Is Hamilton’s principle valid for systems that are not monogenic?

Question

I have read in Goldstein that Hamilton’s principle works only for monogenic systems. Is it true? I thought that the action principle is universal?

Answer 1

1. We can of course implement any set of EOMs (e.g. not necessarily monogenic systems) via Lagrange multipliers in an action principle.

2. If we are not allowed to introduce auxiliary variables, then an action principle may not exist, cf. e.g. this Phys.SE, i.e. then it is not universal.

3. However, often we assume that the EOMs of the system equate kinetic and dynamical terms (think Newton's 2nd law), and typically we assume that the kinetic terms of the EOMs come from positive definite kinetic terms in the action. Then in order for the stationary action/Hamilton's principle to reproduce the dynamical terms of the EOMs (i.e. generalized forces) they must similarly come from dynamical terms in the action (aka. generalized potentials), i.e. the generalized forces should be monogenic.