Maupertuis' principle states in symplectic formulation that the integral of the tautological 1-form is extremal among its integrals on other phase space curves on the given level set of the Hamiltonian, connecting the same starting end ending fiber. Specifically, for closed phase trajectories this is equivalent of the extremallity of the symplectic area of a surface expanded on this closed curve compared to the simplectic area of the surfaces expanded on other closed phase space curves on the same level set and passing across two different fibers from the fibers intersected by the phase trajectory.(adsbygoogle = window.adsbygoogle || []).push({});

For example, in the case of a 2-dimensional harmonic oscillator, the Hamiltonian is:

[tex]H=\frac{1}{2}(x^2+y^2+p_x^2+p_y^2)[/tex].

The level sets are spheres with radius R = 2H.

Now take a level set belonging to R = 1, and the following two points on this level set:

a = (-1, 0, 0, 0) and b = (1, 0, 0, 0).

The reduced action integral (i.e., the integral of the tautological 1-form) on curves (cos t, sin t, 0, 0) and ( cos t, 0, sin t, 0) both will be extremal because the first circle lies in an isotropic plane while the second one in a symplectic plane, hence the action integral (i.e. the symplectic area of the disk bounded by these curves) are 0, and π respectivelly. The firs one is a minimum, while the second one is a maximum. And here comes the paradox. Arnold proves that the Mapertuis' principle yields unique solution (Arnold: Mathematical Methods of Classial Mechanics, p.244).

Could anybody tell me, what is the resolution of this paradox?

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# A paradox about Maupertuis' principle (symplectic viewpoint)

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