Ivancevic_Applied-Diff-Geom

338 Applied Differential Geometry: A Modern Introduction3.12.3 Application: Autonomous Hamiltonian Mechanics3.12.3.1 Basics of Hamiltonian MechanicsIn this section we present classical Hamiltonian dynamics. Let (M, ω) be asymplectic manifold and H ∈ C ∞ (M, R) a smooth real valued function onM. The vector–field X H determined by the conditioni XH ω + dH = 0,is called Hamiltonian vector–field with Hamiltonian energy function H. Atriple (M, ω, H) is called a Hamiltonian mechanical system [Marsden andRatiu (1999); Puta (1993)].Nondegeneracy of ω guarantees that X H exists, but only in the nD case.Let {q 1 , ..., q n , p 1 , ..., p n } be canonical coordinates on M, i.e., ω = dp i ∧dq i . Then in these coordinates we have( ∂HX H =∂p i∂∂q i − ∂H∂q i)∂.∂p iAs a consequence, ( (q i (t)), (p i (t)) ) is an integral curve of X H (for i =1, ..., n) iff Hamiltonian equations hold,˙q i = ∂ pi H, ṗ i = −∂ q iH. (3.164)Let (M, ω, H) be a Hamiltonian mechanical system and let γ(t) be anintegral curve of X H . Then H (γ(t)) is constant in t. Moreover, if φ t is theflow of X H , then H ◦ φ t = H for each t.Let (M, ω, H) be a Hamiltonian mechanical system and φ t be the flow ofX H . Then, by the Liouville Theorem, for each t, φ ∗ t ω = ω, ( d dt φ∗ t ω = 0, soφ ∗ t ω is constant in t), that is, φ t is symplectic, and it preserves the volumeΩ ω .A convenient criterion for symplectomorphisms is that they preserve theform of Hamiltonian equations. More precisely, let (M, ω) be a symplecticmanifold and f : M → M a diffeomorphism. Then f is symplectic iff forall H ∈ C ∞ (M, R) we have f ∗ (X H ) = X H◦f .A vector–field X ∈ X (M) on a symplectic manifold (M, ω) is calledlocally Hamiltonian iff L X ω = 0, where L denotes the Lie derivative. Fromthe equality L [X,Y ] ω = L X L Y ω−L Y L X ω, it follows that the locally Hamiltonianvector–fields on M form a Lie subalgebra of X (M).Let (M, ω) be a symplectic manifold and f, g ∈ C ∞ (M, R). The Poisson