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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."

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The main point that Goldstein here tries to convey is that the Lagrangian $L(q,\dot{q},t)$ is invariant under passive change of generalized coordinates. See also this, this & this related posts.

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  • $\begingroup$ Thanks, if my understanding is correct passive change corresponds to change of coordinates (s) depends only on (q,t), and change in coordinates of (q) depends on (s,t), that is no velocities involved?? $\endgroup$ – user199996 Jan 24 '20 at 13:42
  • $\begingroup$ I think i get it, thanks $\endgroup$ – user199996 Jan 24 '20 at 13:44

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