Let M be differentiable manifold of class C^{∞}, denote by π^{∗} :
T^{∗}M → M the canonical projection of the cotangent space T^{∗}M
onto M and denote by T π^{∗} : T T^{∗}M → T M the derivative of π^{∗}.
On T^{∗}M , we can define a canonical differential 1-form α called
the Liouville form. This form is thus for a given (x, p) ∈ T^{∗}M
of a linear map α_{(x,p)} : T_{(x,p)}(T^{∗}M ) → R. To define it, we just
remark that the two linear maps T_{(x,p)}π^{∗} : T_{(x,p)}(T^{∗}M ) → T_{x}M ,
and p : T_{x}M → R can be composed, and thus we can define α_{(x,p)}
by

∀W ∈ T_{(x,p)}(T^{∗}M ), α_{(x,p)}(W ) = p[T_{(x,p)}π^{∗}(W )].

To understand this differential 1-form α on the T^{∗}M manifold, we
take a chart θ : U → θ(U ) ⊂ R^{n}, we can consider the associated
chart T^{∗}θ : T^{∗}U → T^{∗}(θ(U )) = θ(U ) × R^{n∗}. In these charts the
canonical projection π^{∗}is nothing but the projection θ(U )×R^{n∗} →
θ(U ) on the first factor. This gives us coordinates (x1, . . . , xn) on
U , and therefore coordinates (x_{1}, · · · , x_{n}, p_{1}, · · · , p_{n}) on T^{∗}U such
that the projection π^{∗} is nothing but (x_{1}, · · · , x_{n}, p_{1}, · · · , p_{n}) →
(x1, · · · , xn). A vector W ∈ T_{(x,p)}(T^{∗}M ) has therefore coordinates
(X_{1}, · · · , X_{n}, P_{1}, · · · , P_{n}), and the coordinates of T_{(x,p)}π^{∗}(W ) ∈
T_{x}U are (X_{1}, · · · , X_{n}). It follows that α_{(x,p)}(W ) = Pn

i=1p_{i}X_{i}.
Since X_{i} is nothing but the differential form dx_{i} evaluated on W ,
we get that

α|T^{∗}U =
Xn
i=1

p_{i}dx_{i}.
We therefore conclude that α is of class C^{∞}.

Let ω be a differential 1-form defined on the open subset U of
M . This 1-form is a section ω : U → T^{∗}U, x 7→ ωx. The graph of
ω is the set

Graph(ω) = {(x, ω_{x}) | x ∈ U } ⊂ T^{∗}M.

Lemma 2.5.1. Let ω be a differential 1-form defined on the open subset U of M . We have

ω^{∗}α = ω,

where ω^{∗}α is the pull-back of the form of Liouville α on T^{∗}M by
the map ω : U → T^{∗}U

55
Proof. Using coordinates charts, it suffices to verify in the case
where U is an open subset of R^{n}. Using the canonical coordinates
on U ⊂ R^{n}, we can write ω_{x} =Pn

i=1ω_{i}(x)dx_{i}. As a map ω : U →
T^{∗}U it is thus given in these coordinates by

(x_{1}, . . . , x_{n}) 7→ (x_{1}, . . . , x_{n}, ω_{1}(x), . . . , ω_{n}(x)).

But in these coordinates, it is clear that the pull-back ω^{∗}α is
Pn

i=1ωi(x)dxi = ω.

By taking, the exterior derivative Ω of α we can define a
sym-plectic structure on T^{∗}M . To explain what that means, let us
recall that a symplectic form on a vector space E is an alternate
(or antisymmetric) bilinear form a : E × E → R which is
non-degenerate as a bilinear form, i.e. the map a^{♯} : E → E^{∗}, x 7→

a(x, ·) is an isomorphism.

Lemma 2.5.2. If the finite dimensional vector space E admits a symplectic form, then its dimension is even.

Proof. We choose a basis on E. If A is the matrix of a in this
base, its transpose^{t}A is equal to −A (this reflects the
antisymme-try). Therefore taking determinants, we get det(A) = det(^{t}A) =
det(−A) = −1^{dim E}det(A). The matrix of a^{♯} : E → E^{∗}, using
on E^{∗} the dual basis, is also A. the non degeneracy of a^{♯} gives
det(A) 6= 0. It follows that −1^{dim E} = 1, and therefore dim E is
even.

Definition 2.5.3 (Symplectic Structure). A symplectic structure
on a C^{∞} differentiable manifold V is a C^{∞} closed differential
2-form Ω on V such that, for each x ∈ V , the bilinear 2-form Ω_{x} :
T_{x}V × T_{x}V → R is a symplectic form on the vector space T_{x}V .

As an exterior derivative is closed, to check that Ω = −dα is a
symplectic form on T^{∗}M , it is enough to check the non-degeneracy
condition. We have to do it only in an open subset U of R^{n}, with
the notations introduced higher, we see that

Ω = −dα = − Xn

i=1

dp_{i}∧ dx_{i} =
Xn

i=1

dx_{i}∧ dp_{i},

it is, then, easy to check the non-degeneracy condition. In fact,
using coordinates we can write a W ∈ T_{(x,p)}(T^{∗}M ) as

W = Xn

i=1

X_{i} ∂

∂xi

+ Xn i=1

P_{i} ∂

∂pi

,

Therefore Ω_{(x,p)}(W, ·) =Pn

i=1X_{i}dp_{i}−Pn

i=1P_{i}dx_{i}, and Ω_{(x,p)}(W, ·) =
0 implies X_{i} = P_{i} = 0/, for i = 1, . . . , n, by the independence of
the family (dx_{1}, . . . , dx_{n}, dp_{1}, . . . , dp_{n}).

In the following, we will suppose that V is a manifold provided
with a symplectic structure Ω. If H is a C^{r} function defined on
the open subset O of V , By the fact that Ω^{♯}x is an isomorphism,
we can associate to H a vector field X_{H} on O well defined by

Ωx(XH(x), ·) = dxH(·).

Since Ω is then X_{H} is as smooth as the derivative of H, therefore it
is C^{r−1}. In particular, if H is C^{2}, then the solutions of the vector
field X_{H} define a partial flow φ^{H}_{t} : O → O.

Definition 2.5.4 (Hamiltonian Flow). Suppose H : O → R is
a C^{1} function, defined on the open subset O of the symplectic
manifold V . The Hamiltonian vector of H is the vector field X_{H}
on O, uniquely defined by

Ω_{x}(X_{H}(x), ·) = d_{x}H(·),

where Ω is the symplectic form on V . If, moreover, the function
H is C^{r}, the vector X_{H} field is C^{r−1}. Therefore for r ≥ 2, the
partial flow φ^{H}_{t} generated by H exists, it is called the Hamitonian
flow of H, and H is called the Hamiltonian of the flow φ^{H}_{t} .
Lemma 2.5.5. H : O → R is a C^{2} function, defined on the open
subset O of the symplectic manifold V , then H is constant on the
orbits of its Hamiltonian flow φ^{H}_{t} .

Proof. We must check that d_{x}H(X_{H}(x)) is 0, for all x ∈ O. But
d_{x}H(X_{H}(x)) is Ω_{x}(X_{H}(x), X_{H}(x)) vanishes because the bilinear
form Ω_{x} is alternate.

57 Definition 2.5.6 (Lagrangian Subspace). In a vector space E en-dowed with a symplectic bilinear form a : E×E → R, a Lagrangian subspace is a vector subspace F of E with dim E = 2 dim F and a is identically 0 on F × F .

Lemma 2.5.7. Let F be a subspace of the vector space E which Lagrangian for the symplectic form a on E. If x ∈ E is such that a(x, y) = 0, for all y ∈ F , then, the vector x is itself in F .

Proof. Define F^{⊥} = {x ∈ E | ∀y ∈ F, a(x, y) = 0}. We have F^{⊥} ⊃
F , since a is 0 on F ×F . Since a^{♯}: E → E^{∗}is an isomorphism, the
dimension of F^{⊥} is the same as that of its image a^{♯}(F^{⊥}) = {p ∈
E^{∗}| p|F = 0}. This last subspace can be identified with the dual
(E/F )^{∗} of the quotient of E by F . Therefore dim F^{⊥} = dim E −
dim F = 2 dim F − dim F = dim F . Therefore F^{⊥}= F .

Definition 2.5.8 (Lagrangian Submanifold). If V is a symplectic
manifold, a Lagrangian submanifold of V is a submanifold N of
class at least C^{1}, and such that the subspace T_{x}N of T_{x}V is, for
each x ∈ N , a Lagrangian subspace for the symplectic bilinear
form Ω_{x}.

By the lemma 2.5.7 above, if x ∈ N , any vector v ∈ T_{x}V such
that Ω_{x}(v, v^{′}) = 0, for all v^{′} ∈ T_{x}N , is necessarily in T_{x}N .

Lemma 2.5.9. If ω is a C^{1} differential 1-form on the manifold
M , then the graph Graph(ω) of ω is a Lagrangian submanifold of
T^{∗}M if and only if ω is a closed form.

Proof. Indeed,by lemma 2.5.1, we have ω = ω^{∗}α, and thus also
dω = ω^{∗}dα = −ω^{∗}Ω. However, the form ω regarded as map
of M → T^{∗}M induces a diffeomorphism of C^{1} class of M on
Graph(ω), consequently dω = 0 if and only if Ω| Graph(ω) = 0.

Theorem 2.5.10 (Hamilton-Jacobi). Let H : O → R be a C^{2}
function defined on the open subset O of the symplectic manifold
V . If N ⊂ O is a connected C^{1} Lagrangian submanifold of V , it is
locally invariant by the partial flow φ^{H}_{t} if and only if H is constant
on N .

Proof. If H is constant on N , we have

∀_{x} ∈ N, d_{x}H|T_{x}N = 0,

and thus Ω_{x}(X_{H}(x), v) = 0, for all v ∈ T_{x}N , which implies that
X_{H}(x) ∈ T_{x}N . By the theorem of Cauchy-Peano [Bou76], if N is
of class C^{1} (or Cauchy-Lipschitz, if N is of C^{2} class), the
restric-tion X_{H}|N has solutions with values in N . By uniqueness of the
solutions of X_{H} in O (which holds because X_{H} is C^{1} on O), the
solutions with values in N must be orbits of φ^{H}_{t} . We therefore
conclude that N is invariant by φ^{H}_{t} as soon as H is constant on
N . Conversely, if N is invariant by φ^{H}_{t} , the curves t 7→ φ^{H}_{t} (x)
have a speed X_{h}(x) for t = 0 which must be in TxN , therefore
X_{h}(x) ∈ T_{x}N and d_{x}H|N = Ω(X_{H}(x), ·) vanishes at every point
of N , since N is a Lagrangian submanifold. By connectedness of
N , the restriction H|N is constant.