By Paul Bracken

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Bernoulli sequences and trajectories in the anisotropic Kepler problem, Journal of Mathematical Physics 18: 806-823. [2] Devaney, R. L. (1979). Collision Orbits in the Anisotropic Kepler Problem, Inventions math. 45: 221-251. [3] Gutzwiller, M. C. (1981). Periodic orbits in the anisotropic Kepler problem, in Devaney, R. L. & Nitecki, Z. H. ) Classical Mechanics and Dynamical systems, Marcel Dekker, New York, pp. 69-90. [4] Gutzwiller, M. C. (1971). Periodic orbits and classical quantization conditions, Journal of Mathematical Physics 12: 343-358.

The contribution of a periodic orbit to the density of state The contribution of a single periodic orbit r to the DOS is estimated by a resummation of the sum over the repetition j (after the approximation sinh x ≈ e x /2), ρ( E)|r ≈ Tr ∑ m ur h¯ /2 (Sr − 2π¯h(m + l/4))2 + (ur h¯ /2)2 . (26) This gives Lorentzian peak at Sr = 2π¯h(m + l/4) (27) similar to the Bohr-Sommerfeld formula. In AKP the action S is given as Sr ( E ) = √ Tr . −2E (28) Hence the peak position of the Lorentzian form in the energy is given by Er,m = − 1 2 Tr 2π¯h(m + 1) 2 , (29) where Maslov index l = 4 for three dimensions is taken.

By dealing with the other field equations that appear within Carathéodory’s approach, one may hope to gain additional insight. 5772/53843 mechanical: superconductivity and the response of a sample of charged particles to an external magnetic field. The London equations of superconductivity were originally understood as an ad-hoc assumption, with quantum mechanics lying at its roots. On the other hand, according to classical mechanics there can be no diamagnetism and no paramagnetism at all. We will deal with these two issues, showing how it is possible to classically derive the London equations and the existence of magnetic moments.