Symplectic Aspects

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_xM , and p : T_xM → 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 ) ⊂ Rⁿ, 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 (x₁, · · · , x_n, p₁, · · · , p_n) on T^∗U such that the projection π^∗ is nothing but (x₁, · · · , x_n, p₁, · · · , p_n) → (x1, · · · , xn). A vector W ∈ T_(x,p)(T^∗M ) has therefore coordinates (X₁, · · · , X_n, P₁, · · · , P_n), and the coordinates of T_(x,p)π^∗(W ) ∈ T_xU are (X₁, · · · , X_n). It follows that α_(x,p)(W ) = Pn

i=1p_iX_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_idx_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 Rⁿ. Using the canonical coordinates on U ⊂ Rⁿ, 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

(x₁, . . . , x_n) 7→ (x₁, . . . , x_n, ω₁(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^tA is equal to −A (this reflects the antisymme-try). Therefore taking determinants, we get det(A) = det(^tA) = 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_xV × T_xV → R is a symplectic form on the vector space T_xV .

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 Rⁿ, 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_idp_i−Pn

i=1P_idx_i, and Ω_(x,p)(W, ·) = 0 implies X_i = P_i = 0/, for i = 1, . . . , n, by the independence of the family (dx₁, . . . , dx_n, dp₁, . . . , 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 C², 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 C¹ 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_xH(·),

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 C² 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_xH(X_H(x)) is 0, for all x ∈ O. But d_xH(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 C¹, and such that the subspace T_xN of T_xV 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_xV such that Ω_x(v, v^′) = 0, for all v^′ ∈ T_xN , is necessarily in T_xN .

Lemma 2.5.9. If ω is a C¹ 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 C¹ 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 C² function defined on the open subset O of the symplectic manifold V . If N ⊂ O is a connected C¹ 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_xH|T_xN = 0,

and thus Ω_x(X_H(x), v) = 0, for all v ∈ T_xN , which implies that X_H(x) ∈ T_xN . By the theorem of Cauchy-Peano [Bou76], if N is of class C¹ (or Cauchy-Lipschitz, if N is of C² 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 C¹ 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_xN and d_xH|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.