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Read e-book online Regular and Stochastic Motion PDF

By A.J. Lichtenberg

ISBN-10: 1475742576

ISBN-13: 9781475742572

ISBN-10: 1475742592

ISBN-13: 9781475742596

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The above result is interesting, but has not led to new solutions of physical problems. There has, however, been a recent rekindling of interest in constructing integrable Hamiltonians. Hall (1981) has applied the method to the motion of a particle in static electric and magnetic fields, explicitly introducing the vector potential into the problem. He found that Whittaker's solutions were not complete because the conservation of energy was not explicitly considered in the constraints. He also has examined other classes of invariants, not quadratic in the momenta.

Are there any general methods to test for the integrability of a given Hamiltonian? The answer, for the moment, is no. We can tum the question around, however, and ask if methods can be found to construct potentials that give rise to integrable Hamiltonians. The answer here is that a method exists, at least for a restricted class of problems, but the method becomes rapidly very tedious as the forms allowed for the integrals of the motion are expanded. 61) where the coefficients are functions of the qs.

9. , 1973). 38 1. 58), such that the invariant curves in Fig. 9 can be calculated directly. 52). Even in the original coordinates, I is related to no obvious conservation law or symmetry. Finding Integrable Hamiltonians. Are there any general methods to test for the integrability of a given Hamiltonian? The answer, for the moment, is no. We can tum the question around, however, and ask if methods can be found to construct potentials that give rise to integrable Hamiltonians. The answer here is that a method exists, at least for a restricted class of problems, but the method becomes rapidly very tedious as the forms allowed for the integrals of the motion are expanded.

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Regular and Stochastic Motion by A.J. Lichtenberg


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