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Hamiltonian Formalism

A reformulation of classical mechanics using generalized coordinates (q) and momenta (p), and the Hamiltonian function H(q, p, t).
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The statement of the theorem

Consider a physical system whose dynamics are described by the Lagrangian L(q,q˙,t)L(q, \dot{q}, t). The Hamiltonian formalism reformulates the dynamics by defining the conjugate momenta pip_i via the Legendre transformation:\n\npi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}\n\nSubsequently, the Hamiltonian function HH is defined as:\n\nH(q,p,t)=∑i=1Npiq˙i−L(q,q˙,t)H(q, p, t) = \sum_{i=1}^{N} p_i \dot{q}_i - L(q, \dot{q}, t) \n\nThe evolution is then governed by Hamilton's canonical equations.