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An underlying geometrical manifold for Hamiltonian mechanics |
L. P. Horwitz1,2,3( ),A. Yahalom4(),J. Levitan3(),M. Lewkowicz3() |
1. School of Physics, Tel Aviv University, Ramat Aviv 69978, Israel
2. Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
3. Department of Physics, Ariel University, Ariel 40700, Israel
4. Department of Electrical and Electronic Engineering, Ariel University, Ariel 40700, Israel |
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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.
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| Keywords
geometrical Hamiltonian
stability analysis
coordinate mapping
Riemann–Euclidean Hamiltonian equivalence
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Corresponding Author(s):
L. P. Horwitz,A. Yahalom,J. Levitan,M. Lewkowicz
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Issue Date: 17 October 2016
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| 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
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| 3 |
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| 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)
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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)
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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)
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| 7 |
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| 8 |
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| 9 |
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| 11 |
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