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On the Toda criterion. (English) Zbl 0755.58062

Summary: Using the path-integral formulation of classical mechanics, we present a further study of the Toda criterion. This is an approximate criterion to detect local transitions from ordered to stochastic/ergodic motion or vice versa. We analyze the criterion by studying those minima of the classical path-integral weight that are not invariant under a universal supersymmetry present in any classical Hamiltonian system. This analysis relies on a theorem, that we previously proved, which says that systems which are in a phase with this supersymmetry un-broken are also in the ergodic phase, while systems which are in a phase characterized by ordered motion always have, in that phase, the supersymmetry broken. This study confirms that the Toda criterion is neither a sufficient nor a necessary condition for the transition from ordered to stochastic motion. In the conclusions some ideas are put forward to find a true criterion based on our supersymmetry.

MSC:

58Z05 Applications of global analysis to the sciences
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70K50 Bifurcations and instability for nonlinear problems in mechanics
70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics
81S40 Path integrals in quantum mechanics
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