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transform of turns into a function known as the '. The Hamiltonian satisfies

which implies that

where the velocities are found from the (-dimensional) equation which, by assumption, is uniquely solvable for . The (-dimensional) pair is called phase space coordinates. (Also canonical coordinates).

From Euler-Lagrange equation to Hamilton's equations

In phase space coordinates , the (-dimensional) Euler-Lagrange equation

becomes Hamilton's

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