Classical Mechanics (Goldstein) · Problem 9

The Hamilton Equations of Motion — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · canonical-equations-auxiliary-conditions-lagrange-multipliers

Problem

If the canonical variables are not all independent, but are connected by auxiliary conditions of the form ψk(qi,pi,t)=0,\psi_k(q_i, p_i, t) = 0, show that the canonical equations of motion can be written ∂H∂pi+∑kλk∂ψk∂pi=q˙i,∂H∂qi+∑kλk∂ψk∂qi=−p˙i,\frac{\partial H}{\partial p_i} + \sum_k \lambda_k \frac{\partial \psi_k}{\partial p_i} = \dot{q}_i, \qquad \frac{\partial H}{\partial q_i} + \sum_k \lambda_k \frac{\partial \psi_k}{\partial q_i} = -\dot{p}_i, where the λk\lambda_k are the undetermined Lagrange multipliers. The formulation of the Hamiltonian equations in which tt is a canonical variable is a case in point, since a relation exists between pn+1p_{n+1} and the other canonical variables: H(q1,…,qn+1; p1,…,pn)+pn+1=0.H(q_1, \ldots, q_{n+1};\ p_1, \ldots, p_n) + p_{n+1} = 0. Show that as a result of these circumstances the 2n+22n+2 Hamilton's equations of this formulation can be reduced to the 2n2n ordinary Hamilton's equations plus Eq. (8.41) and the relation λ=dtdθ.\lambda = \frac{dt}{d\theta}. Note that while these results are reminiscent of the relativistic covariant Hamiltonian formulation, they have been arrived at entirely within the framework of nonrelativistic mechanics.

Related theory

Browse the problem library