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I'll take off from the questioner's suggesting that maybe it's better to say what is a NON-integrable system is. The Newtonian planar three body problem, for most masses, has been proven to be non-in …

You may be interested in Shapere, A., and F. Wilczek. 1987. Self-propulsion at low Reynolds number. Phys. Rev. Lett. 58: 2051–2054 where they use gauge theory to describe micro-swimming. Because the …

The answers to question (1) for the 2 body problem are fine, and complete enough. Regarding (2). The 3 body problem (and N-body) with p =3 is significantly simpler than with $p \ne 3$. The added …

To me the answer to your question is clear: Mackey's Mathematical Foundations of Quantum Mechanics either in conjunction with, or followed by Dirac's Principles of Quantum Mechanics. After those …

It is hard to imagine that Eddington's numbers could be anything but the imaginary part of the Clifford algebra $C$ of Minkowski space. Recall that the Clifford algebra for an n-dimensional real vec …

Expanding on Chervov's comment: the Jacobian conjecture for two variables conjectures that if a polynomial map $(x,y) \to (X,Y)$ has for its Jacobian $\partial(X,Y)/\partial(x,y)$ a nonzero const …

I find the parallax effect parallax effect especially convincing evidence. Parallax is the shifting of lines of sight due to translation, eg by waiting half an earth year at which point theory tel …

Take the planar three-body problem. Or, said a bit differently, take that 'cat' to consist of three point masses moving about in the plane -- a triangle! Fix the center of the mass at the origin by …

Feynman, in his lecture notes, argues convincingly that you will understand the guts of Quantum Mechanics [QM] as best you can by looking at the two-slit experiment whose Hilbert space is two-dimensi …

Here are two nice, natural, examples of Lagrangians not of the form $T-U$ which occur naturally in physics. For a relativistic particle of charge e, mass m, travelling under the influence of an el …

Take the planar three-body problem. Or, said a bit differently, take that 'cat' to consist of three point masses moving about in the plane -- a triangle! Fix the center of the mass at the origin by …

I think a good starting place for your question regarding the moduli space for a flat 4-torus is the Fourier-Mukai' correspondence which came out of work of Nahm and which relates the moduli space o …

Following the lines Willie followed, but allowing for unequal masses $m_i$, set $I(x) = \langle x, x \rangle $ where $\langle v, w \rangle = \Sigma m_i v_i \cdot w_i$ is the so-called mass metric …

A rather silly but perhaps useful necessary condition to get $[h, g] = 0$ is that the Hamiltonian vector fields $X_h, X_g$ span an {\it isotropic} two-plane: one on which the symplectic form vanishes. …

{\bf Counterexample.} (But look at my comment above please.) Take your $B$ dead zero: no magnetic field, or friction (however you are thinking of it). Your force field is now pure potential. Your …