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9). 6). 11 The set of Hamiltonian vector fields on P is a Lie subalgebra of the set of the vector fields on P because [XF , XG ] = X−{F,G} . Proof As derivations, [XF , XG ][H] = XF [XG [H]] − XG [XF [H]] = XF [{H, G}] − XG [{H, F }] = {{H, G}, F } − {{H, F }, G} = − {H, {F, G}} = −X{F,G} [H], where we have applied the Jacobi identity in the fourth equality. The next corollary gives Hamilton’s equations in Poisson bracket form. 12 If φt is the flow of XH and F : U → R is an arbitrary smooth function defined on the open subset U of P then d (F ◦ φt ) = {F ◦ φt , H} = {F, H} ◦ φt .

Proof As derivations, [XF , XG ][H] = XF [XG [H]] − XG [XF [H]] = XF [{H, G}] − XG [{H, F }] = {{H, G}, F } − {{H, F }, G} = − {H, {F, G}} = −X{F,G} [H], where we have applied the Jacobi identity in the fourth equality. The next corollary gives Hamilton’s equations in Poisson bracket form. 12 If φt is the flow of XH and F : U → R is an arbitrary smooth function defined on the open subset U of P then d (F ◦ φt ) = {F ◦ φt , H} = {F, H} ◦ φt . 7) Proof We have d (F ◦ φt )(z) = dt dF (φt (z)), dφt (z) dt = dF (φt (z)), XH (φt (z)) = {F, H}(φt (z)).

N, where g ij are the entries of the inverse matrix of (gij ). 3) of a particle in a potential field with which we began our discussion in the Introduction. 3 Hyperregular Lagrangians We shall summarize here the precise equivalence between the Lagrangian and Hamiltonian formulation for hyperregular Lagrangians and Hamiltonians. The proofs are easy lengthy verifications; see [AbMa78] or [MaRa94]. (a) Let L be a hyperregular Lagrangian on T Q and H = E ◦ (FL)−1 , where E is the energy of L. Then the Lagrangian vector field Z on T Q and the Hamiltonian vector field XH on T ∗ Q are related by the identity (FL)∗ XH = Z.

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