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Hamiltonian

Definition (Hamiltonian): A smooth function \(H: M \to {\mathbf{R}}\) will be referred to as an energy functional or a Hamiltonian. If we have \(H: M\times I \to {\mathbf{R}}\), we’ll refer to this as a time-dependent Hamiltonian, i.e. the time slices \(H_t: M \to {\mathbf{R}}\) given by \(H_t(p) = H(p, t)\) are Hamiltonians. ^1c3cf2

Remark: If \((M, \omega)\) is a interior product.

Hamiltonian Vector Field

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Definition (Hamiltonian vector field): Given a smooth functional \(H: (M, \omega) \to {\mathbf{R}}\), the associated Hamiltonian vector field is the unique field \(X_H\) satisfying \(\omega(X_H, {-}) = dH\).

Remark: Conservation of energy Since \(\omega\) is alternating, \begin{align*} X_H(H) = dH(X_H) = \omega(X_H, X_H) = 0 .\end{align*}