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When reading about the Calculus of Variation and Hamilton's principle I come across quotes like this

Hamilton's principle states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models.

This has me curious. What systems submit to this transformation from differential equations to functionals? Can the calculus of variations be applied to any differential equation, or are there restrictions? What is special about "physical systems" which permit this sort of manipulation?

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Lagrangian mechanics is equivalent to Newtonian mechanics, so every problem doable with Newton you can do with Lagrange and Hamilton. Of course, each one of the types are easier to work with in different situations (dissipative forces work best with Newton, for example), but ultimately all three give the same answers, and can be used for the same systems.

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  • $\begingroup$ Tesseract is perfectly true. $\endgroup$
    – Alfred
    Commented Nov 10, 2019 at 5:26

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