In advanced mechanics, we learn about the variational principle, the principle of stationary action, and the Hamilton's principle. I feel that the difference between them is not very clearly organized in my mind. Can somebody explain them that will help me organize them? For example, whether one principle is more basic and general than the other two, whether one can be derived from the other etc.
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The principle of stationary/least action usually$^1$ refers to Hamilton's principle $\delta S=0$. Hamilton's principle is an example of a variational principle.
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$^1$NB: Note that e.g. Goldstein calls the principle of abbreviated action/Maupertuis's principle $\delta \int {\bf p}\cdot \mathrm{d}{\bf q}=0$ for the principle of least action, cf. e.g. this Phys.SE post.
Qmechanic
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Isn't Hamilton's principle more general than the principle of stationary action which applies to the Lagrangian $L(q,\dot{q}, t)$ only? I thought Hamilton's principle also applies to some other function of the form $f(q,p,\dot{q},\dot{p},t)$. – Solidification Mar 16 '22 at 12:05
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So does the principle of stationary action, cf. e.g. my Phys.SE answer here, so no difference there. – Qmechanic Mar 16 '22 at 12:10