Transformation from Lagrangian to Hamiltonian

Alternatively, an auxiliary variable wk can be introduced, constrained to be dynamically equal to qk using a Lagrange multiplier that turns out to be the substituted variable pk. The constraint condition in this ingenious procedure is Xk = Pk wk = 0. The modified Lagrangian is 

The first and second of these equations are Hamilton s equations of motion. The first and third establish wk = qk and pk = as dynamical conditions, equivalent 

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