# Local intersections of Lagrangian manifolds correspond to catastrophe theory

Christian Offen *  — School of Fundamental Science, Massey University, Private
Bag 11 222, Palmerston North, 4442, New Zealand<EMAIL_ADDRESS>
###### Abstract.

Two smooth map germs are right-equivalent if and only if they generate two
Lagrangian submanifolds in a cotangent bundle which have the same contact with
the zero-section. In this paper we provide a reverse direction to this
classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a
symplectic manifold have the same contact with a third Lagrangian submanifold
if and only if the intersection problems correspond to stably right equivalent
map germs. We, therefore, obtain a correspondence between local Lagrangian
intersection problems and catastrophe theory while the classical version only
captures tangential intersections. The correspondence is defined independently
of any Lagrangian fibration of the ambient symplectic manifold, in contrast to
other classical results. Moreover, we provide an extension of the
correspondence to families of local Lagrangian intersection problems. This
gives rise to a framework which allows a natural transportation of the notions
of catastrophe theory such as stability, unfolding and (uni-)versality to the
geometric setting such that we obtain a classification of families of local
Lagrangian intersection problems. An application is the classification of
Lagrangian boundary value problems for symplectic maps.

###### Key words and phrases:

Symplectic Geometry, Lagrangian manifolds, contact equivalence, catastrophe
theory

###### 2010 Mathematics Subject Classification:

53D12, 58K35, 37J20, 58K05

##  1\. Introduction

Local singularities of smooth, scalar-valued maps have been studied
extensively under the headlines catastrophe theory and singularity theory
because the local behaviour of the set of critical points of a smooth map
under perturbations is related to bifurcation phenomena in dynamical systems
[1, 2]. Thanks to the work of Whitney, Thom, Mather, Arnold and others,
classification results for singularities are known [3, 13, 18]. Of fundamental
importance for the classification results is the notion of right equivalence
of map germs.

######  Definition 1.1 (right equivalence of map germs).

Two germs of smooth maps $f,g\colon(\mathbb{R}^{k},0)\to(\mathbb{R},0)$ are
right equivalent if there exists a germ of a diffeomorphism
$h\colon(\mathbb{R}^{k},0)\to(\mathbb{R}^{k},0)$ such that $f=g\circ h$.

In [7] Golubitsky and Guillemin show that the question whether two map germs
are right equivalent has a geometric analogue, namely whether two Lagrangian
submanifolds in a cotangent bundle have the same contact with the zero-
section.

######  Definition 1.2 (tangential contact of Lagrangian intersections (in
cotangent bundles)).

Let $\Lambda$, $\Lambda^{\prime}$ be two Lagrangian submanifolds in the
cotangent bundle $T^{\ast}X$ intersecting a point $z\in X\subset T^{\ast}X$
tangentially. The submanifolds $\Lambda$, $\Lambda^{\prime}$ have the same
contact with $X$ at $z$ if and only if there exists a symplectomorphism $\Phi$
defined on an open neighbourhood $U$ of $z\in T^{\ast}X$ such that

$\Phi(\Lambda\cap U)=\Lambda^{\prime}\cap U,\quad\Phi(z)=z,\quad\Phi(X\cap
U)=X\cap U.$

They prove the following theorem.

######  Theorem 1.1 ([7]).

Let $U$ be an open neighbourhood of 0 in $\mathbb{R}^{k}$. The germs at $0$ of
two smooth maps $f,g\colon U\to\mathbb{R}$ with $f(0)=0=g(0)$, vanishing
gradients at 0, i.e. $\mathrm{d}f|_{0}=0=\mathrm{d}g|_{0}$, and vanishing
Hessian matrices at $0$ (in any local coordinate system) are right equivalent
if and only if the Lagrangian manifolds $\Lambda=\mathrm{d}f(U)$ and
$\Lambda^{\prime}=\mathrm{d}g(U)$ have the same tangential contact with the
zero-section at 0 in $T^{\ast}U$.

In the above theorem $\mathrm{d}f(U)$ and $\mathrm{d}g(U)$ denote the image of
$U\subset\mathbb{R}^{k}$ under the 1-form $\mathrm{d}f$ or $\mathrm{d}g$,
respectively, where 1-forms are interpreted as maps $U\to T^{\ast}U$, i.e. as
sections of the cotangent bundle $T^{\ast}U\to U$.

The strength of Theorem 1.1 is related to the parametric Morse lemma, which
says that up to a local change of coordinates any function germ
$F\colon(\mathbb{R}^{K},0)\to(\mathbb{R},0)$ with critical point at $0$ can be
split into a nondegenerate quadratic map $Q_{F}$ and a fully degenerate part
$f$, i.e. a representative of the germ $F$ can be brought into the form
$f(x^{1},\ldots,x^{k(F)})+Q_{F}(x^{k(F)+1},\ldots,x^{K})$, where $K-k(F)$ is
the rank of the Hessian matrix of $F$ at 0 [5, §14.12]. Therefore,
$F,G\colon(\mathbb{R}^{K},0)\to(\mathbb{R},0)$ are right equivalent if and
only if the signatures of $Q_{F}$ and $Q_{G}$ coincide and their fully reduced
parts $f$ and $g$ fulfil the geometric condition in Theorem 1.1.

Thus, Theorem 1.1 is very appealing because it allows us to turn an analysis
problem into a geometric problem. The geometric problem itself, i.e. the
description of intersecting Lagrangian manifolds, is, however, important in
its own right, for instance, in dynamical systems, where intersections of
Lagrangian invariant manifolds in phase spaces encode important information
about the dynamics of the system [8, 11]. For global aspects of Lagrangian
intersections we refer to [6] and references therein. Moreover, boundary value
problems in Hamiltonian systems can be phrased as Lagrangian intersection
problems and local properties of the intersections are of high significance
for a description of the bifurcation behaviour of solutions [16, 17, 20]. It
is, therefore, desirable, to obtain a reverse direction of Theorem 1.1, i.e. a
statement which allows to turn the description of (possibly non-tangentially)
intersecting Lagrangian manifolds in arbitrary symplectic manifolds into a
problem in classical singularity theory or catastrophe theory.

In this paper we will show that we can assign smooth function germs to local
Lagrangian intersection problems such that the following theorem holds.

######  Theorem 1.2.

Let $X,\Lambda$ and $X^{\prime},\Lambda^{\prime}$ be Lagrangian submanifolds
of a symplectic manifold $Z$ such that $\Lambda$ intersects $X$ in an isolated
point $z$ and $\Lambda^{\prime}$ intersects $X^{\prime}$ in an isolated point
$z$. Let $f$ be the function germ assigned to the problem $z\in X\cap\Lambda$
and $f^{\prime}$ be the function germ assigned to the problem $z^{\prime}\in
X^{\prime}\cap\Lambda^{\prime}$. The germs $f$ and $f^{\prime}$ are stably
right equivalent111Two function germs are stably right equivalent if they
become right equivalent after adding nondegenerate quadratic forms in new
variables (see Definition 2.3). if and only if there exists a local
symplectomorphism $\Phi$ mapping an open neighbourhood $U$ of $z$ to an open
neighbourhood $U^{\prime}$ of $z^{\prime}$ with

$\Phi(z)=z^{\prime},\quad\Phi(X\cap U)=X^{\prime}\cap
U^{\prime},\quad\Phi(\Lambda\cap U)=\Lambda^{\prime}\cap U^{\prime}.$

In other words (notions will be made precise later):

theoremCorrSingularities There exists a 1-1 correspondence between Lagrangian
contact problems modulo stably contact equivalence and smooth real-valued
function germs up to stably right equivalence.

The theorems overcome the following issues which occur when trying to reverse
Theorem 1.1.

  * •

Let $X,\Lambda$ be Lagrangian submanifolds of a symplectic manifold $Z$ such
that $\Lambda$ intersects $X$ in an isolated point $z$. After shrinking all
manifolds around $z$, if necessary, there are many ways of mapping $Z$
symplectically to a neighbourhood of the zero-section $X\subset T^{\ast}X$. We
will refer to a particular choice of such an identification as a choice of a
cotangent bundle structure. For most structures, $\Lambda$ is the image of a
section $\mathrm{d}f$ in the bundle $T^{\ast}X\to X$. The map $f$ is not
defined independently of the auxiliary cotangent bundle structure but we will
show that its stably right equivalence class is well-defined.

  * •

The Lagrangian manifold $\mathrm{d}f(U)$ from Theorem 1.1 intersects the zero-
section of $U\subset T^{\ast}U$ tangentially at 0. Golubitsky and Guillemin’s
proof of the only if direction fails when the intersection of $U$ with
$\mathrm{d}f(U)$ is not tangential. Indeed, the equivalence relation right
equivalence considered in [7] needs to be relaxed to stably right equivalence
when non-tangential intersections are allowed.

The results may be compared with the correspondence of embeddings of a
Lagrangian submanifold into a cotangent bundle with catastrophe theory [4].
There, singularities occur because Lagrangian submanifolds fail to be
projectable and intersect non-transversally with fibres. In contrast, in this
chapter cotangent bundle structures are just of an auxiliary nature.

Furthermore, we provide a parameter-dependent version of the results and
relate families of intersecting Lagrangian manifolds to unfoldings of singular
map germs.

theoremCorBifur There exists a 1-1 correspondence between parameter-dependent
Lagrangian contact problems up to stably right equivalence and unfoldings of
smooth, real-valued function germs up to stably right equivalence as
unfoldings.

This allows transporting the highly-developed notions and algebraic framework
of catastrophe theory to Lagrangian contact problems and bifurcations of
Lagrangian intersection problems. As a corollary, classification results for
singularities apply to Lagrangian contact problems.

Moreover, intersection problems of Lagrangian manifolds are often subject to
symmetry constraints which have an effect on which singularities occur
generically and how the intersections unfold when parameters are present. Such
symmetry constraints appear, for instance, in Hamiltonian boundary value
problems as seen in [16, 17]. We will show under which conditions symmetries
in Lagrangian contact problems and in boundary value problems for symplectic
maps, in particular, yield invariances of the assigned function germs.

The remainder of the paper is structured as follows. In Section 2 we first
review some of Golubitsky and Guillemin’s results and then prove that not
necessarily tangential Lagrangian contact problems in arbitrary symplectic
manifolds up to contact equivalence correspond to map germs up to stably right
equivalence. In Section 3 we consider a symmetric setting and prove that
invariances of generating functions of Lagrangian contact problems correspond
to cotangent lifted maps leaving the manifolds of the contact problem
invariant. In Section 4 we extend the identification results of Section 2 to
families of Lagrangian contact problems. In Section 5 we conclude the Theorems
1.2 and 1.2 and show an application to boundary value problems for symplectic
maps.

##  2\. Lagrangian contact problems and catastrophe theory

Let us introduce the notion of Lagrangian contact problems and review some
definitions based on [7, 15].

######  Definition 2.1 (Lagrangian contact problem).

Let $X$, $\Lambda$ be two Lagrangian submanifolds of a symplectic manifold $Z$
intersecting in an isolated point $z\in\Lambda\cap X$. Then $(X,\Lambda,z)$ is
called a Lagrangian contact problem (in $Z$). We say $\Lambda$ has contact
with $X$ in $z$. In the special case where $X$ and $\Lambda$ are tangential at
$z$ the problem $(X,\Lambda,z)$ is called a tangential Lagrangian contact
problem.

######  Definition 2.2 (contact equivalence of Lagrangian contact problems).

Let $(X,\Lambda,z)$ and $(X^{\prime},\Lambda^{\prime},z^{\prime})$ be two
Lagrangian contact problems in $Z$ and $Z^{\prime}$, respectively. We say that
$(X,\Lambda,z)$ and $(X^{\prime},\Lambda^{\prime},z^{\prime})$ are contact
equivalent or $\Lambda$ has the same contact with $X$ at $z$ as
$\Lambda^{\prime}$ has contact with $X^{\prime}$ at $z^{\prime}$ if there
exist open neighbourhoods $U\subset Z$ of $z$ and $U^{\prime}\subset
Z^{\prime}$ of $z^{\prime}$ and a symplectic diffeomorphism $\Phi\colon U\to
U^{\prime}$ such that

$\Phi(z)=z^{\prime},\quad\Phi(X\cap U)=X^{\prime}\cap
U^{\prime},\quad\Phi(\Lambda\cap U)=\Lambda^{\prime}\cap U^{\prime}.$

######  Definition 2.3 (stably right equivalence of function germs).

Two germs of smooth maps
$f\colon(\mathbb{R}^{k},0)\to(\mathbb{R},0),\,x\mapsto f(x)$ and
$g\colon(\mathbb{R}^{l},0)\to(\mathbb{R},0),\,y\mapsto g(y)$ are stably right
equivalent if there exist nondegenerate quadratic forms $Q_{F}(u)$ and
$Q_{G}(v)$ such that the germs at $0$ of $F(x,u)\coloneqq f(x)+Q_{F}(u)$ and
$G(y,v)\coloneqq g(y)+Q_{G}(v)$ are right equivalent.

######  Theorem 2.1.

Two germs of smooth maps
$f,g\colon(\mathbb{R}^{k},0)\to(\mathbb{R},0),\,x\mapsto f(x)$ in the same
number of variables are stably right equivalent if and only if they are right
equivalent.

###### Proof.

The statement follows from [19]. ∎

######  Remark 2.1 (Warning).

Contact equivalence for Lagrangian contact problems is not to be confused with
Mather’s notion of contact equivalence for map germs which is related to the
contact of a smooth manifold with a zero section of a smooth bundle (i.e.
without symplectic structure) [15].

######  Definition 2.4 (cotangent bundle structure and projection).

Let $X$ be a Lagrangian submanifold of a symplectic manifold $Z$. A
symplectomorphism $\Phi$ from an open neighbourhood $U\subset Z$ with
$X\subset U$ to a tubular neighbourhood of the zero section of $\pi\colon
T^{\ast}X\to X$ is called a cotangent bundle structure for $Z$ (with respect
to $X$). The map $\pi\circ\Phi$ is called the (cotangent bundle) projection
with respect to the structure $\Phi$.

######  Remark 2.2.

The existence of cotangent bundle structures (for a sufficiently small
neighbourhoods $U$ of $X$) is guaranteed by Weinstein’s generalisation of
Darboux’s theorem, see, for instance, [10, Thm. 15.3]. Notice that, in
comparison to the existence of local Darboux coordinates, Weinstein’s
generalisation has a global character since $X$ does not need to be shrunk for
symplectomorphisms to exist. As all considerations in this chapter are local,
the globality is only exploited for notational convenience in this context.
Moreover, we may refer to a function germ and a representative with the same
symbol where a differentiation is not essential.

######  Remark 2.3.

Cotangent bundle structures for a symplectic manifold $Z$ with respect to a
Lagrangian submanifold $X$ correspond to completely integrable Lagrangian
subbundles of the tangent bundle $TZ$ transverse to $X$. (See [10] for
details.) Moreover, cotangent bundle structures are in 1 to 1 correspondence
with 1-forms whose zero-set is $X$ and whose exterior derivative is the
symplectic structure. The cotangent bundle structure $\Phi\colon U\to V$,
where $U$ is an open neighbourhood of $X$ and $V$ is an open neighbourhood of
the zero section of $T^{\ast}X$, associated to such a 1-form $\alpha$ is such
that $\Phi^{\ast}\lambda=\alpha$ [7, 10].

For reference, let us recall some of Golubitsky and Guillemin’s results and
provide proofs for the cases where we would like to use the statement in a
more general setting than in the original paper.

######  Lemma 2.2 ([7, Lemma 3.1]).

Let $X$ be a Lagrangian submanifold of a symplectic manifold $Z$. Let
$x_{1},\ldots,x_{n}$ be local coordinates on $X$ centred at $z$. Consider two
cotangent bundle structures $\Phi^{\alpha}$ and $\Phi^{\beta}$ on $Z$ with
respect to $X$ around $z$. Let $\xi_{1},\ldots,\xi_{n}$ be the conjugate
momenta to $x_{1},\ldots,x_{n}$ with respect to the first structure. Let
$\lambda$ denote the canonical 1-form on $T^{\ast}X$,
$\alpha\coloneqq(\Phi^{\alpha})^{\ast}\lambda$,
$\beta\coloneqq(\Phi^{\beta})^{\ast}\lambda$. The closed 1-form $\alpha-\beta$
can locally be written as $\mathrm{d}H$ with

(2.1) $H(x,\xi)=\sum_{i,j=1}^{n}h_{ij}(x,\xi)\xi_{i}\xi_{j}.$

If the Lagrangian submanifold $\Lambda\subset Z$ is the image of the section
$\mathrm{d}\phi^{\alpha}$ with respect to the $\Phi^{\alpha}$-structure as
well as the image of the section $\mathrm{d}\phi^{\beta}$ with respect to the
$\Phi^{\beta}$-structure then

(2.2) $(\phi^{\beta}\circ
k_{\alpha\beta})(x)=\phi^{\alpha}(x)+H(x,\nabla\phi^{\alpha}(x))+\mathrm{const.},$

where $k_{\alpha\beta}$ is the diffeomorphism
$k_{\alpha\beta}=\pi^{\beta}\circ(\pi^{\alpha}|_{\Lambda})^{-1}$. Here
$\pi^{\alpha}$ and $\pi^{\beta}$ are the projections corresponding to
$\Phi^{\alpha}$ and $\Phi^{\beta}$, respectively.

###### Proof.

The 1-forms $\alpha$ and $\beta$ are primitives of the symplectic form on $Z$.
Moreover, $\alpha|_{z}=0$ if and only if $z\in X$. Analogously for $\beta$.
Therefore, $\alpha-\beta$ is closed and has a local primitive $H$. The
primitive must be of the form (2.1): in local coordinates we have

$\alpha=\sum_{j}\xi_{j}\mathrm{d}x_{j},\quad\beta=\sum_{j}(b_{j}\mathrm{d}x_{j}+c_{j}\mathrm{d}\xi_{j}),\quad\mathrm{d}H=\sum_{j}\left(\frac{\partial
H}{\partial x_{j}}\mathrm{d}x_{j}+\frac{\partial
H}{\partial\xi_{j}}\mathrm{d}\xi_{j}\right).$

Now

(2.3)
$\sum_{j}((\xi_{j}-b_{j})\mathrm{d}x_{j}+c_{j}\mathrm{d}\xi_{j})=\alpha-\beta=\mathrm{d}H=\sum_{j}\left(\frac{\partial
H}{\partial x_{j}}\mathrm{d}x_{j}+\frac{\partial
H}{\partial\xi_{j}}\mathrm{d}\xi_{j}\right).$

Setting $\xi=0$ in (2.3) and using $c_{j}(x,0)\equiv 0$ (which follows from
$\beta|_{(x,\xi)=(x,0)}=0$) we deduce that $\frac{\partial
H}{\partial\xi_{j}}(x,0)\equiv 0$. By Taylor’s theorem there exists a
primitive of the form (2.1).

Let $\iota\colon\Lambda\hookrightarrow Z$ denote the embedding of $\Lambda$
into $Z$. The 1-forms $\iota^{\ast}\alpha$ and $\iota^{\ast}\beta$ are closed
since $\Lambda$ is Lagrangian. Locally around a point of interest
$\iota^{\ast}\alpha$ and $\iota^{\ast}\beta$ have primitives which we denote
by $\phi^{\alpha}$ and $\phi^{\beta}$, respectively. Due to
$\iota^{\ast}\alpha-\iota^{\ast}\beta=\mathrm{d}(H\circ\iota)$ we have

(2.4) $\phi^{\beta}=\phi^{\alpha}+H\circ\iota+\mathrm{const.}$

on $\Lambda$. Expressing relation (2.4) in the canonical coordinates $(x,\xi)$
of the $\alpha$-cotangent bundle structure yields (2.2). ∎

######  Remark 2.4.

If the manifolds $X$ and $\Lambda$ intersect nontrivially and
$x_{1},\ldots,x_{n}$ are centred coordinates at an intersection point of
$\Lambda$ and $X$ then the constant in (2.2) vanishes if we choose
$\phi^{\alpha}(0)=\phi^{\beta}(0)$.

######  Lemma 2.3 ([7, Prop.4.2]).

Let

$H(x,\xi)=\sum_{i,j=1}^{n}h_{ij}(x,\xi)\xi_{i}\xi_{j}$

be defined on an open neighbourhood of the origin in
$\mathbb{R}^{n}\times\mathbb{R}^{n}$. Consider a real valued map $\phi$
defined on an open neighbourhood of 0 in $\mathbb{R}^{n}$ such that
$\phi(0)=0$, $\nabla\phi(0)=0$, $\mathrm{Hess}\,\phi(0)=0$. The map

(2.5) $\psi(x)=\phi(x)+H(x,\nabla\phi(x))$

is right equivalent to $\phi$ on a neighbourhood of the origin in
$\mathbb{R}^{n}$ and the right equivalence fixes the origin.

###### Proof.

To simplify notation, we set

$\overline{H}(x)=H(x,\nabla\phi(x)),\quad\overline{h}_{ij}(x)=h_{ij}(x,\nabla\phi(x)).$

We prove the assertion using the homotopy method: define

(2.6) $\psi_{t}(x)=\phi(x)+t\overline{H}(x).$

We seek a family of local diffeomorphisms $f_{t}$ fixing 0 such that

(2.7) $\psi_{t}\circ f_{t}=\phi.$

Differentiating (2.7) with respect to $t$ we find

$\frac{\mathrm{d}}{\mathrm{d}t}(\psi_{t})\circ
f_{t}+\left\langle\nabla\psi_{t}\circ
f_{t},\frac{\mathrm{d}}{\mathrm{d}t}f_{t}\right\rangle=0.$

Here, $\langle\cdot,\cdot\rangle$ denotes the scalar product in
$\mathbb{R}^{n}$. An evaluation at $f_{t}^{-1}(x)$ yields

(2.8)
$\frac{\mathrm{d}}{\mathrm{d}t}\psi_{t}+\langle\nabla\psi_{t},w(x,t)\rangle=0$

with

(2.9) $w(x,t)=\frac{\mathrm{d}}{\mathrm{d}t}(f_{t})(f_{t}^{-1}(x)).$

We will show that the ODE (2.8) is solvable for $w$ around $x=0$ with
$w(0,t)=0$. Then there exists an open neighbourhood $U\subset\mathbb{R}^{n}$
of $0$ such that the initial value problem

(2.10) $\frac{\mathrm{d}}{\mathrm{d}t}f_{t}(x)=w(f_{t}(x),t),\quad f_{0}(x)=x$

can be solved for all $x\in U$ on the interval $t\in[0,1]$. The obtained
family of functions $f_{t}$ fulfils
$\frac{\mathrm{d}}{\mathrm{d}t}(\psi_{t}\circ f_{t})=0$ with
$f_{0}=\mathrm{id}$ and, therefore, (2.7). Moreover, $f_{t}(0)\equiv 0$ such
that $f_{1}$ is the required right equivalence.

We now show that near $x=0$ the ODE (2.8) is solvable for $w$ with $w(0,t)=0$.
Differentiating (2.6) with respect to $x$ yields

(2.11) $\begin{split}\frac{\partial\psi_{t}}{\partial
x_{l}}&=\frac{\partial\phi}{\partial
x_{l}}+t\sum_{i,j}\left(\frac{\partial\overline{h}_{ij}}{\partial
x_{l}}\frac{\partial\phi}{\partial x_{i}}\frac{\partial\phi}{\partial
x_{j}}+2\overline{h}_{ij}\frac{\partial^{2}\phi}{\partial x_{i}\partial
x_{l}}\frac{\partial\phi}{\partial x_{j}}\right)\\\
&=\sum_{j}\underbrace{\left(\delta_{lj}+t\sum_{i}\left(\frac{\partial\overline{h}_{ij}}{\partial
x_{l}}\frac{\partial\phi}{\partial
x_{i}}+2\overline{h}_{ij}\frac{\partial^{2}\phi}{\partial x_{i}\partial
x_{l}}\right)\right)}_{=:B_{lj}(t,x)}\frac{\partial\phi}{\partial
x_{j}}\end{split}$

The maps $B_{lj}(t,x)$ form a matrix $B$ with $B(t,0)=\mathrm{Id}$. Therefore,
there exists a neighbourhood of $x=0$ such that $B$ is invertible for all
$t\in[0,1]$. We have

(2.12) $\nabla\phi=B^{-1}\nabla\psi_{t}.$

The functions $\overline{h}_{ij}$ constitute a matrix which we denote by
$\mathcal{H}$. Differentiating (2.6) with respect to $t$ and using (2.12) we
get

$\frac{\mathrm{d}}{\mathrm{d}t}\psi_{t}=\nabla\phi^{\top}\mathcal{H}\nabla\phi\stackrel{{\scriptstyle\eqref{eq:RelateDpsiDpsit}}}{{=}}\nabla\psi_{t}^{\top}\underbrace{B^{-T}\mathcal{H}\nabla\phi}_{=:-w}.$

Now $w(0,t)=0$ and $w$ solves (2.8). This completes the proof.

∎

The Lemmas 2.2 and 2.3 imply the following proposition.

######  Proposition 2.4.

Let $(X,\Lambda,z)$ be a tangential Lagrangian contact problem in $Z$.
Consider two cotangent bundle structures over $X$ near $z$ such that $\Lambda$
is the image of the section $\mathrm{d}\phi^{\alpha}$ and
$\mathrm{d}\phi^{\beta}$ with $\phi^{\alpha}(z)=0=\phi^{\beta}(z)$. Then
$\phi^{\alpha}$ and $\phi^{\beta}$ are right equivalent.

######  Remark 2.5.

As $X$ and $\Lambda$ are tangential at $z$, for any cotangent bundle structure
the submanifold $\Lambda$ is the image of a section of the cotangent bundle
locally around the point of contact. This is not true for general Lagrangian
contact problems.

######  Theorem 2.5.

Let $(X,\Lambda,z)$ and $(X,\Lambda^{\prime},z)$ be two tangential Lagrangian
contact problems in $Z$. For any cotangent bundle structure over $X$ near $z$
such that $\Lambda$ is the image of the section $\mathrm{d}\phi$ and
$\Lambda^{\prime}$ the image of $\mathrm{d}\phi^{\prime}$ with
$\phi(z)=0=\phi^{\prime}(z)$, the map germs $\phi$ and $\phi^{\prime}$ are
right equivalent if and only if the tangential Lagrangian intersection
problems are contact equivalent.

###### Proof.

In the following, we may shrink $Z,X,\Lambda,\Lambda^{\prime}$ repeatedly to
intersections with open neighbourhoods of $z$ without mentioning. Assume
$\phi=\phi^{\prime}\circ r$ for a right equivalence $r$ such that $r$ fixes
$z$. Its cotangent lifted map $R$ (see [14, §6.3] for definitions) fixes $X$
and

$R(\mathrm{d}\phi^{\prime}|_{x})=\mathrm{d}\phi^{\prime}|_{x}\circ\mathrm{d}r|_{r^{-1}(x)}=\mathrm{d}(\phi^{\prime}\circ
r)|_{r^{-1}(x)}=\mathrm{d}\phi|_{r^{-1}(x)},\qquad x\in X.$

Therefore, the symplectic diffeomorphism $R$ maps $\Lambda^{\prime}$ to
$\Lambda$ and, thus, provides a contact equivalence between $(X,\Lambda,z)$
and $(X,\Lambda^{\prime},z)$.

Now assume there exists a symplectic diffeomorphism with $\Phi(X)=X$,
$\Phi(z)=z$ and $\Phi(\Lambda)=\Lambda^{\prime}$. Choose a cotangent bundle
structure with projection denoted by $\pi\colon Z\to X$ such that $\Lambda$ is
the image of the section $\mathrm{d}\phi$ and $\Lambda^{\prime}$ the image of
the section $\mathrm{d}\phi^{\prime}$ around $z$ with
$\phi(z)=0=\phi^{\prime}(z)$. Using $\Phi$ we can obtain a new cotangent
bundle structure with bundle projection $\pi^{\prime}\colon Z\to X$ such that
$\pi^{\prime}=\pi\circ\Phi^{-1}$. The map $\Phi\circ\mathrm{d}\phi$ maps $X$
onto $\Lambda^{\prime}$ and is a section of $\pi^{\prime}\colon Z\to X$. This
means $\Lambda^{\prime}$ can be represented by $\mathrm{d}\phi$ in the new
structure. Therefore, by Proposition 2.4, the map germs $\phi$ and
$\phi^{\prime}$ must be right equivalent. ∎

We recall the well-known parametric Morse Lemma or Splitting Lemma and
establish some notation.

######  Lemma 2.6 (parametric Morse lemma).

Let $\phi\colon(\mathbb{R}^{n},0)\to(\mathbb{R},0)$ be a function germ with
critical point at the origin 0. Consider the decomposition
$\mathbb{R}^{n}=\overline{X}\oplus\underline{X}$ for two linear subspaces
$\overline{X}$ and $\underline{X}$ such that the Hessian matrix $B$ of the
restriction $\phi|_{\underline{X}}\colon(\underline{X},0)\to(\mathbb{R},0)$ is
invertible. There exists a change of coordinates $K$ on a neighbourhood of
$0\in\mathbb{R}^{n}$ of the form
$K(\overline{x},\underline{x})=(\overline{x},\kappa(\overline{x},\underline{x}))$
with $K(0,0)=(0,0)$ such that

$(\phi\circ
K)(\overline{x},\underline{x})=f(\overline{x})+{\underline{x}}^{\top}B\underline{x}.$

If we choose the dimension of $\underline{X}$ to be maximal222We can choose
$\dim\underline{X}$ to be the rank of the Hessian matrix of $\phi$ at $0$ then
the 2-jet of $f$ vanishes.

A proof is given in [5, §14.12] or in [18, Lemma 5.12]. A fibred version
including uniqueness results can be found in [19].

We leave the setting of tangential Lagrangian contact problems and extend
Proposition 2.4: a function germ assigned to a (not necessarily tangential)
Lagrangian intersection problem using any cotangent bundle structure for which
the intersection problem is graphical is well-defined up to stably right
equivalence. For this we first prove the following lemma.

######  Lemma 2.7.

On $\mathbb{R}^{n}=\overline{X}\oplus\underline{X}$ consider

  * •

coordinates
$x=(\overline{x},\underline{x})=((x^{1},\ldots,x^{k}),(x^{k+1},\ldots,x^{n}))$,

  * •

a nondegenerate symmetric matrix $Q\in\mathbb{R}^{(n-k)\times(n-k)}$,

  * •

a map $g\colon\mathbb{R}^{k}\to\mathbb{R}^{k}$ whose 2-jet vanishes at 0,

  * •

and a matrix valued function
$\mathcal{H}\colon\mathbb{R}^{n}\to\mathrm{Sym}(n)$ with

$\mathcal{H}(x)=(h_{ij}(x))_{i,j=1,\ldots,n}=\begin{pmatrix}H^{11}(x)&H^{12}(x)\\\
{H^{12}}(x)^{\top}&0\end{pmatrix}\in\mathrm{Sym}(n)\subset\mathbb{R}^{n\times
n}$

for $H^{11}(x)\in\mathrm{Sym}(k)$ and $H^{12}(x)\in\mathbb{R}^{k\times(n-k)}$.

For $t\in\mathbb{R}$ let

$\displaystyle\psi(x)$
$\displaystyle=g(\overline{x})+\underline{x}^{\top}Q\underline{x}$
$\displaystyle\psi_{t}(x)$
$\displaystyle=g(\overline{x})+\underline{x}^{\top}Q\underline{x}+t(\nabla\psi(x)^{\top}\mathcal{H}(x)\nabla\psi(x)).$

Then $\psi_{t}$ is right equivalent to $\psi=\psi_{0}$ around $x=0$ and the
right equivalence fixes 0.

###### Proof.

Motivated by the proof of Lemma 2.3 we define the components $B_{lj}(t,x)$ of
a matrix $B(t,x)\in\mathbb{R}^{n\times n}$ as

$B_{lj}=\delta_{lj}+t\sum_{i}\left(\frac{\partial h_{ij}}{\partial
x_{l}}\frac{\partial\psi}{\partial
x_{i}}+2h_{ij}\frac{\partial^{2}\psi}{\partial x_{i}\partial x_{l}}\right).$

We have

(2.13)
$B(t,0)=\mathrm{Id}_{n}+2t\mathrm{Hess}\,\psi(0)\mathcal{H}(0)=\begin{pmatrix}\mathrm{Id}_{k}&0\\\
4tQ{H^{12}}(0)^{\top}&\mathrm{Id}_{n-k},\end{pmatrix}$

which is invertible for all $t$. Using the same calculation as in (2.11) we
obtain

$\nabla\psi_{t}(x)=B(t,x)\nabla\psi(x).$

Therefore,

(2.14)
$\frac{\mathrm{d}}{\mathrm{d}t}\psi_{t}=\nabla\psi^{\top}\mathcal{H}\nabla\psi=\nabla\psi_{t}^{\top}\underbrace{B^{-T}\mathcal{H}\nabla\psi}_{=:-\omega}=-\langle\nabla\psi_{t},\omega\rangle.$

There exists a neighbourhood $U$ of $0\in\mathbb{R}^{n}$ such that the initial
value problem

$\frac{\mathrm{d}}{\mathrm{d}t}f_{t}(x)=\omega(f_{t}(x),t),\quad f_{0}(x)=x$

is solvable for all $x\in U$ and all $t\in[0,1]$. Since $\omega(0,t)=0$ we
have $f_{t}(0)=0$ and

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(\psi_{t}\circ f_{t})$
$\displaystyle\;=\frac{\mathrm{d}}{\mathrm{d}t}(\psi_{t})\circ
f_{t}+\left\langle\nabla\psi_{t}\circ
f_{t},\frac{\mathrm{d}}{\mathrm{d}t}f_{t}\right\rangle$
$\displaystyle\;=\frac{\mathrm{d}}{\mathrm{d}t}(\psi_{t})\circ
f_{t}+\left\langle\nabla\psi_{t},\omega\right\rangle\circ f_{t}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:ddtpsit}}}{{=}}0.$

Since $f_{0}=\mathrm{id}_{U}$ we have

$\psi_{t}\circ f_{t}=\psi_{0}\circ f_{0}=\psi$

and $f_{t}$ is the required right equivalence. ∎

######  Proposition 2.8.

Let $(X,\Lambda,z)$ be a Lagrangian contact problem in $Z$. After shrinking
$Z$ to a sufficiently small neighbourhood of $z$, consider two cotangent
bundle structures over $X$ with projections $\pi^{\alpha}\colon Z\to X$,
$\pi^{\beta}\colon Z\to X$ such that $\Lambda$ is given as the image of the
section $\mathrm{d}\phi^{\alpha}$ and $\mathrm{d}\phi^{\beta}$, respectively,
whereas $\phi^{\alpha}(z)=0=\phi^{\beta}(z)$. Then the germs of
$\phi^{\alpha}$ and $\phi^{\beta}$ at $z$ are stably right equivalent.

###### Proof.

By the parametric Morse Lemma (Lemma 2.6) there exist coordinates
$x=(\overline{x},\underline{x})=((x^{1},\ldots,x^{k}),(x^{k+1},\ldots,x^{n}))$
on $X$ centred at $z$ such that

$\phi^{\alpha}(x)=f(\overline{x})+\underline{x}^{\top}B\underline{x}$

for a smooth function germ $f$ with vanishing 2-jet at $\overline{x}=0$ and an
invertible symmetric matrix $B$. The map $\phi^{\alpha}$ is stably right
equivalent to $f$. By Lemma 2.2 we have

(2.15) $\phi^{\beta}\circ
k_{\alpha\beta}=\phi^{\alpha}+H(x,\nabla\phi^{\alpha}),$

for a map

$H(x,\xi)=\sum_{i,j=1}^{n}h_{ij}(x,\xi)\xi_{i}\xi_{j}=\xi^{\top}\begin{pmatrix}H^{11}(x,\xi)&H^{12}(x,\xi)\\\
H^{12}(x,\xi)^{\top}&H^{22}(x,\xi)\end{pmatrix}\xi$

with matrices $H^{11}(x,\xi)\in\mathrm{Sym}(k)$,
$H^{22}(x,\xi)\in\mathrm{Sym}(n-k)$, and
$H^{12}(x,\xi)\in\mathbb{R}^{k\times(n-k)}$ and with
$k_{\alpha\beta}=\pi^{\beta}\circ(\pi^{\alpha}|_{\Lambda})^{-1}$. For
$i,j\in\\{1,2\\}$ define

$\mathcal{H}(x)\coloneqq\mathcal{H}(\overline{x},\underline{x})\coloneqq
H(x,\nabla\phi^{\alpha}(x)),\quad\mathcal{H}^{ij}(x)\coloneqq\mathcal{H}^{ij}(\overline{x},\underline{x})\coloneqq
H^{ij}(x,\nabla\phi^{\alpha}(x)).$

Therefore,

$\mathcal{H}(x)=\nabla_{\overline{x}}f(\overline{x})^{\top}\mathcal{H}^{11}(x)\nabla_{\overline{x}}f(\overline{x})+4\underline{x}^{\top}B\mathcal{H}^{12}(x)^{\top}\nabla_{\overline{x}}f(\overline{x})+4\underline{x}^{\top}B\mathcal{H}^{22}(x)B\underline{x}.$

We calculate

(2.16)
$\begin{split}\phi^{\beta}(k_{\alpha\beta}(\overline{x},\underline{x}))&\stackrel{{\scriptstyle\eqref{eq:ApplyCompCot}}}{{=}}\phi^{\alpha}(\overline{x},\underline{x})+\mathcal{H}(x)\\\
&\;\;\;=f(\overline{x})+\underline{x}^{\top}(B+4B\mathcal{H}^{22}(x)B)\underline{x}\\\
&\;\;\;\phantom{=}+\nabla_{\overline{x}}f(\overline{x})^{\top}\mathcal{H}^{11}(x)\nabla_{\overline{x}}f(\overline{x})+4\underline{x}^{\top}B\mathcal{H}^{12}(x)^{\top}\nabla_{\overline{x}}f(\overline{x}).\end{split}$

Let us define

$B^{\prime}(x)=B+4B\mathcal{H}^{22}(x)B.$

The kernel of $\mathrm{Hess}\,(\phi^{\alpha})$ and the kernel of
$\mathrm{Hess}\,(\phi^{\beta})$ at $(\overline{x},\underline{x})=(0,0)$ both
describe the intersection $T_{z}X\cap T_{z}\Lambda$ (but in different
coordinates). Indeed, the kernel of $\mathrm{Hess}\,(\phi^{\beta}\circ
k_{\alpha\beta})$ must coincide with the kernel of
$\mathrm{Hess}\,(\phi^{\alpha})$ which is
$\overline{X}=\\{\underline{x}=0\\}$. We calculate the Hessian matrix of
$\phi^{\beta}\circ k_{\alpha\beta}$ at $(\overline{x},\underline{x})=(0,0)$
using (2.16) and obtain

$\mathrm{Hess}\,(\phi^{\beta}\circ k_{\alpha\beta})(0,0)=\begin{pmatrix}0&0\\\
0&2B^{\prime}(0)\end{pmatrix}.$

Since $\Lambda$ is graphical in both cotangent bundle structures,
$B^{\prime}(0)$ must be invertible by a dimension argument. The function
$\phi^{\alpha}$ is stably right equivalent to

$\phi^{\alpha}_{1}(x):=f(\overline{x})+\underline{x}^{\top}B^{\prime}(0)\underline{x}.$

For $x$ in a sufficiently small neighbourhood of $0$ the signature of
$B^{\prime}(x)$ is constant. By Sylvester’s law of inertia, there exists a
smooth family of invertible matrices $A(x)$ such that

(2.17) $A(x)^{-T}B^{\prime}(x)A^{-1}(x)=B^{\prime}(0)$

for all $x$ near 0. Consider

$r(\overline{x},\underline{x})=(\overline{x},A(x)\underline{x}).$

The map $r$ fixes $x=0$ and is a diffeomorphism on a neighbourhood of $x=0$:
the Jacobian matrix of $r$ is given by the block matrix

$\mathrm{D}r(x)=\begin{pmatrix}\mathrm{Id}_{k}&0\\\ (\partial
A(x))_{l=1,\ldots,k}&(\partial A(x))_{l=k+1,\ldots,n}+A(x)\end{pmatrix},$

where $(\partial A(x))_{l=1,\ldots,k}$ denotes the first $k$ columns and
$(\partial A(x))_{l=k+1,\ldots,n}$ the remaining $n-k$ columns of an
$(n-k)\times n$ matrix $\partial A(x)$ whose $l$-th column is given as

$(\partial A(x))_{l}=\frac{\partial A}{\partial x^{l}}(x)\underline{x},$

where the derivative $\frac{\partial A}{\partial x^{l}}$ is taken component-
wise. Now the determinant of $\mathrm{D}r(0)$ coincides with the determinant
of $A(0)$ which is non-zero, so $r$ is indeed a right equivalence.

Let us define

(2.18)
$\begin{split}\tilde{\mathcal{H}}^{11}(x)&=\mathcal{H}^{11}(r^{-1}(x))\\\
\tilde{\mathcal{H}}^{12}(x)&=\mathcal{H}^{12}(r^{-1}(x))BA(r^{-1}(x))^{-1}B^{\prime}(0)^{-1}\end{split}$

By Lemma 2.7 the map $\phi^{\alpha}_{1}$ is right equivalent to

$\displaystyle\phi^{\alpha}_{2}(x):$
$\displaystyle=f(\overline{x})+\underline{x}^{\top}B^{\prime}(0)\underline{x}$
$\displaystyle+\nabla_{\overline{x}}f(\overline{x})^{\top}\tilde{\mathcal{H}}^{11}(x)\nabla_{\overline{x}}f(\overline{x})+4\underline{x}^{\top}B^{\prime}(0)\tilde{\mathcal{H}}^{12}(x)^{\top}\nabla_{\overline{x}}f(\overline{x}).$

We have

(2.19) $\begin{split}(\phi^{\alpha}_{2}\circ
r)(x)&=f(\overline{x})+\underline{x}^{\top}A(x)^{\top}B^{\prime}(0)A(x)\underline{x}+\nabla_{\overline{x}}f(\overline{x})^{\top}\tilde{\mathcal{H}}^{11}(r(x))\nabla_{\overline{x}}f(\overline{x})\\\
&\phantom{=}+4\underline{x}^{\top}A(x)^{\top}B^{\prime}(0)\tilde{\mathcal{H}}^{12}(r(x))^{\top}\nabla_{\overline{x}}f(\overline{x}).\end{split}$

Comparing (2.19) with (2.16) shows that the function $\phi^{\alpha}_{2}\circ
r$ coincides with $\phi^{\beta}\circ k_{\alpha\beta}$. Thus, the maps
$\phi^{\alpha}$ and $\phi^{\beta}$ are stably right equivalent. ∎

######  Remark 2.6.

We see from the proof of Proposition 2.8 that if the intersection of $\Lambda$
and $X$ is not tangential then the dimension of $\underline{X}$ is greater
than 0 and there exist two cotangent bundle structures such that
$\phi^{\alpha}$ and $\phi^{\beta}$ are stably right equivalent but not right
equivalent: to a cotangent bundle structure over $X$ defined333Recall from
Remark 2.3 that suitable 1-forms define cotangent bundle structures. by a
canonical 1-form $\alpha$ for which $\Lambda$ is graphical choose another
cotangent bundle structure $\beta=\alpha+\mathrm{d}H$ with

$H(x,\xi)=\xi^{\top}\begin{pmatrix}0&0\\\
0&\frac{1}{4}(B^{-1}DB^{-1}-B^{-1})\end{pmatrix}\xi,$

where $D$ is an invertible symmetric matrix which has a different signature
then $B$. As in the proof of Proposition 2.8, the coordinates $(x,\xi)$ refer
to Darboux coordinates with respect to the $\alpha$-structure. We get
$B^{\prime}=D$ which is invertible such that $\Lambda$ is graphical for the
cotangent bundle structure defined by $\beta$. However, the signatures of
$\mathrm{Hess}\,\phi^{\alpha}(0)$ and $\mathrm{Hess}\,\phi^{\beta}(0)$ do not
coincide.

######  Remark 2.7 (existence of cotangent bundle structures).

Let $(X,\Lambda,z)$ be a Lagrangian contact problem in a symplectic manifold
$Z$. It is easy to see from a perturbation argument or from the classification
of intersections of Lagrangian linear subspaces of finite dimensional
symplectic vector spaces [12] that there is always a contangent bundle
structure $\pi\colon Z\to X$ such that $\Lambda$ is graphical, i.e. the image
of a smooth section of $\pi\colon Z\to X$ locally around $z$ or, equivalently,
s.t. $\pi|_{\Lambda}\colon\Lambda\to X$ is an immersion around $z$. A proof
can be found in [16, Lemma 2.1].

######  Definition 2.5 ((fully reduced) generating function).

Let $(X,\Lambda,z)$ be a Lagrangian contact problem in $Z$. Consider a
cotangent bundle structure such that $\Lambda$ is given as the image of the
section $\mathrm{d}\phi$ locally around $z\in Z$ with $\phi(z)=0$. We refer to
$\phi$, to the germ of $\phi$ at $z$, and to (the germ at 0 of) a coordinate
expression of $\phi$ with centred coordinates at $z$ as a generating function
of $(X,\Lambda,z)$. By the parametric Morse Lemma (Lemma 2.6) there exist
coordinates
$x=(\overline{x},\underline{x})=((x^{1},\ldots,x^{k}),(x^{k+1},\ldots,x^{n}))$
on $X$ centred at $z$ such that

$\phi(x)=f(\overline{x})+\underline{x}^{\top}B\underline{x}$

for a smooth function germ $f$ with vanishing 2-jet and an invertible matrix
$B$. The map germ $f$ is called a fully reduced generating function of
$(X,\Lambda,z)$.

Now Proposition 2.8 can be phrased as follows.

######  Corollary 2.9.

Generating functions of Lagrangian contact problems are defined up to stably
right equivalence.

Using Theorem 2.1 we obtain the following corollary.

######  Corollary 2.10.

Fully reduced generating functions of Lagrangian contact problems are defined
up to right equivalence.

######  Lemma 2.11.

Consider a symplectic diffeomorphism $\Phi\colon Z\to Z^{\prime}$. Any
generating function of a Lagrangian contact problem $(X,\Lambda,z)$ in $Z$ is
right equivalent to any generating function of the Lagrangian contact problem
$(\Phi(X),\Phi(\Lambda),\Phi(z))$ in $Z^{\prime}$ when shrinking all involved
manifolds to intersections with open neighbourhoods of $z\in Z$ or $\Phi(z)\in
Z^{\prime}$, if necessary.

###### Proof.

Cotangent bundle structures on $Z$ over $X$ correspond to cotangent bundle
structures on $Z^{\prime}$ over $\Phi(X)$, whereas the corresponding canonical
1-forms $\lambda$ and $\lambda^{\prime}$ relate by
$\lambda=\Phi^{\ast}\lambda^{\prime}$. Therefore, if
$\iota\colon\Lambda\hookrightarrow Z$ is the embedding of $\Lambda$ into $Z$
and $\iota^{\prime}=\Phi\circ\iota\circ(\Phi^{-1})|_{\Phi(\Lambda)}$ the
embedding of $\Phi(\Lambda)$ into $Z^{\prime}$ then

${\iota^{\prime}}^{\ast}\lambda^{\prime}=(\Phi^{-1})|_{\Phi(\Lambda)}^{\ast}{\iota}^{\ast}\lambda.$

Thus, the primitive of ${\iota^{\prime}}^{\ast}\lambda^{\prime}$ around
$\Phi(z)$ which vanishes at $\Phi(z)$ and the primitive of
${\iota}^{\ast}\lambda$ around $z$ which vanishes at $z$ relate by
$(\Phi^{-1})|_{\Phi(\Lambda)}$. Expressing the primitives in coordinates on
$X$ or $\Phi(X)$, we obtain generating functions which are right equivalent. ∎

We can now extend Theorem 2.5 to non-tangential Lagrangian contact problems.

######  Theorem 2.12.

Two Lagrangian contact problems $(X,\Lambda,z)$ in $Z$ and
$(X^{\prime},\Lambda^{\prime},z^{\prime})$ in $Z^{\prime}$ with $\dim Z=\dim
Z^{\prime}$ are contact equivalent if and only if their generating functions
are stably right equivalent.

###### Proof.

By Lemma 2.11 and the local nature of the problem it is sufficient to consider
$Z^{\prime}=Z$, $X^{\prime}=X$ and $z^{\prime}=z$. In the following we may
shrink $Z$ and all embedded manifolds to their intersections with open
neighbourhoods of $z$ repeatedly without mentioning. Assume there exists a
symplectic diffeomorphism with $\Phi(X)=X$, $\Phi(z)=z$ and
$\Phi(\Lambda)=\Lambda^{\prime}$. As justified in Remark 2.7, there exists a
cotangent bundle structure over $X$ with projection $\pi\colon Z\to X$ such
that $\Lambda$ is the image of the section $\mathrm{d}\phi$ and
$\Lambda^{\prime}$ the image of the section $\mathrm{d}\phi^{\prime}$ around
$z$ with $\phi(z)=0=\phi^{\prime}(z)$. Consider the cotangent bundle structure
$\pi^{\prime}\colon Z\to X$ with $\pi^{\prime}=\pi\circ\Phi^{-1}$. The map
$\Phi\circ\mathrm{d}\phi$ maps $X$ onto $\Lambda^{\prime}$ and is a section of
$\pi^{\prime}\colon Z\to X$. This means that $\Lambda^{\prime}$ is the image
of the section $\mathrm{d}\phi$ with respect to the new structure. Therefore,
$\phi$ and $\phi^{\prime}$ are both generating functions of
$\Lambda^{\prime}$. By Corollary 2.9, the map germs $\phi$ and $\phi^{\prime}$
are stably right equivalent.

Now let $\phi$ be a generating function of $(X,\Lambda,z)$ and $\phi^{\prime}$
of $(X,\Lambda^{\prime},z)$ and assume that the function germs $\phi$ and
$\phi^{\prime}$ are stably right equivalent. By Proposition 2.8, without loss
of generality we may assume that $\phi$, $\phi^{\prime}$ refer to the same
cotangent bundle structure $\pi\colon Z\to X$.

By the parametric Morse Lemma (Lemma 2.6) there exist coordinates
$(\overline{x},\underline{x})=((x^{1},\ldots,x^{k}),(x^{k+1},\ldots,x^{n}))$
and
$(\overline{x^{\prime}},\underline{x^{\prime}})=(({x^{\prime}}^{1},\ldots,{x^{\prime}}^{k}),({x^{\prime}}^{k+1},\ldots,{x^{\prime}}^{n}))$
on $X$ centred at $z$, function germs $f$, $f^{\prime}$ with vanishing 2-jets
at 0 and invertible, symmetric matrices $B$ and $B^{\prime}$ such that

$\displaystyle\phi(\overline{x},\underline{x})$
$\displaystyle=f(\overline{x})+{\underline{x}}^{\top}B\underline{x}$
$\displaystyle\phi^{\prime}(\overline{x^{\prime}},\underline{x^{\prime}})$
$\displaystyle=f^{\prime}(\overline{x^{\prime}})+{\underline{x^{\prime}}}^{\top}B^{\prime}\underline{x^{\prime}}.$

Let $D$ be an invertible, symmetric, $n-k$-dimensional matrix such that the
matrix $-B^{-1}+DB^{\prime}D$ is invertible. Define the maps

$\displaystyle\phi^{(0)}(\overline{x},\underline{x})$
$\displaystyle=f^{\prime}(\overline{x})+{\underline{x}}^{\top}B^{\prime}\underline{x}$
$\displaystyle\phi^{(1)}(\overline{x},\underline{x})$
$\displaystyle=f(\overline{x})+{\underline{x}}^{\top}B^{\prime}\underline{x}$
$\displaystyle\phi^{(2)}(\overline{x},\underline{x})$
$\displaystyle=f(\overline{x})+{\underline{x}}^{\top}BDB^{\prime}DB\underline{x}.$

We now show that we can map $\Lambda^{\prime}=\mathrm{d}\phi^{\prime}(X)$ to
$\mathrm{d}\phi^{(0)}(X)$, then to $\mathrm{d}\phi^{(1)}(X)$, then to
$\mathrm{d}\phi^{(2)}(X)$, and finally to $\Lambda=\mathrm{d}\phi(X)$ with
symplectic diffeomorphisms which fix $X$ and $z$.

  * •

The change of coordinates between $(\overline{x},\underline{x})$ and
$(\overline{x^{\prime}},\underline{x^{\prime}})$ induces a diffeomorphism
$\chi$ locally defined around $z\in X$ with $\chi(z)=z$. The cotangent lift of
$\chi$ fixes $z$ and provides a symplectomorphism $\Phi^{(0)}$ between
$\Lambda^{\prime}$ and $\mathrm{d}\phi^{(0)}(X)$ on an open neighbourhood of
$z$ in $Z$.

  * •

Since $\phi$ and $\phi^{\prime}$ are stably right equivalent, there exists a
right equivalence $r$ such that $f=f^{\prime}\circ r$ (Theorem 2.1). Define
$\tilde{r}(\overline{x},\underline{x})=(r(\overline{x}),\underline{x})$ and
denote the cotangent lift of $\tilde{r}$ by $\Phi^{(1)}$. Now $\Phi^{(1)}$
maps $\mathrm{d}\phi^{(0)}(X)$ to a manifold which coincides with
$\mathrm{d}\phi^{(1)}(X)$ on an open neighbourhood of $z$: indeed, denote
$\overline{\pi}(x)=\overline{x}$, $\underline{\pi}(x)=\underline{x}$. There
exist appropriate choices of open neighbourhoods $V^{\prime}\subset Z$ of $z$,
$\tilde{V}\subset X$ of $z$, and $U,U^{\prime}\subset\mathbb{R}^{n}$ of 0 such
that

$\displaystyle\Phi^{(1)}(\Lambda^{\prime}\cap V^{\prime})$
$\displaystyle=\Phi^{(1)}\left(\left\\{\mathrm{d}(f^{\prime}\circ\overline{\pi}+\underline{\pi}^{\top}B^{\prime}\underline{\pi})|_{(\overline{x},\underline{x})}\,|\,(\overline{x},\underline{x})\in
U^{\prime}\right\\}\right)$
$\displaystyle=\left(\left\\{\mathrm{d}(f\circ\overline{\pi}+\underline{\pi}^{\top}B^{\prime}\underline{\pi})|_{(r^{-1}(\overline{x}),\underline{x})}\,|\,(\overline{x},\underline{x})\in
U^{\prime}\right\\}\right)$
$\displaystyle=\left(\left\\{\mathrm{d}(f\circ\overline{\pi}+\underline{\pi}^{\top}B^{\prime}\underline{\pi})|_{(\overline{x},\underline{x})}\,|\,(\overline{x},\underline{x})\in
U\right\\}\right)$ $\displaystyle=\mathrm{d}\phi^{(1)}(X\cap\tilde{V}).$

  * •

Denote the cotangent lift of the map
$(\overline{x},\underline{x})\mapsto(\overline{x},DB\underline{x})$ by
$\Phi^{(2)}$. The symplectomorphism $\Phi^{(2)}\circ\Phi^{(1)}$ maps
$\Lambda^{\prime}$ to a manifold which coincides with
$\mathrm{d}\phi^{(2)}(X)$ on a neighbourhood of $z$.

  * •

It remains to show that there exists a symplectomorphism mapping
$\mathrm{d}\phi^{(2)}(X)$ to $\Lambda$ which fixes $X$ and $z$. Let $\lambda$
denote the canonical 1-form of the cotangent bundle structure for which
$\pi\colon Z\to X$ is the projection. We define another cotangent bundle
structure over $X$ by setting its canonical 1-form to
$\lambda^{\prime}=\lambda+\mathrm{d}H$ with

$H(x,\xi)=\xi^{\top}\begin{pmatrix}0&0\\\
0&\frac{1}{4}(-B^{-1}+DB^{\prime}D)\end{pmatrix}\xi.$

The manifold $\Lambda$ is graphical with respect to the cotangent bundle
structure defined by $\lambda^{\prime}$ by the choice of $D$. Let
$\phi^{\lambda^{\prime}}$ denote the generating function of $\Lambda$ with
respect to the new structure $\lambda^{\prime}$. Applying Lemma 2.2 we get

$(\phi^{\lambda^{\prime}}\circ
k)(\overline{x},{\underline{x}})=f(\overline{x})+{\underline{x}}^{\top}BDB^{\prime}DB\underline{x}$

for a diffeomorphism $k$ on $X$. The cotangent lifted map $K$ of $k$ via the
$\lambda$-structure maps $\mathrm{d}\phi^{\lambda^{\prime}}|_{x}$ to
$\mathrm{d}(\phi^{\lambda^{\prime}}\circ
k)|_{k^{-1}(x)}=\mathrm{d}(\phi^{(2)})|_{k^{-1}(x)}$ for each $x\in X$.
Therefore,

$\mathrm{d}\phi^{\lambda^{\prime}}(X)\quad\text{ and
}\quad(K^{-1}\circ\Phi^{(2)}\circ\Phi^{(1)})(\Lambda^{\prime})$

coincide near $z$. Locally around $z$ the symplectormorphism relating the
cotangent bundle structures $\lambda$ and $\lambda^{\prime}$ maps
$\mathrm{d}\phi^{\lambda^{\prime}}(X)$ to $\Lambda$ and fixes $X$ and $z$.

It follows that $(X,\Lambda,z)$ and $(X,\Lambda^{\prime},z)$ are contact-
equivalent. ∎

######  Corollary 2.13.

The local algebras of generating functions (see [3, 13, 18] for a definition
and the meaning for catastrophe theory) of contact equivalent Lagrangian
contact problems are isomorphic.

Rather than using $X$ as a zero section for a cotangent bundle structure to
describe the Lagrangian contact problem $(X,\Lambda,z)$ in $Z$ we can use any
other Lagrangian submanifold $L\subset Z$ as a reference manifold as explained
in the following proposition. This gives us some flexibility when computing
generating functions which we will exploit in a parameter-dependent setting.

######  Proposition 2.14.

Let $(X,\Lambda,z)$ be a Lagrangian contact problem in the symplectic manifold
$Z=T^{\ast}L$. Assume that $X$ and $\Lambda$ are graphical and given as the
images of $\mathrm{d}\phi^{X}$ and $\mathrm{d}\phi^{\Lambda}$, for
$\phi^{X},\phi^{\Lambda}\colon L\to\mathbb{R}$ with
$\phi^{X}(\pi(z))=\phi^{\Lambda}(\pi(z))$, where $\pi\colon Z\to L$ is the
bundle projection. The map

(2.20) $\phi=\phi^{\Lambda}-\phi^{X}$

expressed in local coordinates on $L$ centred at $\pi(z)$ is stably right
equivalent to any generating function of $(X,\Lambda,z)$.

###### Proof.

Denote the canonical 1-form on $Z=T^{\ast}L$ by $\lambda$ and the cotangent
bundle projection by $\pi\colon Z\to L$. Consider the fibre preserving
diffeomorphism

$\chi\colon Z\to Z,\quad z\mapsto z-\mathrm{d}\phi^{X}|_{\pi(z)}.$

We have

$\lambda^{\prime}\coloneqq\chi^{\ast}\lambda=\lambda-\mathrm{d}(\phi^{X}\circ\pi).$

The 1-form $\lambda^{\prime}$ on $Z$ is a primitive of the symplectic
structure $\mathrm{d}\lambda$ and vanishes exactly at the points on $X$. By
Remark 2.3, $\lambda^{\prime}$ corresponds to a cotangent bundle structure on
$Z$ over $X$ with the same Lagrangian fibres as the original structure. Let
$\pi^{\prime}\colon Z\to X$ denote the cotangent bundle projection. In the
following we may localise the problem by intersecting $L$ with open
neighbourhoods of $\pi(z)$ in $L$ and shrink $Z=T^{\ast}L$ and all embedded
submanifolds accordingly repeatedly without mentioning. A generating function
for $(X,\Lambda,z)$ with respect to the structure $\lambda^{\prime}$ can be
obtained as follows: let $S^{\prime}\colon\Lambda\to\mathbb{R}$ be a primitive
of the closed 1-form $\iota^{\ast}_{\Lambda}\lambda^{\prime}$ around $z$ with
$S^{\prime}(z)=0$, where $\iota_{\Lambda}\colon\Lambda\hookrightarrow Z$ is
the inclusion. A generating function for $(X,\Lambda,z)$ with respect to
$\lambda^{\prime}$ is given as
$\phi^{\prime}=S^{\prime}\circ(\pi^{\prime}\circ\iota_{\Lambda})^{-1}\colon
X\to\mathbb{R}$. An expression of $\phi^{\prime}$ in local coordinates on $X$
centred at $z$ is, therefore, right equivalent to an expression of
$S^{\prime}$ in local coordinates on $\Lambda$ centred at $z$.

To show that a coordinate expression of $\phi$ centred at $\pi(z)\in L$ and a
coordinate expression of $\phi^{\prime}$ centred at $z$ are right equivalent,
it suffices to verify that the pullback of $\mathrm{d}S^{\prime}$ to $L$ via
$\mathrm{d}\phi^{\Lambda}\colon L\to\Lambda$ coincides with $\mathrm{d}\phi$.
Here and in the following calculation we neglect to differentiate between
$\mathrm{d}\phi^{\Lambda}\colon L\to\Lambda$ and
$\mathrm{d}\phi^{\Lambda}\colon L\to Z$. Indeed,

$\displaystyle(\mathrm{d}\phi^{\Lambda})^{\ast}(\mathrm{d}S^{\prime})$
$\displaystyle=(\mathrm{d}\phi^{\Lambda})^{\ast}(\iota^{\ast}_{\Lambda}\lambda^{\prime})=(\mathrm{d}\phi^{\Lambda})^{\ast}(\chi^{\ast}\lambda)=(\chi\circ\mathrm{d}\phi^{\Lambda})^{\ast}\lambda$
$\displaystyle=(\mathrm{d}\phi^{\Lambda}-\mathrm{d}\phi^{X}|_{\pi\circ\mathrm{d}\phi^{\Lambda}})^{\ast}\lambda=(\mathrm{d}\phi^{\Lambda}-\mathrm{d}\phi^{X})^{\ast}\lambda=\mathrm{d}\phi^{\ast}\lambda=\mathrm{d}\phi.$

Since generating functions are well-defined up to stably right equivalence by
Corollary 2.9, this shows that $\phi$ expressed in local coordinates centred
at $\pi(z)$ is stably right equivalent to any generating function of
$(X,\Lambda,z)$. ∎

##  3\. Symmetries

If a fully reduced generating function $f$ of a Lagrangian contact problem
$(X,\Lambda,z)$ is invariant under a diffeomorphism $\overline{h}$ defined on
a neighbourhood of $0$, i.e. $f\circ\overline{h}=f$, then any other fully
reduced generating function is of the form $f\circ r$ for a right equivalence
$r$ (Corollary 2.10) and is, therefore, invariant under
$r^{-1}\circ\overline{h}\circ r$. Invariance of fully reduced generating
functions is, thus, a well-defined concept. The fully reduced generating
function $f$ arises as the restriction of a generating function

$\phi(x)=f(\overline{x})+\underline{x}^{\top}B\underline{x}$

for $(X,\Lambda,z)$ to some appropriate submanifold
$\overline{X}=\\{(\overline{x},\underline{x})\,|\,\underline{x}=0\\}\subset X$
(obtained by the parametric Morse lemma (Lemma 2.6), for instance). The map
$h(\overline{x},\underline{x})=(\overline{h}(\overline{x}),\underline{x})$
extends $\overline{h}$ to a diffeomorphism on a neighbourhood of $z$ in $X$.
We have $\phi\circ h=\phi$.

Let us understand the geometric meaning of invariance of generating functions
of Lagrangian contact problems.

######  Lemma 3.1.

Consider the cotangent bundle $\pi\colon T^{\ast}X\to X$, let $\phi\colon
X\to\mathbb{R}$ be a smooth function, and denote $\Lambda=\mathrm{d}\phi(X)$.
If $\phi\circ h=\phi$ for some $h\colon X\to X$ then the cotangent lifted
action $H\colon T^{\ast}X\to T^{\ast}X$ of $h$ fulfills $H(X)=X$ and
$H(\Lambda)=\Lambda$.

###### Proof.

As $H$ is a cotangent lifted map $H(X)=X$. Let
$\mathrm{d}\phi|_{x}\in\Lambda$. We have

$H(\mathrm{d}\phi|_{x})=\mathrm{d}\phi|_{x}\circ\mathrm{d}h|_{h^{-1}(x)}=\mathrm{d}(\phi\circ
h)|_{h^{-1}(x)}=\mathrm{d}\phi|_{h^{-1}(x)}.$

Clearly $\mathrm{d}\phi|_{h^{-1}(x)}\in\Lambda$. Since $H$ is a diffeomorphism
it follows that $H(\Lambda)=\Lambda$. ∎

In the following, it is convenient to view a generating functions of a
Lagrangian contact problem $(X,\Lambda,z)$ in $Z$ as functions
$S\colon\Lambda\to\mathbb{R}$ obtained as a primitive of the closed 1-form
$\iota_{\Lambda}^{\ast}\lambda$, where $\iota_{\Lambda}\colon\Lambda\to Z$ is
the inclusion of $\Lambda$ and the 1-form $\lambda$ on $Z$ defines a cotangent
bundle structure over $X$ (Remark 2.3). Provided that $\Lambda$ is graphical
for the cotangent bundle structure, the function $S$ can be related to a
generating function $\phi$ defined locally on $X$ by
$\phi=S\circ\pi|_{\Lambda}^{-1}$, where $\pi$ denotes the cotangent bundle
projection and $\pi|_{\Lambda}\colon\Lambda\cap V\to\pi(\Lambda\cap V)$ the
restriction of $\pi$ for an open neighbourhood $V\subset\Lambda$ containing
$z$.

######  Lemma 3.2.

Let $Z$ be a symplectic manifold with cotangent bundle structure over a simply
connected Lagrangian submanifold $X$ defined by a 1-form $\lambda$. Let
$\iota_{\Lambda}\colon\Lambda\hookrightarrow Z$ be another embedded, simply
connected Lagrangian submanifold. Consider a symplectmorphism $F\colon Z\to Z$
with $F(X)=X$ and $F(\Lambda)=\Lambda$. The 1-form
$\lambda^{\prime}=F^{\ast}\lambda$ defines another cotangent bundle structure
over $X$. If $\mathrm{d}S=\iota^{\ast}_{\Lambda}\lambda$ and
$\mathrm{d}S^{\prime}=\iota^{\ast}_{\Lambda}\lambda^{\prime}$ for
$S,S^{\prime}\colon\Lambda\to\mathbb{R}$ then

$S^{\prime}=S\circ F|_{\Lambda}+\text{const.},$

where $F|_{\Lambda}\colon\Lambda\to\Lambda$ is the restriction of $F$ to
$\Lambda$.

###### Proof.

Indeed, the 1-form $\lambda^{\prime}=F^{\ast}\lambda$ defines a cotangent
bundle structure over $X$ because $F$ is symplectic and $\lambda^{\prime}$
vanishes exactly at points on $X$. Moreover, we have

$\mathrm{d}S^{\prime}=\iota^{\ast}_{\Lambda}\lambda^{\prime}=\iota^{\ast}_{\Lambda}F^{\ast}\lambda=(F\circ\iota_{\Lambda})^{\ast}\lambda=(\iota_{\Lambda}\circ
F|_{\Lambda})^{\ast}\lambda=F|_{\Lambda}^{\ast}\iota_{\Lambda}^{\ast}\lambda=F|_{\Lambda}^{\ast}\mathrm{d}S$

such that $S^{\prime}$ and $S$ relate as claimed. ∎

######  Lemma 3.3.

If in the setting of Lemma 3.2 we have

$S=S\circ F|_{\Lambda},$

then $F$ coincides on $\Lambda$ with the cotangent lifted action of

$h\colon X\to X,\quad h\coloneqq\pi\circ F^{-1}\circ\pi|_{\Lambda}^{-1}.$

provided that $\Lambda$ is graphical in the cotangent bundle structure
$\lambda$.

###### Proof.

Notice that $\pi|_{\Lambda}^{-1}=\mathrm{d}\phi$ for a map $\phi\colon
X\to\mathbb{R}$ with $\phi(X)=\Lambda$. Denote the cotangent lifted action of
$h$ by $H$. For $\mathrm{d}\phi|_{x}\in\Lambda$ we have

(3.1) $\pi(H(\mathrm{d}\phi|_{x}))=h^{-1}(x)=(\pi\circ
F\circ\mathrm{d}\phi)(x)=\pi(F(\mathrm{d}\phi|_{x})),$

i.e. the basepoints of $H(\mathrm{d}\phi|_{x})$ and $F(\mathrm{d}\phi|_{x})$
coincide. The function $\phi$ is invariant under $h$ such that
$H(\mathrm{d}\phi|_{x})\in\Lambda$ by Lemma 3.1. Since
$F(\mathrm{d}\phi|_{x})\in\Lambda$ and since the restriction of $\pi$ to
$\pi|_{\Lambda}=\pi\circ\iota_{\Lambda}\colon\Lambda\to X$ is a diffeomorphism
we can conclude from (3.1) that

$H(\mathrm{d}\phi|_{x})=F(\mathrm{d}\phi|_{x}).$

Therefore, $H\circ\iota_{\Lambda}=F\circ\iota_{\Lambda}$. ∎

We conclude from Lemmas 3.1, 3.2 and 3.3 the following theorem.

######  Theorem 3.4.

Consider a Lagrangian contact problem $(X,\Lambda,z)$ in $Z$, where
$X,\Lambda$ are simply connected. Fix a cotangent bundle structure over $X$
near $z$ defined by the 1-form $\lambda$ such that $\Lambda$ is graphical. We
denote the bundle projection by $\pi$. Consider
$\mathrm{d}S=\iota_{\Lambda}^{\ast}\lambda$. We have $S\circ F_{\Lambda}=S$
for a diffeomorphism $F_{\Lambda}\colon\Lambda\to\Lambda$ if and only if
$F_{\Lambda}$ coincides on $\Lambda$ with the cotangent lifted map $H$ of
$h=\pi\circ F_{\Lambda}^{-1}\circ\pi|_{\Lambda}^{-1}$.

In other words, an invariance of a Lagrangian contact problem $(X,\Lambda,z)$
under a symplectic map $F$ yields an invariance of the generating function
related to a cotangent bundle structure if and only if $F$ coincides on
$\Lambda$ with the cotangent lifted action for the invariance.

######  Remark 3.1.

The symmetries characterised in Theorem 3.4 yield symmetries of the reduced
generating function if $F_{\Lambda}$ leaves a submanifold of
$\mathrm{d}\phi(\overline{X})\subset\Lambda$ invariant, where $\overline{X}$
is such that $\phi$ is stably right equivalent to $\phi|_{\overline{X}}$.

##  4\. Parameter dependent Lagrangian contact problems

Let us extend our analysis of Lagrangian contact problems to parameter
dependent problems. This will help us to deepen the relation to catastrophe
theory. A parameter dependent version of Lemma 2.7 holds true.

######  Lemma 4.1.

On $\mathbb{R}^{n}=\overline{X}\oplus\underline{X}$ consider coordinates
$x=(\overline{x},\underline{x})$ with $\overline{x}=(x^{1},\ldots,x^{k})$ and
$\underline{x}=(x^{k+1},\ldots,x^{n})$, a smooth family of nondegenerate
symmetric matrices $Q_{\mu}\in\mathbb{R}^{(n-k)\times(n-k)}$, a smooth family
of maps $g_{\mu}$ defined on an open neighbourhood of $0\in\mathbb{R}^{k}$
such that the 2-jet of $g_{0}$ vanishes at 0. Here $\mu\in I$ is the family
parameter and $I$ is an open neighbourhood of $0\in\mathbb{R}^{l}$. Moreover,
consider a smooth family of matrix valued functions
$\mathcal{H}_{\mu}\colon\mathbb{R}^{n}\to\mathrm{Sym}(n)$ with

$\mathcal{H}_{\mu}(x)=(h_{\mu}^{ij}(x))_{i,j=1,\ldots,n}=\begin{pmatrix}H^{11}_{\mu}(x)&H_{\mu}^{12}(x)\\\
{H_{\mu}^{12}}^{\top}(x)&0\end{pmatrix}\in\mathrm{Sym}(n)\subset\mathbb{R}^{n\times
n}.$

for $H^{11}_{\mu}(x)\in\mathrm{Sym}(k)$ and
$H_{\mu}^{12}(x)\in\mathbb{R}^{k\times(n-k)}$. For $t\in\mathbb{R}$ let

$\displaystyle\psi(\mu,x)$
$\displaystyle=g_{\mu}(\overline{x})+\underline{x}^{\top}Q_{\mu}\underline{x}$
$\displaystyle\psi_{t}(\mu,x)$
$\displaystyle=g_{\mu}(\overline{x})+\underline{x}^{\top}Q_{\mu}\underline{x}+t\nabla\psi(\mu,x)^{\top}\mathcal{H}_{\mu}(x)\nabla\psi(\mu,x).$

Then $\psi_{t}$ is right equivalent to $\phi$ around $(\mu,x)=(0,0)$. The
right equivalence is fibred, i.e. of the form
$(\mu,x)\mapsto(\mu,r_{\mu}(x))$, and fixes $(\mu,x)=(0,0)$.

###### Proof.

The proof is almost analogous to the proof of Lemma 2.7 with parameter-
dependent data. Rather than repeating the calculations and arguments, we point
out the two minor modifications that need to be made for the parameter
dependent setting. (1) The 2-jet of $g_{\mu}$ does not necessarily vanish at
$\overline{x}=0$ unless $\mu=0$. However, this is not an issue: in (2.13) the
invertibility of $B_{\mu}(t,0)$ follows from the invertibility of $B_{0}(t,0)$
for all $\mu$ close to 0 and $t\in[0,1]$, which is sufficient for the
argument. (2) The other difference is that $\omega_{\mu}(0,t)$ from (2.14)
does not necessarily vanish if $\mu\not=0$ and, therefore, the values
$f_{\mu,t}(0)$ are only guaranteed to vanish if $\mu=0$. However, this is
sufficient as the sought right equivalence needs to fix $x=0$ only at $\mu=0$.
∎

######  Definition 4.1 (smooth Lagrangian family).

Let $I\subset\mathbb{R}^{l}$ be an open neighbourhood of the origin. A family
$(\Lambda_{\mu})_{\mu\in I}$ of Lagrangian submanifolds of a symplectic
manifold $Z$ is smooth at $\mu=0$ around $z\in\Lambda_{0}$ if there exists an
open neighbourhood $\tilde{Z}$ of $z$, a cotangent bundle structure
$\pi\colon\tilde{Z}\to\Lambda_{0}\cap\tilde{Z}$, an open neighbourhood
$\tilde{I}\subset I$ of 0 and a smooth family of maps
$(\phi_{\mu})_{\tilde{I}}$ such that $\Lambda_{\mu}\cap\tilde{Z}$ is the image
of the section $\mathrm{d}\phi_{\mu}\colon\Lambda_{0}\to\tilde{Z}$ for all
$\mu\in\tilde{I}$.

Definition 4.1 is independent of the cotangent bundle structure
$\pi\colon\tilde{Z}\to\Lambda_{0}\cap\tilde{Z}$ because two different
structures relate by a smooth transition (Lemma 2.2).

######  Definition 4.2 (parameter-dependent Lagrangian contact problem).

Let $(X_{\mu})_{\mu\in I}$ and $(\Lambda_{\mu})_{\mu\in I}$ be two smooth
Lagrangian families in a symplectic manifold $Z$ such that
$X_{0}\cap\Lambda_{0}$ intersects in an isolated point $z_{0}$ and such that
the set $X_{\mu}\cap\Lambda_{\mu}$ is discrete for all $\mu\in I$. Then
$((X_{\mu})_{\mu\in I},(\Lambda_{\mu})_{\mu\in I},z_{0})$ is called a
(parameter-dependent) Lagrangian contact problem in $Z$.

######  Definition 4.3 (constant unfolding).

Let $\mathcal{L}=((X_{\mu})_{\mu\in I},(\Lambda_{\mu})_{\mu\in I},z_{0})$ be a
Lagrangian contact problem. Let $\tilde{I}\subset\mathbb{R}^{\tilde{l}}$ be an
open neighbourhood of 0. The Lagrangian contact problem
$((X_{\mu})_{(\mu,\tilde{\mu})\in
I\times\tilde{I}},(\Lambda_{\mu})_{(\mu,\tilde{\mu})\in
I\times\tilde{I}},z_{0})$ is called a constant unfolding of $\mathcal{L}$.
Similarly, if $\mathcal{F}=(\phi_{\mu})_{\mu\in I}$ is a smooth family of
scalar-valued maps then $(\phi_{\mu})_{(\mu,\tilde{\mu})\in I\times\tilde{I}}$
is called a constant unfolding of $\mathcal{F}$.

######  Definition 4.4 (Morse-reduced form).

Consider an open neighbourhood $I$ of $0\in\mathbb{R}^{l}$ and a family of
scalar-valued maps $(\phi_{\mu})_{\mu\in I}$ defined around $z_{0}\in X$ with
$\phi_{0}(z_{0})=0$ and $\mathrm{d}\phi_{0}|_{z_{0}}=0$, where $X$ is an
$n$-dimensional manifold. Consider coordinates
$(\overline{x},\underline{x})=((x^{1},\ldots,x^{k}),(x^{k+1},\ldots,x^{n}))$
centred at 0 such that $\phi_{\mu}$ is of the form

$\phi_{\mu}(x)=f_{\mu}(\overline{x})+\underline{x}^{\top}B\underline{x}$

for a symmetric, nondegenerate matrix $B$ and a smooth family of maps
$(f_{\mu})_{\mu\in I}$ such that $\nabla_{\overline{x}}f_{0}(0)=0$ and
$\mathrm{Hess}\,f_{0}(0)=0$. Then $(f_{\mu})_{\mu\in I}$ is a Morse-reduced
form of $(\phi_{\mu})_{\mu\in I}$.

######  Remark 4.1.

A potential parameter-dependence of the coordinates
$x=(\overline{x},\underline{x})$ is suppressed in our notation.

######  Lemma 4.2 (existence and uniqueness of Morse-reduced forms).

A family $(\phi_{\mu})_{\mu\in I}$ as in Definition 4.4 has a Morse-reduced
form $(f_{\mu})_{\mu\in I}$ and $(f_{\mu})_{\mu\in I}$ is locally around
$(\mu,x)=(0,0)$ determined up to a right-action with a diffeomorphism of the
form $K(\mu,\overline{x})=(\mu,r_{\mu}(\overline{x}))$ with $K(0,0)=0$ and
addition of a term $\chi(\mu)$, where $\chi$ is smooth and $\chi(0)=0$.

###### Proof.

The lemma follows from the splitting theorem as formulated in [19]. Compared
with Lemma 2.6 it incorporates uniqueness properties. Let us indicate a way to
obtain the claimed existence result form Lemma 2.6. Let the dimension of the
kernel of the Hessian matrix of $\phi_{0}$ at $z_{0}$ be $k$. We find an
$n-k$-dimensional submanifold $\underline{X}\subset X$ containing $z_{0}$ such
that the Hessian matrix of $\phi_{0}|_{\underline{X}}$ at $z_{0}$ is
nondegenerate. Consider a submanifold $\overline{X}$ containing $z_{0}$ that
is transversal to $\underline{X}$. We apply the parametric444In this context
the parameters are $(\mu,\overline{x})$ Morse Lemma (Lemma 2.6) to

$(\mu,x)\mapsto\phi_{\mu}(x)-\underbrace{\langle\nabla_{\mu}\phi_{\mu}(0)|_{\mu=0},\mu\rangle}_{=:\chi(\mu)}-\phi_{0}(0)$

with respect to the splitting $(I\oplus\overline{X})\oplus\underline{X}$.
Notice that the 1-jet of the above map with respect to the coordinates
$((\mu,\overline{x}),\underline{x})$ vanishes. A fibred change of coordinates
yields coordinates $((\mu,\overline{x}),\underline{x})$ with
$x=(\overline{x},\underline{x})=((x^{1},\ldots,x^{k}),(x^{k+1},\ldots,x^{n}))$
on $X$ centred at $z_{0}$ such that $(\overline{x},0)$ are coordinates on
$\overline{X}$ and $(0,\underline{x})$ are coordinates on $\underline{X}$ and

$\phi_{\mu}(x)=\tilde{f}_{\mu}(\overline{x})+\phi_{0}(0)+\chi(\mu)+\underline{x}^{\top}B\underline{x}=:f_{\mu}(\overline{x})+\underline{x}^{\top}B\underline{x}$

The function $\overline{x}\mapsto f_{0}(\overline{x})-\phi_{0}(0)$ has a
vanishing 2-jet at $\overline{x}=0$ and
$B=\mathrm{Hess}\,\phi_{0}|_{\underline{X}}$ is invertible. ∎

######  Proposition 4.3.

Let $(X,(\Lambda_{\mu})_{\mu\in I},z_{0})$ be a parameter dependent Lagrangian
contact problem in $Z$. Let $I\subset\mathbb{R}^{k}$ denote an open
neighbourhood of $0$. Consider smooth families of cotangent bundle structures
such that $\Lambda_{\mu}$ is given as the image of the section
$\mathrm{d}\phi_{\mu}^{\alpha}$ and $\mathrm{d}\phi_{\mu}^{\beta}$ locally
around $z_{0}\in Z$ with $\phi_{0}^{\alpha}(z_{0})=0=\phi_{0}^{\beta}(z_{0})$,
respectively. Then $(\phi^{\alpha}_{\mu})_{\mu}$ and
$(\phi^{\beta}_{\mu})_{\mu}$ admit the same Morse-reduced forms up to an
addition of a smooth map $\chi(\mu)$ with $\chi(0)=0$.

###### Proof.

The following proof is a parameter-dependent version of the proof of
Proposition 2.8 using Lemma 4.1 instead of Lemma 2.7.

Let the dimension of the kernel of the Hessian matrix of $\phi_{0}^{\alpha}$
at $z_{0}$ be $k$ and the dimension of $X$ $n$. By Lemma 4.2 there exist a
local coordinate system
$x=(\overline{x},\underline{x})=((x^{1},\ldots,x^{k}),(x^{k+1},\ldots,x^{n}))$
on $X$ centred at $z_{0}$ such that

$\phi^{\alpha}_{\mu}(x)=f_{\mu}(\overline{x})+\underline{x}^{\top}B\underline{x},$

the function $\overline{x}\mapsto f_{0}(\overline{x})$ has a vanishing 2-jet
at $\overline{x}=0$, and $B=\mathrm{Hess}\,\phi^{\alpha}_{0}|_{\underline{X}}$
is invertible. A fibre-wise application of Lemma 2.2 yields

(4.1) $\phi_{\mu}^{\beta}\circ
k^{\mu}_{\alpha\beta}=\phi_{\mu}^{\alpha}+H_{\mu}(x,\nabla\phi_{\mu}^{\alpha})+\chi(\mu),$

for a smooth map $\chi$ with $\chi(0)=0$ and

$H_{\mu}(x,\xi)=\sum_{i,j=1}^{n}h^{\mu}_{ij}(x,\xi)\xi_{i}\xi_{j}=\xi^{\top}\begin{pmatrix}H_{\mu}^{11}(x,\xi)&H_{\mu}^{12}(x,\xi)\\\
H_{\mu}^{12}(x,\xi)^{\top}&H_{\mu}^{22}(x,\xi)\end{pmatrix}\xi$

with $H_{\mu}^{11}(x,\xi)\in\mathrm{Sym}(k)$,
$H_{\mu}^{12}(x,\xi)\in\mathbb{R}^{k\times(n-k)}$,
$H_{\mu}^{22}(x,\xi)\in\mathrm{Sym}(n-k)$, and with
$k^{\mu}_{\alpha\beta}=\pi^{\beta}\circ(\pi^{\alpha}|_{\Lambda_{\mu}})^{-1}$.
For $i,j\in\\{1,2\\}$ define

$\mathcal{H}_{\mu}(x)=\mathcal{H}_{\mu}(\overline{x},\underline{x})=H_{\mu}(x,\nabla\phi_{\mu}^{\alpha}(x)),\quad\mathcal{H}_{\mu}^{ij}(x)=\mathcal{H}_{\mu}^{ij}(\overline{x},\underline{x})=H_{\mu}^{ij}(x,\nabla\phi_{\mu}^{\alpha}(x)).$

Define

$B^{\prime}_{\mu}(x)=B+4B\mathcal{H}_{\mu}^{22}(x)B.$

We calculate

(4.2)
$\begin{split}\phi_{\mu}^{\beta}(k^{\mu}_{\alpha\beta}(\overline{x},\underline{x}))&\stackrel{{\scriptstyle\eqref{eq:ApplyCompCotParam}}}{{=}}\phi_{\mu}^{\alpha}(\overline{x},\underline{x})+\mathcal{H}_{\mu}(x)+\chi(\mu)\\\
&\;\;\;=f_{\mu}(\overline{x})+\underline{x}^{\top}B_{\mu}^{\prime}(x)\underline{x}+\nabla_{\overline{x}}f_{\mu}(\overline{x})^{\top}\mathcal{H}_{\mu}^{11}(x)\nabla_{\overline{x}}f_{\mu}(\overline{x})\\\
&\;\;\;\phantom{=}+4\underline{x}^{\top}B\mathcal{H}_{\mu}^{12}(x)^{\top}\nabla_{\overline{x}}f_{\mu}(\overline{x})+\chi(\mu).\end{split}$

We calculate the Hessian matrix of $\phi_{0}^{\beta}\circ k^{0}_{\alpha\beta}$
at $(\overline{x},\underline{x})=(0,0)$ using (4.2) and obtain

$\mathrm{Hess}\,(\phi_{0}^{\beta}\circ
k_{\alpha\beta})(0,0)=\begin{pmatrix}0&0\\\
0&2B_{0}^{\prime}(0)\end{pmatrix}.$

Since $\Lambda_{0}$ is graphical in both cotangent bundle structures,
$B^{\prime}_{0}(0)$ must be invertible by a dimension argument. For $(\mu,x)$
in a sufficiently small neighbourhood of $(0,0)$ the matrices
$B_{\mu}^{\prime}(x)$ are invertible and have constant signature. By
Sylvester’s law of inertia, there exists a smooth family of invertible
matrices $A_{\mu}(x)$ such that

(4.3) $A_{\mu}^{-T}(x)B^{\prime}_{\mu}(x)A_{\mu}^{-1}(x)=B^{\prime}_{\mu}(0)$

for all $(\mu,x)$ near $(0,0)$. Consider

$r_{\mu}(\overline{x},\underline{x})=(\overline{x},A_{\mu}(x)\underline{x}).$

For all $\mu$ near 0 the map $r_{\mu}$ fixes $x=0$ and is a diffeomorphism on
a neighbourhood of $x=0$. Let us define

(4.4)
$\begin{split}\tilde{\mathcal{H}}_{\mu}^{11}(x)&=\mathcal{H}_{\mu}^{11}(r_{\mu}^{-1}(x))\\\
\tilde{\mathcal{H}}_{\mu}^{12}(x)&=\mathcal{H}_{\mu}^{12}(r_{\mu}^{-1}(x))BA^{-1}_{\mu}(r_{\mu}^{-1}(x))B_{\mu}^{\prime}(0)^{-1}.\end{split}$

By Lemma 4.1 the map

$\phi^{\alpha}_{1}\coloneqq
f_{\mu}(\overline{x})+\underline{x}^{\top}B^{\prime}_{\mu}(0)\underline{x}$

is right equivalent to

$\displaystyle\phi^{\alpha}_{2}(\mu,x)$ $\displaystyle\coloneqq
f_{\mu}(\overline{x})+\underline{x}^{\top}B^{\prime}_{\mu}(0)\underline{x}$
$\displaystyle\phantom{=}+\nabla_{\overline{x}}f_{\mu}(\overline{x})^{\top}\tilde{\mathcal{H}}_{\mu}^{11}(x)\nabla_{\overline{x}}f_{\mu}(\overline{x})$
$\displaystyle\phantom{=}+4\underline{x}^{\top}B^{\prime}_{\mu}(0)\tilde{\mathcal{H}}_{\mu}^{12}(x)^{\top}\nabla_{\overline{x}}f_{\mu}(\overline{x}).$

We have

$\displaystyle(\phi^{\alpha}_{2}\circ r_{\mu})(x)$
$\displaystyle=f_{\mu}(\overline{x})+\underline{x}^{\top}A_{\mu}(x)^{\top}B^{\prime}_{\mu}(0)A_{\mu}(x)\underline{x}$
$\displaystyle\phantom{=}+\nabla_{\overline{x}}f_{\mu}(\overline{x})^{\top}\tilde{\mathcal{H}}_{\mu}^{11}(r_{\mu}(x))\nabla_{\overline{x}}f_{\mu}(\overline{x})$
$\displaystyle\phantom{=}+4\underline{x}^{\top}A_{\mu}(x)^{\top}B_{\mu}^{\prime}(0)\tilde{\mathcal{H}}_{\mu}^{12}(r_{\mu}(x))^{\top}\nabla_{\overline{x}}f_{\mu}(\overline{x}).$

Comparing the above with (4.2) shows that the map $\phi^{\alpha}_{2}\circ
r_{\mu}$ coincides with $\phi_{\mu}^{\beta}\circ
k^{\mu}_{\alpha\beta}-\chi(\mu)$. This proves the claim. ∎

######  Definition 4.5 (stably equivalent as unfoldings).

Consider open neighbourhoods $I$ of $0\in\mathbb{R}^{l}$ and $I^{\prime}$ of
$0\in\mathbb{R}^{l^{\prime}}$ and two families of scalar-valued maps
$(\phi_{\mu})_{\mu\in I}$ and $(\phi^{\prime}_{\mu^{\prime}})_{\mu^{\prime}\in
I^{\prime}}$ defined around $0\in\mathbb{R}^{n}$ or
$0\in\mathbb{R}^{n^{\prime}}$, respectively. Assume that $\phi_{0}(0)=0$ and
$\phi^{\prime}_{0}(0)=0$. If

  * •

there exists an extension of the parameter spaces $I$ to
$I\times\mathbb{R}^{l+(L-l)}$ and $I^{\prime}$ to
$I^{\prime}\times\mathbb{R}^{l^{\prime}+(L-l^{\prime})}$ for some
$L\in\mathbb{N}$, where the additional parameters enter trivially in the map
families (constant unfolding),

  * •

there exist reparametrisations of both parameter spaces fixing $0\in
I\times\mathbb{R}^{l+(L-l)}$ or $0\in
I^{\prime}\times\mathbb{R}^{l^{\prime}+(L-l^{\prime})}$, respectively,

  * •

and there exists a smooth map $\chi$ in the extended parameter
$\hat{\mu}\in\hat{I}\subset\mathbb{R}^{L}$

such that the families
$(\phi_{\hat{\mu}}+\chi(\hat{\mu}))_{\hat{\mu}\in\hat{I}}$ and
$(\phi_{\hat{\mu}})_{\hat{\mu}\in\hat{I}}$ admit the same Morse-reduced forms
then $(\phi_{\mu})_{\mu\in I}$ and
$(\phi^{\prime}_{\mu^{\prime}})_{\mu^{\prime}\in I^{\prime}}$ are stably right
equivalent as unfoldings.

######  Remark 4.2.

Definition 4.5 corresponds to the equivalence relation reduction in definition
5.3 (p.124) of Wassermann’s dissertation [18]. The necessary notions are
developed in [18, §4, §5]. Here, however, we use right equivalence where the
reference uses right-left equivalence.

######  Proposition 4.4.

Let $((X_{\mu})_{\mu\in I},(\Lambda_{\mu})_{\mu\in I},z_{0})$ be a parameter-
dependent Lagrangian contact problem in $Z$. Consider two cotangent bundle
structures on $Z$ over $X_{0}$ locally around $z_{0}\in X_{0}\cap\Lambda_{0}$
such that for each $\mu\in I$ near 0 after shrinking all involved manifolds
around $z_{0}$, if necessary, the submanifold $\Lambda_{\mu}$ is the image of
the section $\mathrm{d}\phi_{\mu}$ and $X_{\mu}$ the image of the section
$\mathrm{d}\psi_{\mu}$ with respect to the first cotangent bundle structure
and $\Lambda_{\mu}$ is the image of the section
$\mathrm{d}\phi^{\prime}_{\mu}$ and $X_{\mu}$ is the image of the section
$\mathrm{d}\psi^{\prime}_{\mu}$ with respect to the second cotangent bundle
structure such that $\phi_{0},\psi_{0},\phi^{\prime}_{0},\psi^{\prime}_{0}$
vanish at $z_{0}$. Then the families
$(\rho_{\mu})_{\mu}=(\phi_{\mu}-\psi_{\mu})_{\mu}$ and
$(\rho^{\prime}_{\mu})_{\mu}=(\phi^{\prime}_{\mu}-\psi^{\prime}_{\mu})_{\mu}$
are stably right equivalent as unfoldings.

###### Proof.

We modify the first cotangent bundle structure using the fibre-preserving
symplectic diffeomorphism $\xi\mapsto\xi-\mathrm{d}\psi_{\mu}|_{\pi(\xi)}$ and
the second cotangent bundle structure by
$\xi\mapsto\xi-\mathrm{d}\psi^{\prime}_{\mu}|_{\pi^{\prime}(\xi)}$ fibre-wise.
In the updated structures all $X_{\mu}$ are zero-sections and $\Lambda_{\mu}$
is given as the image of the section
$\mathrm{d}(\phi_{\mu}-\psi_{\mu})=\mathrm{d}\rho_{\mu}$ with respect to the
first structure and as the image of the section
$\mathrm{d}(\phi^{\prime}_{\mu}-\psi^{\prime}_{\mu})=\mathrm{d}\rho^{\prime}_{\mu}$
with respect to the second structure. Now the claim follows by Proposition
4.3. ∎

The smooth family $(\rho_{\mu})_{\mu}$ of maps constructed in Proposition 4.4
is called a generating family for $((X_{\mu})_{\mu\in
I},(\Lambda_{\mu})_{\mu\in I},z_{0})$. More precisely, we can formulate the
following definition.

######  Definition 4.6 (generating family).

Consider a parameter-dependent Lagrangian contact problem $((X_{\mu})_{\mu\in
I},(\Lambda_{\mu})_{\mu\in I},z_{0})$ and a cotangent bundle structure over
$X_{0}$ such that for $\mu$ in a neighbourhood of 0 in $I$ the manifolds
$X_{\mu}$, $\Lambda_{\mu}$ are locally around $z_{0}$ given as the images of
$\mathrm{d}\psi_{\mu}$ and $\mathrm{d}\phi_{\mu}$, respectively, for smooth
function families $(\psi_{\mu})_{\mu\in I}$, $(\phi_{\mu})_{\mu\in I}$ with
$\psi_{0}(z_{0})=0=\phi_{0}(z_{0})$. The family
$\rho_{\mu}=\phi_{\mu}-\psi_{\mu}$ is called generating family for
$((X_{\mu})_{\mu\in I},(\Lambda_{\mu})_{\mu\in I},z_{0})$.

######  Definition 4.7 (contact equivalence of parameter-dependent Lagrangian
contact problems).

Let
$\hat{\mathcal{L}}=((X_{\hat{\mu}})_{{\hat{\mu}}\in\hat{I}},(\Lambda_{\hat{\mu}})_{{\hat{\mu}}\in\hat{I}},z_{0})$
and ${\mathcal{L}}^{\prime}=((X^{\prime}_{\mu^{\prime}})_{\mu^{\prime}\in
I^{\prime}},(\Lambda^{\prime}_{\mu^{\prime}})_{\mu^{\prime}\in
I^{\prime}},z^{\prime}_{0})$ be Lagrangian contact problems in $Z$ and
$Z^{\prime}$ where $\dim Z=\dim Z^{\prime}$. The families are called contact
equivalent if there exist a constant unfolding $((X_{\mu})_{\mu\in
I},(\Lambda_{\mu})_{\mu\in I},z_{0})$ of $\hat{\mathcal{L}}$ and a constant
unfolding $((X^{\prime}_{\mu})_{\mu\in I},(\Lambda^{\prime}_{\mu})_{\mu\in
I},z^{\prime}_{0})$ of ${\mathcal{L}}^{\prime}$ such that after shrinking $Z$
to an open neighbourhood of $z_{0}$ and $Z^{\prime}$ to an open neighbourhood
of $z^{\prime}_{0}$, there exists a smooth family of symplectomorphisms
$\Phi_{\mu}\colon Z\to Z^{\prime}$ such that for all $\mu$ in an open
neighbourhood of $0\in I$

$\Phi_{\mu}(X_{\mu})=X^{\prime}_{\theta(\mu)},\quad\Phi(\Lambda_{\mu})=\Lambda^{\prime}_{\theta(\mu)},\quad\Phi_{0}(z_{0})=z_{0}^{\prime},$

where $\theta$ is a diffeomorphism defined around $0\in I$ fixing $\mu=0$.

We can conclude the section with the following theorem which extends Theorem
2.12 to the parameter-dependent case.

######  Theorem 4.5.

Two parameter-dependent Lagrangian contact problems in a symplectic manifold
are contact equivalent if and only if their generating families are stably
right equivalent as unfoldings.

###### Proof.

Let $((X_{\mu})_{\mu\in I},(\Lambda_{\mu})_{\mu\in I},z_{0})$ and
$((X^{\prime}_{\mu^{\prime}})_{\mu^{\prime}\in
I^{\prime}},(\Lambda^{\prime}_{\mu^{\prime}})_{\mu^{\prime}\in
I^{\prime}},z^{\prime}_{0})$ be parameter dependent Lagrangian contact
problems in $Z$. Since symplectic manifolds of the same dimension are locally
symplectomorphic, we can assume $z_{0}=z^{\prime}_{0}$. (Also see Lemma 2.11.)
Moreover, as the notion of contact equivalence of Lagrangian contact problems
as well as the notion of stably equivalence as unfoldings admits the
extensions of the parameter spaces via constant unfoldings, it is sufficient
to proof the assertion for $I=I^{\prime}$. In the following, we will shrink
the manifold $Z$ to neighbourhoods of $z_{0}$, and $I$ to a neighbourhood of
$0$, repeatedly, without mentioning. As seen in the proof of Proposition 4.4
we can reduce the problem to a problem with a constant family $X_{\mu}\equiv
X$.

For the forward direction, assume that $(X,(\Lambda_{\mu})_{\mu\in I},z_{0})$
and $(X,(\Lambda^{\prime}_{\mu})_{\mu\in I},z_{0})$ are contact equivalent.
There exists a family of symplectomorphisms $\Phi_{\mu}\colon Z\to Z$ with

$\Phi_{\mu}(X)=X,\quad\Phi_{\mu}(\Lambda_{\mu})=\Lambda^{\prime}_{\theta(\mu)},\quad\Phi_{0}(z_{0})=z_{0}.$

for a local diffeomorphism fixing $\mu=0$. After a reparametrisation of the
second family, we can assume $\theta=\mathrm{id}$. Now the proof proceeds
similarly to the first part of the proof of Theorem 2.12: consider a cotangent
bundle structure over $X$ such that $\Lambda_{\mu}$ is given as the image of
$\mathrm{d}\phi_{\mu}$ for a smooth family $(\phi_{\mu})_{\mu\in I}$ with
$\phi_{\mu}(0)=0$. Denote the cotangent bundle projection by $\pi$. Using
$\Phi_{\mu}$ we can construct another cotangent bundle structure over $X$ with
cotangent bundle projection $\pi_{\mu}^{\prime}=\pi\circ\Phi^{-1}_{\mu}$. The
map $\Phi_{\mu}\circ\mathrm{d}\phi_{\mu}$ is a section of $\pi_{\mu}^{\prime}$
and maps $X$ to $\Lambda^{\prime}_{\mu}$. Therefore, in the new structure, the
family $(\Lambda^{\prime}_{\mu})_{\mu\in I}$ can again be represented by the
family $(\phi_{\mu})_{\mu\in I}$.

For the reverse direction, assume that $(X,(\Lambda_{\mu})_{\mu\in I},z_{0})$
and $(X,(\Lambda^{\prime}_{\mu})_{\mu\in I},z_{0})$ have equivalent generating
families. Consider a cotangent bundle structure over $X$ such that
$\Lambda_{\mu}$, $\Lambda^{\prime}_{\mu}$ are graphical and given as

$\phi_{\mu}(X)=\Lambda_{\mu},\quad\phi^{\prime}_{\mu}(X)=\Lambda^{\prime}_{\mu}.$

By Lemma 4.2 there exist coordinates $(\overline{x},\underline{x})$ and
$(\overline{x^{\prime}},\underline{x^{\prime}})$ centred at $z_{0}$ such that

$\displaystyle\phi_{\mu}(\overline{x},\underline{x})$
$\displaystyle=f_{\mu}(\overline{x},\underline{x})+\underline{x}^{\top}B\underline{x}$
$\displaystyle\phi_{\mu}(\overline{x^{\prime}},\underline{x^{\prime}})$
$\displaystyle=f_{\mu}(\overline{x^{\prime}},\underline{x^{\prime}})+\underline{x^{\prime}}^{\top}B^{\prime}\underline{x^{\prime}},$

where the 2-jets of $f$ and $f^{\prime}$ vanish at $\mu=0$ and $B$ and
$B^{\prime}$ are nondegenerate, symmetric matrices. We have
$f_{\mu}(\overline{x})=f^{\prime}_{\theta(\mu)}(r_{\mu}(x))$ for a
diffeomorphism $\theta$ fixing $0$ and a fibred right equivalence $r$ such
that $r_{0}(0)=0$. After a reparametrisation of the second family, we can
assume $\theta=\mathrm{id}$. Now the proof proceeds analogously to the second
part of the proof of Theorem 2.12. Let us sketch the four steps to construct a
family of symplectic maps around $z_{0}$ identifying the two contact problems.

  * •

$\Lambda^{\prime}_{\mu}$ is mapped to a manifold that coincides with
$\mathrm{d}\phi_{\mu}^{(0)}(X)$ on a neighbourhood of $z_{0}$ with

$\phi_{\mu}^{(0)}(\overline{x},\underline{x})=f^{\prime}_{\mu}(\overline{x},\underline{x})+\underline{x}^{\top}B^{\prime}\underline{x}$

using the cotangent lift of the change of coordinates
$(\overline{x},\underline{x})\mapsto(\overline{x^{\prime}},\underline{x^{\prime}})$.

  * •

On an open neighbourhood of $z_{0}$ the submanifold
$\mathrm{d}\phi_{\mu}^{(0)}(X)$ is mapped to $\mathrm{d}\phi_{\mu}^{(1)}(X)$
with

$\phi_{\mu}^{(1)}(\overline{x},\underline{x})=f_{\mu}(\overline{x},\underline{x})+\underline{x}^{\top}B^{\prime}\underline{x}$

using the cotangent lift of
$(\overline{x},\underline{x})\mapsto(r_{\mu}(\overline{x}),\underline{x})$.

  * •

On an open neighbourhood of $z_{0}$ the submanifold
$\mathrm{d}\phi_{\mu}^{(1)}(X)$ is mapped to $\mathrm{d}\phi_{\mu}^{(2)}(X)$
with

$\phi_{\mu}^{(2)}(\overline{x},\underline{x})=f_{\mu}(\overline{x},\underline{x})+\underline{x}^{\top}BDB^{\prime}DB\underline{x}$

for a suitable choice of a symmetric, invertible matrix $D$ using the
cotangent lift of
$(\overline{x},\underline{x})\mapsto(\overline{x},DB\underline{x})$.

  * •

On an open neighbourhood of $z_{0}$ the submanifold
$\mathrm{d}\phi_{\mu}^{(2)}(X)$ is mapped to $\tilde{\Lambda}_{\mu}$ using the
cotangent lift of the fibred right equivalence defined by Lemma 4.1 for

$\mathcal{H}=\begin{pmatrix}0&0\\\
0&\frac{1}{4}(-B^{-1}+DB^{\prime}D)\end{pmatrix}\in\mathbb{R}^{n\times n}.$

Let the original cotangent bundle structure be defined by the 1-form
$\lambda$. $\Lambda_{\mu}$ is mapped to $\tilde{\Lambda}_{\mu}$ using the
symplectomorphism that identifies the cotangent bundle structure $\lambda$
with $\lambda+\mathrm{d}H$, where $H(x,\xi)=\xi^{\top}\mathcal{H}(x,\xi)\xi$
in Darboux coordinates with respect to the original structure.

The procedure constructs the required family of symplectomorphisms. ∎

##  5\. Concluding remarks and application to boundary value problems of
symplectic maps

###  5.1. Stably contact equivalence

It is now justified to extend the notion of contact equivalence of Lagrangian
contact problems to stably contact equivalence. This will allow us to compare
Lagrangian contact problems in symplectic manifolds of different dimensions.

######  Definition 5.1.

Two Lagrangian contact problems (in symplectic manifolds of possibly different
dimensions) are stably contact equivalent if their generating functions are
stably right equivalent. Moreover, two parameter-dependent Lagrangian contact
problems (in symplectic manifolds of possibly different dimensions) are stably
contact equivalent if their generating families are stably right equivalent as
unfoldings.

######  Proposition 5.1.

Two Lagrangian contact problems in symplectic manifolds of the same dimension
are stably contact equivalent if and only if they are contact equivalent.

###### Proof.

The statement follows by Theorems 2.12 and 4.5. ∎

The following two theorems announced in Section 1 now follow trivially from
Definition 5.1. Their geometric meaning is encoded in Theorems 2.12 and 4.5.

*

*

######  Remark 5.1.

The notion of versality and stability can be translated to the setting of
(parameter-dependent) Lagrangian contact problems such that classification
results from catastrophe theory apply (see [3, Part II], for instance).

###  5.2. Boundary value problems for symplectic maps

An application is the classification of singularities and bifurcations which
occur in boundary value problems for symplectic maps [16]: consider a smooth
family of symplectic maps $\phi_{\mu}\colon Z\to Z^{\prime}$ for $\mu\in I$,
where $I\subset\mathbb{R}^{l}$ is an open neighbourhood of the origin. Let us
denote the symplectic form of $Z$ by $\omega$ and the symplectic form of
$Z^{\prime}$ by $\omega^{\prime}$. Let $\mathrm{pr}\colon Z\times
Z^{\prime}\to Z$ and $\mathrm{pr}\colon Z\times Z^{\prime}\to Z^{\prime}$
denote projections to the first or second component of the product. Define the
symplectic form
$\Omega=\mathrm{pr}^{\ast}\omega-{\mathrm{pr}^{\prime}}^{\ast}\omega^{\prime}$
on $Z\times Z^{\prime}$. The graphs of $(\phi_{\mu})_{\mu}$ define a smooth
family $(\Lambda_{\mu})_{\mu}$ of Lagrangian submanifolds in $Z\times
Z^{\prime}$.

The Lagrangian contact problems $((\Lambda_{\mu})_{\mu},(X_{\mu})_{\mu},z)$
for a smooth family $(X_{\mu})_{\mu}$ of Lagrangian submanifolds of $Z\times
Z^{\prime}$ and a point $z\in Z\times Z^{\prime}$ can be interpreted as a
family of boundary value problems for the symplectic maps
$(\phi_{\mu})_{\mu}$, where $(X_{\mu})_{\mu}$ represents the boundary
conditions. The elements of the intersection $\Lambda_{\mu}\cap X_{\mu}$
correspond to solutions to the boundary value problem.

######  Example 5.1 (periodic boundary conditions).

Consider $Z^{\prime}=Z$, let $(\phi_{\mu})_{\mu\in I}$ be a family of
symplectic maps on $Z$, let $\Lambda_{\mu}$ denote the graph of $\phi_{\mu}$
viewed as a subset of $Z\times Z$ and let $X_{\mu}\equiv X$ be the diagonal
embedding of $Z$ into $Z\times Z$. The elements of the intersection
$\Lambda_{\mu}\cap X$ correspond to solutions to the boundary value problem
$\phi_{\mu}(z)=z$, $z\in Z$.

######  Example 5.2 (Dirichlet-type boundary conditions).

Consider a family of Hamiltonians $(H_{\mu})_{\mu\in I}$ on a cotangent bundle
space $T^{\ast}U$ with bundle projection $\pi\colon T^{\ast}U\to U$. Let
$q^{\ast},Q^{\ast}\in U$ and let $\phi_{\mu}$ denote the time-1-map
corresponding to the Hamiltonian system
$(H_{\mu},T^{\ast}U,-\mathrm{d}\lambda)$. Let $(q,p)$ denote local canonical
coordinates on $T^{\ast}U$ around a point of interest where $Z$ ($Z^{\prime}$)
is a coordinate patch such that $q^{\ast}\in\pi(Z)$
($Q^{\ast}\in\pi(Z^{\prime})$). The equation

(5.1) $\pi(\phi_{\mu}(q^{\ast},p))=Q^{\ast}$

is a boundary value problem for $\phi_{\mu}$. This kind of boundary value
problems can occur for first-order formulations of parameter-dependent second-
order systems of ordinary differential equations with Dirichlet boundary
conditions, for instance. Let $\Lambda_{\mu}$ denote the graph of
$\phi_{\mu}\colon Z\to Z^{\prime}$ viewed as a subset of $Z\times Z^{\prime}$
and let $X_{\mu}\equiv
X=\mathrm{pr}^{-1}(q^{\ast})\times{\mathrm{pr}^{\prime}}^{-1}(Q^{\ast})$. The
elements of the intersection $\Lambda_{\mu}\cap X$ correspond to solutions to
the boundary value problem (5.1).

Two parameter dependent boundary value problems for symplectic maps are
equivalent at a solution if and only if their corresponding parameter-
dependent Lagrangian contact problems in $Z\times Z^{\prime}$ are contact
equivalent. Therefore, Remark 5.1 applies to parameter-dependent boundary
value problems for symplectic maps and relates the problem to classical
catastrophe theory. For this also notice that small perturbations of the
corresponding catastrophe theory problems relate to perturbations of the
Lagrangian contact problems which can still be interpreted as boundary value
problems for symplectic maps. Indeed, we obtain a rigorous framework for the
work done in [16], i.e. the analysis of bifurcations in boundary value
problems for symplectic maps. In particular, Theorem 4.5 fills a gap in the
argumentation of proposition 2.1 in [16].

######  Example 5.3.

Singularities in conjugate loci can be viewed as singularities of exponential
maps. Interpreting exponential maps as Lagrangian maps [9], the singularities
can be classified via generating families up to stably $R^{+}$-equivalence [3,
p.304]. Alternatively, elements of conjugate loci can be interpreted as
singularities of boundary value problems for exponential maps. Using a
slightly different notion to identify the singularities, elements in the
conjugate loci correspond to map germs up to right equivalence with certain
symmetries [17].

##  6\. Acknowledgements

I thank Robert McLachlan for many useful discussions. This research was
supported by the Marsden Fund of the Royal Society Te Apārangi

## References

  * [1] Vladimir I. Arnold.  Catastrophe Theory.  Springer Berlin Heidelberg, 1992.  doi:10.1007/978-3-642-58124-3.
  * [2] Vladimir I. Arnold, V.S. Afrajmovich, Y.S. Il’yashenko, and L.P. Shil’nikov.  Dynamical Systems V: Bifurcation Theory and Catastrophe Theory.  Springer Berlin Heidelberg, Berlin, Heidelberg, 1994.  doi:10.1007/978-3-642-57884-7.
  * [3] Vladimir I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko.  Singularities of Differentiable Maps, Volume 1.  Birkhäuser Boston, 2012.  doi:10.1007/978-0-8176-8340-5.
  * [4] Sean Bates and Alan Weinstein.  Lectures on the Geometry of Quantization, volume 8.  American Mathematical Society, 1997.
  * [5] Theodor Bröcker.  Differentiable Germs and Catastrophes.  London Mathematical Society Lecture Note Series. Cambridge University Press, 1975.  doi:10.1017/CBO9781107325418.
  * [6] K. Fukaya.  Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I.  AMS/IP studies in advanced mathematics. American Mathematical Society, 2010.
  * [7] M Golubitsky and V Guillemin.  Contact equivalence for Lagrangian manifolds.  Advances in Mathematics, 15(3):375–387, 1975.  doi:10.1016/0001-8708(75)90143-7.
  * [8] Álex Haro.  The primitive function of an exact symplectomorphism.  Nonlinearity, 13(5):1483–1500, 6 2000.  doi:10.1088/0951-7715/13/5/304.
  * [9] S Janeczko and T Mostowski.  Relative generic singularities of the exponential map.  Compositio Mathematica, 96(3):345–370, 1995.  URL: http://www.numdam.org/item?id=CM_1995__96_3_345_0.
  * [10] P. Libermann and C. M. Marle.  Symplectic manifolds and Poisson manifolds, pages 89–184.  Springer Netherlands, Dordrecht, 1987.  doi:10.1007/978-94-009-3807-6_3.
  * [11] Héctor E Lomelí, James D Meiss, and Rafael Ramírez-Ros.  Canonical Melnikov theory for diffeomorphisms.  Nonlinearity, 21(3):485–508, 2 2008.  doi:10.1088/0951-7715/21/3/007.
  * [12] Jonathan Lorand and Alan Weinstein.  (Co)isotropic pairs in Poisson and presymplectic vector spaces.  SIGMA 11, 11, 2015.  arXiv:1503.00169, doi:10.3842/SIGMA.2015.072.
  * [13] Y.C. Lu.  Singularity theory and an introduction to catastrophe theory.  Universitext (1979). Springer-Verlag, 1976.  doi:10.1007/978-1-4612-9909-7.
  * [14] Jerrold E. Marsden and Tudor S. Ratiu.  Cotangent Bundles, pages 165–180.  Springer New York, New York, NY, 1999.  doi:10.1007/978-0-387-21792-5_6.
  * [15] John N. Mather.  Stability of ${C}^{\infty}$ mappings: III. finitely determined map-germs.  Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 35(1):127–156, 12 1968.  doi:10.1007/BF02698926.
  * [16] Robert I. McLachlan and Christian Offen.  Bifurcation of solutions to Hamiltonian boundary value problems.  Nonlinearity, 31(6):2895–2927, 2018.  arXiv:1710.09991, doi:10.1088/1361-6544/aab630.
  * [17] Robert I. McLachlan and Christian Offen.  Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci.  New Zealand Journal of Mathematics (NZJM), 48:83–99, 2018.  URL: http://nzjm.math.auckland.ac.nz/index.php/Hamiltonian_Boundary_Value_Problems%2C_Conformal_Symplectic_Symmetries%2C_and_Conjugate_Loci, arXiv:1804.07479.
  * [18] Gordon Wassermann.  Stability of Unfoldings.  Springer Berlin Heidelberg, 1974.  doi:10.1007/bfb0061658.
  * [19] Alan Weinstein.  Singularities of families of functions.  Differentialgeometrie im Grossen (Ber. Tagung, Math. Forschungsinst., Oberwolfach, 1969), 4:323–330, 1971.
  * [20] Alan Weinstein.  Lagrangian submanifolds and Hamiltonian systems.  Annals of Mathematics, 98(3):377–410, 1973.  doi:10.2307/1970911.