The role of kinematics in Hamiltonian dynamics: Momentum as a Lagrange multiplier

The domain space of the Lagrangian for n degrees of freedom is formally a 2n‐dimensional space—e.g., for n=1, L=L(x,v)—and the virtual paths associated with Hamilton’s principle are formal paths in that space of states. The requirement that those paths be consistent with physical kinematics can be regarded as a constraint: A particle’s parametric velocity function v(t) and its position function x(t) must satisfy dx/dt−v(t)=0. When the role of this constraint is explicated by maintaining the logical distinction between v and ẋ, the momentum p emerges as a Lagrange multiplier. When the Lagrangian and Hamiltonian are regarded as state functions, it is seen that the integrand in the customary ‘‘modified Hamilton’s principle’’—which involves nonkinematical paths—is not the Lagrangian function L=pv−H but is rather pẋ−H=L+p(ẋ−v). The relationship of that variational condition to canonical transformations is reviewed, and a parsimonious proof is given for the canonical invariance of the Poisson brackets.