Symplectic Aspects

Nel documento Weak KAM Theorem in Lagrangian Dynamics Seventh Preliminary Version (pagine 68-72)

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

∀W ∈ T(x,p)(TM ), α(x,p)(W ) = p[T(x,p)π(W )].

To understand this differential 1-form α on the TM manifold, we take a chart θ : U → θ(U ) ⊂ Rn, we can consider the associated chart Tθ : TU → T(θ(U )) = θ(U ) × Rn∗. In these charts the canonical projection πis nothing but the projection θ(U )×Rn∗ → θ(U ) on the first factor. This gives us coordinates (x1, . . . , xn) on U , and therefore coordinates (x1, · · · , xn, p1, · · · , pn) on TU such that the projection π is nothing but (x1, · · · , xn, p1, · · · , pn) → (x1, · · · , xn). A vector W ∈ T(x,p)(TM ) has therefore coordinates (X1, · · · , Xn, P1, · · · , Pn), and the coordinates of T(x,p)π(W ) ∈ TxU are (X1, · · · , Xn). It follows that α(x,p)(W ) = Pn

i=1piXi. Since Xi is nothing but the differential form dxi evaluated on W , we get that

α|TU = Xn i=1

pidxi. 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 → TU, x 7→ ωx. The graph of ω is the set

Graph(ω) = {(x, ωx) | x ∈ U } ⊂ TM.

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

ωα = ω,

where ωα is the pull-back of the form of Liouville α on TM by the map ω : U → TU

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 ωx =Pn

i=1ωi(x)dxi. As a map ω : U → TU it is thus given in these coordinates by

(x1, . . . , xn) 7→ (x1, . . . , xn, ω1(x), . . . , ωn(x)).

But in these coordinates, it is clear that the pull-back ωα is Pn

i=1ωi(x)dxi = ω.

By taking, the exterior derivative Ω of α we can define a sym-plectic structure on TM . 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 transposetA is equal to −A (this reflects the antisymme-try). Therefore taking determinants, we get det(A) = det(tA) = det(−A) = −1dim Edet(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 −1dim 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 Ωx : TxV × TxV → R is a symplectic form on the vector space TxV .

As an exterior derivative is closed, to check that Ω = −dα is a symplectic form on TM , 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

dpi∧ dxi = Xn

i=1

dxi∧ dpi,

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

W = Xn

i=1

Xi

∂xi

+ Xn i=1

Pi

∂pi

,

Therefore Ω(x,p)(W, ·) =Pn

i=1Xidpi−Pn

i=1Pidxi, and Ω(x,p)(W, ·) = 0 implies Xi = Pi = 0/, for i = 1, . . . , n, by the independence of the family (dx1, . . . , dxn, dp1, . . . , dpn).

In the following, we will suppose that V is a manifold provided with a symplectic structure Ω. If H is a Cr function defined on the open subset O of V , By the fact that Ωx is an isomorphism, we can associate to H a vector field XH on O well defined by

x(XH(x), ·) = dxH(·).

Since Ω is then XH is as smooth as the derivative of H, therefore it is Cr−1. In particular, if H is C2, then the solutions of the vector field XH define a partial flow φHt : 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 XH on O, uniquely defined by

x(XH(x), ·) = dxH(·),

where Ω is the symplectic form on V . If, moreover, the function H is Cr, the vector XH field is Cr−1. Therefore for r ≥ 2, the partial flow φHt generated by H exists, it is called the Hamitonian flow of H, and H is called the Hamiltonian of the flow φHt . 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 φHt .

Proof. We must check that dxH(XH(x)) is 0, for all x ∈ O. But dxH(XH(x)) is Ωx(XH(x), XH(x)) vanishes because the bilinear form Ωx 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 → Eis 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 TxN of TxV is, for each x ∈ N , a Lagrangian subspace for the symplectic bilinear form Ωx.

By the lemma 2.5.7 above, if x ∈ N , any vector v ∈ TxV such that Ωx(v, v) = 0, for all v ∈ TxN , is necessarily in TxN .

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 TM if and only if ω is a closed form.

Proof. Indeed,by lemma 2.5.1, we have ω = ωα, and thus also dω = ωdα = −ωΩ. However, the form ω regarded as map of M → TM 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 φHt if and only if H is constant on N .

Proof. If H is constant on N , we have

x ∈ N, dxH|TxN = 0,

and thus Ωx(XH(x), v) = 0, for all v ∈ TxN , which implies that XH(x) ∈ TxN . 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 XH|N has solutions with values in N . By uniqueness of the solutions of XH in O (which holds because XH is C1 on O), the solutions with values in N must be orbits of φHt . We therefore conclude that N is invariant by φHt as soon as H is constant on N . Conversely, if N is invariant by φHt , the curves t 7→ φHt (x) have a speed Xh(x) for t = 0 which must be in TxN , therefore Xh(x) ∈ TxN and dxH|N = Ω(XH(x), ·) vanishes at every point of N , since N is a Lagrangian submanifold. By connectedness of N , the restriction H|N is constant.

Nel documento Weak KAM Theorem in Lagrangian Dynamics Seventh Preliminary Version (pagine 68-72)