I am reading Goldstein classical mechanics chapter 2 p. 35. Here the author states that the action integral $$\int L(q,\dot q,t)dt$$ is invariant under change in generalized coordinates $$q_i=q_i(s_1,...,s_n,t)\qquad i=1,...,n.$$ I can't proceed from $$\int L(q(s,t),\dot q(s,\dot s ,t),t)dt$$ How to continue from here?
Concretely, Goldstein writes under eq. (2.2):
"the Hamilton's principle is a sufficient condition for deriving the equations of motion enable us to construct mechanics of monogenic systems from Hamilton's principle as the basic postulate rather than Newton's laws of motion. such a formulation as advantages example since the integral $I$ is obviously invariant to the system of generalized coordinates used to express the equations of motion must always have Lagrangian form no matter how the generalized coordinates are transformed."