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Phase Space

A multi-dimensional space where each point represents the state of a system, defined by its position and momentum coordinates.
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The statement of the theorem

Define the phase space M\mathcal{M} as the 2N2N-dimensional manifold parameterized by the generalized coordinates q=(q1,…,qN)\mathbf{q} = (q_1, \dots, q_N) and the conjugate momenta p=(p1,…,pN)\mathbf{p} = (p_1, \dots, p_N). A point x∈M\mathbf{x} \in \mathcal{M} represents the instantaneous state of the system, such that x=(q1,…,qN,p1,…,pN)\mathbf{x} = (q_1, \dots, q_N, p_1, \dots, p_N). The dynamics are described by the flow dxdt=J⋅∇H(x)\frac{d\mathbf{x}}{d t} = \mathbf{J} \cdot \nabla H(\mathbf{x}), where J\mathbf{J} is the symplectic matrix.