Gozzi, E.; Reuter, M.; Thacker, W. D. On the Toda criterion. (English) Zbl 0755.58062 Chaos Solitons Fractals 2, No. 4, 441-458 (1992). 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. Cited in 1 ReviewCited in 3 Documents 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 Keywords:path-integral; Toda criterion; supersymmetry PDFBibTeX XMLCite \textit{E. Gozzi} et al., Chaos Solitons Fractals 2, No. 4, 441--458 (1992; Zbl 0755.58062) Full Text: DOI References: [1] Koopman, B. O., Proc. Nat. Acad. Sc. USA, 17, 315 (1931) [2] von Neumann, J., Ann. Math., 33, 587 (1932) [3] Gozzi, E.; Reuter, M.; Thacker, W. D., Phys. Rev. D, 40, 10, 3363 (1989) [4] Schulman, B. S., Techniques and applications of path-integration (1981), J. Wiley: J. Wiley New York · Zbl 0587.28010 [5] Abraham, R.; Marsden, J., Foundation of Mechanics (1978), Benjamin: Benjamin New York [6] Onofri, E.; Pauri, M., J. Math. Phys., 14, 1106 (1973) [7] Arnold, V. I.; Avez, A., Ergodic problems of classical mechanics (1968), W.A. Benjamin Inc · Zbl 0167.22901 [8] E. PLB, 238, 2,3,4, 451 (1990) [9] Witten, E., J. Diff. Geom., 17, 661 (1982) [10] Arnold, V. I., Mathematical methods of classical mechanics (1978), Springer Verlag: Springer Verlag New York · Zbl 0386.70001 [11] Witten, E., Nucl. Phys., B202, 253 (1982) [12] Kastler, D., (Symposia Mathematica, Vol XX (1976), Istituto Nazionale di Alta Matematica) [13] Toda, M., Phys. Lett., 48A5, 335 (1974) [14] Enz, C. P., Helvetica Physica Acta, Vol 48, 787 (1975) [15] Henon, M.; Heiles, C., Astron J., 69, 73 (1974) [16] Benettin, G., Physica, 87A, 381 (1977) [17] Ruelle, D., Thermodynamic Formalism (1978), Addison Wesley Publ. Company [18] Benettin, G., Nuovo Cimento, 44B, 183 (1978), Among many references see for example This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.