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Hamilton's Equations of Motion

These equations describe the time evolution of a system's phase space coordinates, \dot{q} = \frac{\partial H}{\partial p} and \dot{p} = -\frac{\partial H}{\partial q}, where H is the Hamiltonian.

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The statement of the theorem

Let

H(q, p, t): \mathbb{R}^{2N} \to \mathbb{R}

be the Hamiltonian function, where

q = (q_1, \dots, q_N)

and

p = (p_1, \dots, p_N)

are the generalized coordinates and momenta, respectively. The time evolution of the system's phase space coordinates

(\mathbf{q}, \mathbf{p})

is governed by Hamilton's canonical equations:\n\n

\frac{d q_i}{d t} = \frac{\partial H}{\partial p_i} \quad \text{and} \quad \frac{d p_i}{d t} = -\frac{\partial H}{\partial q_i} \quad \text{for } i = 1, \dots, N

Source: Wikipedia