If the canonical variables are not all independent, but are connected by auxiliary conditi...

If the canonical variables are not all independent, but are connected by auxiliary conditions of the form

show that the canonical equations of motion can be written

where the λk are the undetermined Lagrange multipliers. The formulation of the Hamiltonian equations in which t is a canonical variable is a case in point, since a relation exists between pn+1 and the other canonical variables:

Show that as a result of these circumstances the 2n + 2 Hamilton’s equations of this formulation can be reduced to the 2n ordinary Hamilton’s equations plus Eq. (8.41) and the relation

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.