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Hamiltonian Function

H(q, p, t) = \sum_{i} \pi_i rac{\partial q_i}{\partial \tau} + L(q, rac{\partial q}{\partial \tau}), where \pi_i are the conjugate momenta and L is the Lagrangian.
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The statement of the theorem

Given a system with Lagrangian L(q,q˙,t)L(q, \dot{q}, t), the Hamiltonian function H(q,p,t)H(q, p, t) is defined through the Legendre transformation relating generalized coordinates qiq_i and conjugate momenta pip_i: \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\nWhere the generalized velocities q˙i\dot{q}_i are implicitly determined by the momenta pip_i via the relation pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}. Thus, HH is expressed solely in terms of (q,p,t)(q, p, t).
Source: Wikipedia