An underlying geometrical manifold for Hamiltonian mechanics

Abstract

We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture), that can be put into correspondence with the usual Hamilton–Lagrange mechanics. The requirement of dynamical equivalence of the two types of Hamiltonians, that the momenta generated by the two pictures be equal for all times, is sufficient to determine an expansion of the conformal factor, defined on the geometrical coordinate representation, in its domain of analyticity with coefficients to all orders determined by functions of the potential of the Hamiltonian–Lagrange picture, defined on the Hamilton–Lagrange coordinate representation, and its derivatives. Conversely, if the conformal function is known, the potential of a Hamilton–Lagrange picture can be determined in a similar way. We show that arbitrary local variations of the orbits in the Hamilton–Lagrange picture can be generated by variations along geodesics in the geometrical picture and establish a correspondence which provides a basis for understanding how the instability in the geometrical picture is manifested in the instability of the the original Hamiltonian motion.

Keywords

geometrical Hamiltonian / stability analysis / coordinate mapping / Riemann–Euclidean Hamiltonian equivalence

References

[1] L. P. Horwitz, Y. Ben Zion, M. Lewkowicz, M. Schiffer, and J. Levitan, Geometry of Hamiltonian chaos, Phys. Rev. Lett. 98, 234301 (2007). Geometrical methods of a different form were first introduced by C. G. J. Jacobi, Vorlesungen über Dynamik, Berlin: Verlag Reimer, 1884; J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésiques, J. Math. Pures Appl. 4, 27 (1898), and further developed by L. Casetti and M. Pettini, Analytic computation of the strong stochasticity threshold in Hamiltonian dynamics using Riemannian geometry, Phys. Rev. E 48, 4320 (1993). see also: M. Pettini, Geometery and Topology in Hamiltonian Dynamics and Statistical Mechanics, New York: Springer, 2006, and references therein

[2] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, New York: Springer-Verlag, 1990

[3] W. D. Curtis and F. R. Miller, Differentiable Manifolds and Theoretical Physics, New York: Academic Press, 1985

[4] Y. Ben Zion and L. P. Horwitz, Detecting order and chaos in three-dimensional Hamiltonian systems by geometrical methods, Phys. Rev. E 76(4), 046220 (2007). Y. Ben Zion and L. Horwitz, Applications of geometrical criteria for transition to Hamiltonian chaos, Phys. Rev. E 78(3), 036209 (2008). Y. Ben Zion and L. Horwitz, Controlling effect of geometrically defined local structural changes on chaotic Hamiltonian systems, Phys. Rev. E 81, 046217 (2010)

[5] A. Yahalom, J. Levitan, and M. Lewkowicz, Lyapunov vs. Geometrical stability analysis of the Kepler and the restricted three-body problems, Phys. Lett. A 375, 2111 (2011). See also, J. Levitan, A. Yahalom, L. Horwitz, and M. Lewkowicz, On the stability of Hamiltonian systems with weakly time dependent potentials, Chaos 23(2), 023122 (2013)

[6] L. Horwitz, A. Yahalom, M. Lewkowicz, and J. Levitan, Subtle is the Lord: On the difference between Hamiltonian (Lyapunov) stability analysis and geometrical stability analysis of gravitational orbits, Int. J. Mod. Phys. D 20(14), 2787 (2011)

[7] A. Yahalom, M. Lewkowicz, J. Levitan, G. Elgressy, L. P. Horwitz, and Y. Ben Zion, Uncertainty relations for chaos, International Journal of Geometric Methods in Modern Physics 12(09), 1550093 (2015)

[8] Y. Strauss, L. P. Horwitz, J. Levitan, and A. Yahalom, Quantum field theory of classically unstable Hamiltonian dynamics, J. Math. Phys. 56(7), 072701 (2015)

[9] E. Calderon, L. Horwitz, R. Kupferman, and S. Shnider, On the geometrical formulation of Hamiltonian dynamics, Chaos 23(1), 013120 (2013)

[10] A. Gershon and L. P. Horwitz, Kaluza-Klein theory as a dynamics in a dual geometry, J. Math. Phys. 50(10), 102704 (2009)

[11] L. P. Horwitz, A. Gershon, and M. Schiffer, Hamiltonian map to conformal modification of space-time metric: Kaluza–Klein and TeVeS, Found. Phys. 41(1), 141 (2010)

[12] M. Milgrom, A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis, Astrophys. J. 270, 365 (1983); A modification of the Newtonian dynamics: Implications for Galaxies, 371, A modification of the Newtonian dynamics: Implications for Galaxy Systems, 384 (1983)

[13] J. D. Bekenstein, Relativistic gravitation theory for the MOND paradigm, Phys. Rev. D 70, 083509 (2004)

[14] J. D. Bekenstein and R. H. Sanders, A Primer to Relativistic MOND Theory, EAS Pub. Series 20, 225 (2006)

[15] H. Safaai, M. Hasan, and G. Saadat, On the Prediction of Chaos in the Restricted Three-body Problem, in Understanding Complex Systems, Berlin: Springer, 2006, p. 369

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag Berlin Heidelberg