journals [2] emphasize the importance of combating social isolation
among deafblind people. A survey of 28 deafblind people from Europe [3] found
that many had a strong desire to contribute to society, and were active in
different organizations, including in leadership roles. However, assistive
devices for deafblind communication - e.g. Braille, hearing aids, text-to-
speech - require further improvement [3]. High costs are a key factor
restricting access to assistive devices [3]. The WFDB emphasizes that a lack
of access to affordable assistive technology and support contributes to
deafblind people’ “poverty trap” [1].

This paper aims to present a low-cost, easy-to-learn vibrotactile interface
for deafblind users’ interaction with computers. It uses Morse code - the
encoding of the Latin alphabet and special characters as sequences of dots and
dashes (examples in Fig. 1) [4]. The Morse Input/Output (MIO) interface
enables entry of strings of any length into any text field on an Android
device, with haptic feedback to users. A client-side application was developed
as a learning environment and for gathering data about system usage.

The principle of “Nothing about us without us” [5] promotes the inclusion of
people with disabilities in decision-making processes in any area that has an
impact on their lives. In keeping with this, deafblind users evaluated the MIO
application. The design, implementation, and evaluation processes in this
study were informed by feedback from over two dozen organizations, academics,
and professionals.

Overall, the main contributions of this paper are: the MIO vibrotactile
interface, interface evaluations specifically with deafblind users, and an
evaluation of practical implications about MIO’s usability and real-world
applicability.

Figure 1: Chart showing the Morse encodings of the alphabet and decimal
numbers per the International Telecommunication Union (ITU) standard [4]

###  I-A Related Work

Applications of haptic devices designed for deafblind people have been the
focus of some research in the last decade. However, due to the difficulties
with recruiting deafblind people, these studies often had sighted and/or
hearing participants evaluate the technology in question [6] [7] [8] [9]. For
example, a 2020 study [7] aimed to evaluate the learnability of Morse code
without a visual reference. The study’s 10 participants (all with normal
vision) succeeded in learning at least 15 (and up to 24) letters in Morse code
within just 30 minutes. Noteworthy is the study’s method of introducing a
small group of letters at a time and teaching the recognition of words
composed of these letters before the entire alphabet is learned. However, this
study was limited to feedback only from sighted participants, with dots and
dashes applied to the left and right arms, respectively, a technique that is
not possible to implement on a single phone.

Implementation of a vibrotactile Morse-based system for deafblind
communication has already been researched. In 2014, Norberg et al. [8]
developed a web navigation system using an Xbox 360 controller to “read” text
to a user operating an otherwise unmodified computer. The study was carried
out with “simulated” deafblind participants, who wore noise-canceling
headphones and blindfolds. They found that three out of four participants were
successful in completing a series of simple navigation tasks with just 15
minutes of training in Morse code. The study’s conclusions were undermined by
the subjects’ likely prior experience using a mouse and cursor.

Samsung Good Vibes is an application for deafblind communication that has been
available since 2019 [10]. It has two interfaces - one for the deafblind, the
other for sighted users. It allows for two-way communication, much as a
messaging application would, using vibrotactile Morse to relay message
contents to a deafblind user. There are no sources confirming its use outside
Asia, and unfortunately, at the time of writing, no studies on usability or
learnability of this App were publicly available.

In 2014, Arato et al. [11] developed, for a single deafblind individual (PG),
an interface for communicating by SMS on an Android phone. Because PG was
already familiar with Braille, he was enabled to enter text in Braille.
However, recognizing Morse characters by vibrations was more difficult. PG was
able to sense 15 characters per minute, or 10 characters per minute for
unknown texts, and continued to use his Morse-Braille vibrating phone in his
day-to-day life after the study. This hybrid Morse-Braille approach was
considered to be low-cost and usable.

Overall, there has been scant research into vibrotactile assistive technology
for the deafblind. The only reference to an interface implemented on a
smartphone, using Morse code for text entry, was Samsung Good Vibes [10],
although no experimental results were available. Common to all of these
studies, with the exception of [11], is the absence of deafblind people from
the design and evaluation process.

##  II Morse Input/Output

In this section, we present the feedback from professionals and academics and
the design and development of the MIO interface.

###  II-A Interviews with professionals

> “The average person pays $400 to $1000 for a mobile phone - I don’t mind
> paying that, but my Braille display is a minimum $1800 to as much as $5000.”
> \- Scott Davert, 2021

Our study applied the user-centered design approach [12]. Academics and
professionals who have worked with deafblind people thus participated in the
design and evaluation of the MIO application.

Dr. Marion Hersh (Glasgow University) has published over a dozen papers on
deafblindness and assistive technology, for which she interviewed deafblind
people across Europe [3]. In a discussion for this paper, Dr. Hersh raised the
concern that “Morse has been developed for convenience of transmission rather
than convenience of understanding”, as well as the possibility that longer
patterns (i.e., more dots and dashes) might be harder to learn. Consequently,
in the experiment, we used more complex letters (e.g., C, -•-•) as well as
concisely encoded ones (e.g., A, •-). She also acknowledged the benefits of a
smartphone application because of the wide availability of smartphones, which
are more affordable and also more discreet than assistive devices for use in
public.

Heather Colson-Osborne has 13 years of experience at the Anne Sullivan Centre
in Ireland, which supports deafblind individuals. She interpreted for one of
our participants and contributed to the design of the interface. The idea of
“touch recording” was introduced as the Centre’s staff pointed out that many
deafblind people are illiterate, having learned a limited tactile vocabulary,
e.g. a spoon to refer to mealtime. With touch recording, a user could tap or
hold their finger to the screen, and the smartphone would register the timings
and replay the message as vibrations of the lengths of time pressed.

Scott Davert has previously worked with hundreds of blind and deafblind people
as a Braille teacher and the coordinator for New York State’s federally funded
iCanConnect program. He is himself deafblind and proficient in Morse code,
thus uniquely suited as a contributor to this project. He was particularly
interested in integrating MIO as an alternative interface to the main method
on his phone. On Android devices, the only way to connect a Braille display is
through Bluetooth, which is not consistently reliable. According to Mr.
Davert, a Morse vibration-based interface would give deafblind people “a
chance to repair [their] device without having to ask someone who can either
see or hear to do it. It’s something I would definitely use”.

Mr. Davert further explained Android’s lack of popularity among blind people:
“The Braille access is bad - it’s really bad. And they know; I actually went
to Mountain View […] and told them!”. The accessibility features of the latest
Android version are comparable to those of iOS 4. Therefore, he warned that it
would be very challenging to find deafblind participants with access to an
Android phone. Nevertheless, for the purposes of MIO, he acknowledged that it
might be more fruitful to integrate the interface into an open-source Android
operating system than iOS, which has major restrictions for new software
development.

###  II-B MIO design

Based on the interviews with professionals detailed above, MIO was designed
for individuals whose disabilities range from total deafness and blindness to
a severe impairment in hearing and vision. A Graphical User Interface (GUI)
was included because some deafblind users have partial vision, and to
facilitate development. MIO is an interface which uses vibrations for text
output and a large keyboard for text input. It was implemented in Java for
Android OS.

Although the professional participants’ input (Section II-A) implied that
deafblind people are more likely to own iOS devices, an Android platform was
chosen for MIO, as it is more conducive to receiving experimental software.
Moreover, bespoke setups, e.g. using a Raspberry Pi, were excluded for reasons
of cost. Cost is an important factor restricting access to assistive
technology among deafblind people [1] [3]. For this study, which was based on
user-centered design [12], one of the crucial requirements was accessibilty of
the application.

Finally, we have chosen Morse code over an encoding derived from Braille (or a
hybrid one, as in [11]) primarily for the conciseness and simplicity of the
Morse alphabet. Although Braille is much more popular among blind people, it
is more complex and less time-efficient for use through a touchscreen. The
number of dots in a Braille character depend on its position in the alphabet.
The letter T, for instance, is a single dash in Morse; it is represented by t
in Braille, and would therefore require more time to both read and input.

####  II-B1 The MIO interface

The MIO interface consists of an output method \- Morse vibrations, and an
input method \- the MIO keypad. A brightly-colored GUI (Fig. 2) is also
implemented for ease of use for the partially sighted.

Morse vibrations. Text output is performed by emitting vibrations of different
lengths, directly applying Morse code according to the ITU standard [4] (Fig.
1). After some calibration with a user, the length of one unit (a dot) was set
at 200 ms. Thus, e.g., the word “TEA”, which is encoded as - • •- in Morse, is
emitted as: 600 ms vibration (-); 600 ms pause (between letters); 200 ms
vibration (•); 600 ms pause (between letters); 200 ms vibration (•); 200 ms
pause (within a letter); 600 ms vibration (-).

(a) Landing screen.

(b) Words practice.

Figure 2: The MIO application on Android.

MIO keypad. The input method is a keypad divided horizontally into three
buttons on the bottom half of a smartphone screen (Fig. 2(b)). From left to
right, the buttons are dot, square and dash. The button outlines and the
symbols are in yellow to maximize readability [13] for use by partially
sighted persons and sighted carers. We added a square key for ease of use and
to minimise input mistakes. The square key is pressed once to “submit” a
letter, twice to enter a space, and three times to create a new line. A user
wishing to type the phrase “HI PAL” would thus enter the sequence
••••$\square$••$\square\square$•--•$\square$•-$\square$•-••. For dots and
dashes, the length of input and output vibrations is identical. Pressing the
dot key produces a 200 ms vibration, while the dash key vibrates for 600 ms.
When pressed, the square button vibrates for 100 ms, the shortest length
perceptible by our participants. Thus, all keys produce unique haptic
feedback, so that users can easily differentiate between the buttons. Swiping
the keyboard from right to left resets the current input, emitting a 1200 ms
vibration to distinguish the action from an accidental button touch.

####  II-B2 MIO activities

The MIO application includes several activities intended to familiarize users
with the interface. Fig. 2 shows the landing screen (Fig. 2(a)) and a practice
activity (Fig. 2(b)). The layout is intended to enable a deafblind user to
easily find buttons from the edge of the screen.

We included a unique vibration that is emitted at the start of an activity, at
the suggestion of our participants. One 100 ms vibration is emitted for Words
Practice, two for Exercise, three for ABC Practice, and four for Playback.
Swiping backwards from right to left in any activity returns the user to the
main screen. Navigating to the main screen emits the string “MIO” in Morse.

The Settings activity is intended for a sighted helper. It enables access to
the system logs and customization. The Words Practice, ABC Practice and
Exercise activities are all Practice activities (Fig. 2(b)), for learning to
recognize and type a prompt in Morse. Tapping the top half of the screen
vibrates a word displayed there. Successfully entering the word progresses the
user to the next word. An incorrect entry resets the input and emits a 1200 ms
vibration to prompt the user to re-attempt entry.

To learn and practice individual letters, users can open the ABC Practice
activity, which cycles through the alphabet. The Exercise activity presents a
specially designed series of prompts building up to the phrase “THE CAT EATS
THE CAKE”. This design, using just 6 letters, is in line with the Plaisier et
al’s [7] approach of first teaching users a small number of letters so that
they can read and enter entire words, before they learn and memorize all the
letters of the alphabet.

The Words Practice activity, by default, displays random words from Basic
English [14]. However, it can be configured in the settings. Consequently,
users can practice through arbitrary lists of strings and learn Morse code by,
e.g., using a poem they are already familiar with.

Finally, with the Playback activity, users can go through the alphabet without
typing it in Morse. Simply tapping the screen progresses the user to the next
prompt and emits it as a Morse vibration. The application can also be
configured to playback lines from a familiar text.

##  III User Studies

One of the main contributions of this paper was that we specifically recruited
deafblind people to participate in the design and evaluation of the MIO
application. User-centered design [12] “assures that the product will be
suitable for its intended purpose in the environment in which it will be
used”. The selection of participants was reinforced by the feedback received
in the pre-study interviews with professionals (II-A). Moreover, the sampling
decision was consistent with the literature on assessment of assistive device
usability for people with disabilities [15], and the “Nothing about us without
us” approach promoted by their community [5].

Therefore, a purposive sampling approach [16] was adopted. We reached out to
more than 90 deafblind organizations worldwide, of which 26 responded.
However, in the final study, only two deafblind individuals (AC and WK)
participated, both recruited through the Anne Sullivan Foundation in Ireland.

Particular attention was given to protecting the participants’ privacy and
obtaining their informed consent for data collection. The project received
ethics approval from the Durham University Department of Computer Science.
Both participants agreed to having their interviews video-recorded and to
reporting of their direct quotes. Each video call lasted a few hours, with
approximately one week between calls (i.e., study stages).

Interview and experiment design was semi-structured. The small number of
participants made it possible to change the order of questions and tasks on-
the-fly, tailored to the interests and engagement of the participants. Semi-
structured interviews with an open response format grant both the assurance
that relevant topics are covered and the flexibility to allow participants to
point out issues [3].

The user study had three stages:

  1. 1.

Background and preliminary questions. The participants were asked to report
on: their experience with deafblindness (e.g., at what age their impairments
developed); methods of communication they had previously used; assistive
technologies they were familiar with, and how these might be improved. The
informed consent to being recorded for the study was also gathered at this
stage, and the correct configuration of the application was verified.

  2. 2.

Application description, experiment, and evaluation. In the second stage, the
MIO application was presented. The interviewer provided instructions directly
to one participant (WK), as the MIO application was designed for use and
navigation without visualization of the screen. The other participant (AC)
received instructions through her interpreter.

First, the ABC Practice activity was used to demonstrate the letter A output
as vibrations and how to input it. Participants were then encouraged to
continue practicing with successive letters until comfortable to attempt the
Experiment activity. The experiment required attempting a sequence of prompts,
building up to the sentence “THE CAT EATS”; starting with one letter at a
time, then whole words, and finally the sentence itself, using 5 letters of
the alphabet.

Following the experiment, the MIO application was evaluated. Participants
rated (Likert scale: 1 - strongly disagree to 5 - strongly agree) the generic
10 questions from the System Usability Scale (SUS) [17] and elaborated on
their answers.

We also asked an additional set of questions, based on SUS, however, tailored
to the MIO application. These were:

    1. (1)

I understand how the app is intended to work; it was easy to conceptualize it.

    2. (2)

Typing words using this method is physically difficult.

    3. (3)

I could find keys without issue.

    4. (4)

I often pressed the wrong button by accident.

    5. (5)

It was always immediately clear to me what button I had just pressed.

    6. (6)

The idea of learning Morse code seems easy to me.

    7. (7)

I often had to remind myself of what I had to type by tapping the top of the
screen.

At the end of the experiment, the participants’ ability to recognize and
recall letters was tested. They were asked to play the letter A vibration,
then to give the Morse representation of letter C. Lastly, they evaluated the
MIO interface by reporting on: the perceived speed of communication in
comparison with other means of communication; potential obstacles to using the
interface; ways to usefully integrate MIO to facilitate day-to-day activities;
and perceived ease of learning Morse code. As a preparation for the follow-up
interview, participants were encouraged to practice using the MIO interface.

  3. 3.

Follow-up interviews. A week later, participants were asked to reflect on the
frequency of MIO use since the last interview, to test if their recognition of
letters in vibrotactile Morse improved. The participant WK re-attempted
entering the phrase “THE CAT EATS”, and completed the SUS questionnaire again.

##  IV Results

This section presents qualitative and quantitative results from the MIO
interface evaluation.

###  IV-A Participant background information

Participant AC, a 50-55 year old female, was born deaf, and lost her vision at
the age of 12. She learned to lipread and Irish Sign Language (ISL) at school
as a child. She also knows and uses Braille. AC normally lives at a care
center. She has a support worker for daily interactions and assistance in
using her phone. Therefore, interviews with AC were relatively slow due to the
delay in interpretation between English and ISL. Moreover, there were
difficulties in conveying certain concepts, such as the level of agreement on
a numerical scale. According to her support worker, answering questions in
this manner was completely new to her.

Participant WK, a 20-25 year old female, had 10% vision from birth; she was
able to read text and see pictures with a magnifier, but could only see nearby
objects. At the age of 10, her vision deteriorated, hence she had to learn to
read Braille. Her hearing was above 60% until the age 17. It is now at 0%,
although she can still communicate through speech by using special hearing
aids. WK is proficient with assistive technology; she makes use of both
Braille displays and text-to-speech technologies in her day-to-day life,
enabling her to navigate complex systems. However, she said she experienced
frustration as the software that she needed often lacked accessibility
features.

###  IV-B MIO usability results

Participant WK provided two sets of answers. The first time she used the
system, she evaluated it with an SUS score of 57.5. After four days of further
practice, she was more familiar with the system, and rated it with an above
average SUS score of 75 (SUS = 68 is an average experience, and SUS=80 is a
good experience [18]). Participant AC’s numerical scores were estimated with
the help of her support worker on the basis of natural-language (ISL)
responses to the prompts. Only one set of answers was collected, with a low
total score of 10 [18]. Participants’ detailed feedback to the questions is
summarized below.

Participant AC: “If staff help me, I can learn to do it.”

AC was at times frustrated and confused by the questionnaire, possibly more so
than by the MIO application. AC would like to use MIO “a little bit” going
forward. She considered that it had a steep learning curve, and that most
people would be slow to adapt to it. For her, the MIO interface was “hard to
use,” and she would need staff to help her. Her support worker reported she
had difficulties finding and pressing the right keys at first, although this
improved over the course of the experiment. AC had difficulty counting
vibrations and differentiating between dots (200 ms) and dashes (600 ms).
Nevertheless, she was enthusiastic to participate in the study and eager to
learn. Upon her request, the experiment was modified on-the-fly to allow her
to feel a word she chose. In the end she expressed the willingness to continue
learning to use the MIO interface.

Participant WK

> I feel very confident because I know where everything is.

WK evaluated the MIO application twice. After the first experiment, WK
reported: “kind of 50-50, but once you start using it you get quicker”. She
added that she would be using the application more after the experiment, and
emphasized that it is important to have “a bit of time to get used to it”. In
the second evaluation, WK’s answers implied an improvement to the
application’s perceived usability.

WK found the MIO application broadly understandable, consistent, well-
integrated, and not cumbersome. However, she noted that the application’s
feedback was unclear. At WK’s request, the application was modified to only
play the suffix to the current input, enabling the user to quickly identify
the next letter in the prompt.

Overall, WK expressed an enthusiasm to continue using the MIO interface. She
found it would be useful as a fallback method for answering messages, for
instance if a Braille display was not available. She perceived the MIO’s
haptic feedback as very useful for avoiding typos and checking input.

###  IV-C System log analysis

The MIO log included the following data about each prompt attempt: time,
whether it was successful, and playback frequency. AC attempted 23 prompts
during a single experiment, while WK attempted 74, over two experiments and in
her own time.

Figure 3: Time taken by AC and WK to enter prompts using the MIO application

Fig. 3 shows a scatter plot of the time taken to successfully enter a prompt
against user experience. There is no clear trend in the users’ first session
with the application (blue crosses for WK, red triangles for AC). However,
subsequent uses of the application (yellow diamonds and green circles for WK)
indicate a significant decrease in the input time; analysis by linear
regression gives us $\beta_{1}=-0.727$ $(p<0.0002)$.

##  V Discussion and Future Work

This paper aimed to introduce new means of interaction with computers for
deafblind people by developing a vibrotactile interface. The involvement of
the target users contrasts with the majority of previous work (Section I-A),
which recruited sighted and hearing users, hence limiting the applicability of
their results in practice. An additional contribution is the adoption of user-
centered design and the involvement of professionals and academics with prior
experience in deafblind culture and assistive technology. Thus, the MIO design
choices were informed by users’ needs.

Based on our findings, the advantages of the MIO application include: low
cost, (relative) learnability, and simplicity. Moreover, the MIO interface
requires only three buttons for text input, and can be integrated into any
Android application. Hence, it is cheaper than Braille displays or specially-
built haptic devices, although devices such as DB-Hand [6] (with a
manufacturing cost of around $150) may become a viable option in the future.

The transmission bandwidth (200 ms per unit, or 30-35 characters per minute)
is greater than that of Dual Tactile Morse-Braille [11], which transmitted 15
characters per minute. MIO is especially appropriate for short messages,
consistent with [7]. Our results additionally confirm that the relative ease
with which Morse was learned without visual reference by sighted participants
in [7] is likely to extend to deafblind people with high literacy.

Furthermore, our findings show that the MIO interface was sufficiently easy to
learn for a deafblind user who is proficient with other assistive technology,
even if the user did not have experience with Morse code or haptic feedback.
However, MIO’s perceived usability is inconclusive. The SUS results indicated
a wide gap in the perceived usability of the interface, which may be due to
the difference in the two participants’ past experience. This study focused on
measuring users’ ability to recognize and input strings from a subset of the
alphabet.

Prior in-depth case studies for usability involving participants with
disabilities typically had between three and 10 participants. This allowed for
more generalizable findings [15]. However, similar to our research, the sole
previous study that evaluated smartphone vibration-based haptics for the
deafblind had only one deafblind participant [11]. Future work might include
an increased participant sample size and study duration (to allow more time
for users to become familiar with the interface).

In future work, longitudinal studies can be considered, comparing MIO to
deafblind users’ usual text entry methods, after reaching similar familiarity
with both methods. Moreover, MIO should be implemented on iOS devices to
enable a greater numbers of deafblind participants, and the collection of
interface usage data (e.g., usage time, rate of sending/receiving text). As
per WK’s recommendation, the application could be further extended to
intercept notifications (e.g., Whatsapp and SMS messages), and vibrate the
identity of the sender.

## Acknowledgments

We thank Heather Colson-Osborne (Anne Sullivan Foundation, Ireland); Mr. Scott
Davert; Silvia Fajard Flores (Universidad de Colima); Dr. Marion Hersh
(Glasgow University); Charmaine Tang (Singapore Association for the Deaf);
Katie Trowsdale and Holly Whittome (proofreading); and the study’s two
participants, AC and WK, without whom there would be little to write about!

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