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6

The more common names for what you are talking about are the abbreviated action $$S_0[q] := \int p \mathrm{d}q$$ versus the action $$S[q] := \int_{t_1}^{t_2}L(q,\dot q,t)\mathrm{d}t$$ Both are used in different formulations of classical mechanics, and deliver a different "flavor" of solutions. On both one can do variations calculus and obtains the ...

5

Explicitly proving non-integrability of an arbitrary Hamiltonian system is an open problem. For some classes of Hamiltonian systems (e.g systems on a plane) is possible to prove explicitly the non-integrability of the system, using theorems of Poincare, Burns, Ziglin and Yoshida (and generalizations). For example there is a theorem of Poincare: For a ...

2

As user ACuriousMind correctly writes: What Goldstein calls the principle of least action $\int p~\mathrm{d}q$ is usually called Maupertuis' principle or the principle of abbreviated action. What Goldstein calls the Hamilton's variational principle is often also called the the principle of least/extremal/stationary action $\int L~\mathrm{d}t$. This is ...

2

This is low-Reynolds number particle sedimentation. It turns out that the problem is strikingly difficult, despite the simplicity of the setup and even of the equations (Stokes plus dynamics of pointwise solid particles). Check the webpage of E Guazzelli who's been working a lot on this. However, I believe you can get a fair rendering with simply a ...

1

Before we begin, note first of all, that there exist various definitions of a canonical transformation (CT) in the literature, cf. e.g. this Phys.SE post. For instance, OP's last equation (v1) is called an extended canonical transformation (ECT) in Ref. 1. OP is essentially asking (v1): If we have a transformation \tag{A} (q,p)~\longrightarrow~ ...

1

First things first: waves that have already reached close vicinity to the beach DO displace water towards the shoreline - just notice how the water moves back and forth at the point it's ankle-deep. This is related to the phenomenon by which they lose their wave form and get a crest. There are many forces acting on a surfer, but two of them are the ...

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Yes, take the orbits of Neptune and Pluto, for example, which are inclined to each other. Furthermore, it is possible for both satellites to influence each other's orbit and in doing so, to run along chaotic trajectories.

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