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Open AccessArticle

Canonical Transformation of Potential Model Hamiltonian Mechanics to Geometrical Form I

1
Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva 84105, Israel
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Department of Physics, Ariel University, Ariel 40700, Israel
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Department of Physics, Tel Aviv University, Ramat Aviv 69978, Israel
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Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
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Department of Electrical & Electronic Engineering, Ariel University, Ariel 40700, Israel
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PPPL, Princeton University, Princeton, NJ 08543, USA
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(6), 1009; https://doi.org/10.3390/sym12061009
Received: 29 April 2020 / Revised: 7 June 2020 / Accepted: 10 June 2020 / Published: 14 June 2020
Using the methods of symplectic geometry, we establish the existence of a canonical transformation from potential model Hamiltonians of standard form in a Euclidean space to an equivalent geometrical form on a manifold, where the corresponding motions are along geodesic curves. The advantage of this representation is that it admits the computation of geodesic deviation as a test for local stability, shown in recent previous studies to be a very effective criterion for the stability of the orbits generated by the potential model Hamiltonian. We describe here an algorithm for finding the generating function for the canonical transformation and describe some of the properties of this mapping under local diffeomorphisms. We give a convergence proof for this algorithm for the one-dimensional case, and provide a precise geometric formulation of geodesic deviation which relates the stability of the motion in the geometric form to that of the Hamiltonian standard form. We apply our methods to a simple one-dimensional harmonic oscillator and conclude with a discussion of the relation of bounded domains in the two representations for which Morse theory would be applicable. View Full-Text
Keywords: classical Hamiltonian dynamics; symplectomorphism; geometric representation; geodesic deviation; stability classical Hamiltonian dynamics; symplectomorphism; geometric representation; geodesic deviation; stability
MDPI and ACS Style

Strauss, Y.; Horwitz, L.P.; Levitan, J.; Yahalom, A. Canonical Transformation of Potential Model Hamiltonian Mechanics to Geometrical Form I. Symmetry 2020, 12, 1009.

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