Virtual work of constraints in Hamilton‘s principle

Question

Goldstein 2ed pg 36

So in the case of holonomic constraints we can move back and forth between Hamilton's principle and Lagrange equations given as $$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)-\frac{\partial L}{\partial q_{j}}=0.\tag{1-57}$$ But the Lagrange equations were derived from d'Alemberts principle assuming that the virtual work of constraints is zero. But in the Hamilton‘s principle, we don’t see that the constraints should be workless.

Is it implicitly assumed that the virtual work of constraints is zero in Hamilton‘s principle? How can we prove it?

Answer 1

• For what it's worth, the holonomic constraints (that Ref. 1 is here talking about) are the ones used to define the generalized coordinates $q^1,\ldots, q^n,$ from the positions ${\bf r}_1, \ldots, {\bf r}_N,$ in the first place, cf. eq. (1-38).

• Ref. 1 is (at this point in the text) not considering additional constraints, i.e. there are no remaining constraints in the action (2-2), i.e. the generalized coordinates $q^1,\ldots, q^n$ are independent variables. So the stationary action principle/Hamilton's principle (2-2) is equivalent to Euler-Lagrange equations (1-57).

• Concerning the relation between d'Alembert's principle and Lagrange equations, see e.g. this related Phys.SE post.

• Note in particular that constraint forces are here non-applied forces in d'Alembert's principle, cf. e.g. this related Phys.SE post.

• Moreover, the non-applied forces are not modelled in the Lagrangian. In this sense the Lagrange equations are consistent with d'Alembert's principle, i.e. that non-applied forces produce no virtual work, cf. OP's question.

References:

1. H. Goldstein, Classical Mechanics; Section 2.1.