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Legendre Transformations

A method for converting between Hamiltonian and Lagrangian formulations of a system, essential for solving various problems in classical mechanics.
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The statement of the theorem

Given the Lagrangian L(q,q˙,t)L(q, \dot{q}, t) and the generalized momentum pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}, the Hamiltonian H(q,p,t)H(q, p, t) is obtained from the Legendre transformation: H(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) where q˙i\dot{q}_i is implicitly defined by the relation pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}. Furthermore, the transformation of the time derivative is dHdt=∂L∂t+∂∂t(∑piq˙i−L)\frac{\text{d}H}{\text{d}t} = \frac{\partial L}{\partial t} + \frac{\partial}{\partial t} \left( \sum p_i \dot{q}_i - L \right).
Source: Wikipedia