Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems

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December 2020, 12(4): 563-583. doi: 10.3934/jgm.2020024

Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems

Sergey Rashkovskiy

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Vernadskogo Ave., 101/1, Moscow, 119526, Russia

Received April 2018 Revised July 2020 Published December 2020 Early access September 2020

Fund Project: This work was done on the theme of the State Task No. AAAA-A20-120011690135-5

A generalization of the Hamilton-Jacobi theory to arbitrary dynamical systems, including non-Hamiltonian ones, is considered. The generalized Hamilton-Jacobi theory is constructed as a theory of ensemble of identical systems moving in the configuration space and described by the continual equation of motion and the continuity equation. For Hamiltonian systems, the usual Hamilton-Jacobi equations naturally follow from this theory. The proposed formulation of the Hamilton-Jacobi theory, as the theory of ensemble, allows interpreting in a natural way the transition from quantum mechanics in the Schrödinger form to classical mechanics.

Keywords: Hamiltonian and non-Hamiltonian systems, ensemble, configuration space, continual description, the Hamilton-Jacobi equation, transition from quantum mechanics to classical mechanics.

Mathematics Subject Classification: Primary: 70H20, 70S05; Secondary: 37J05.

Citation: Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

References:

[1]	G. Auletta, Foundations and Interpretation of Quantum Mechanics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789810248215.
[2]	D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden variables", Ⅰ, Physical Review, 85 (1952), 166-179. doi: 10.1103/PhysRev.85.166.
[3]	D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden variables", Ⅱ, Physical Review, 85 (1952), 180-193. doi: 10.1103/PhysRev.85.180.
[4]	D. Dürr and S. Teufel, Bohmian Mechanics: The Physics and Mathematics of Quantum Theory, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/b99978.
[5]	H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 3$^nd$ edition, Addison Wesley, 2001.
[6]	P. Holland, Computing the wavefunction from trajectories: Particle and wave pictures in quantum mechanics and their relation, Annals of Physics, 315 (2005), 505-531. doi: 10.1016/j.aop.2004.09.008.
[7]	H. A. Kramers, Quantum Mechanics, North-Holland Publishing Company, Amsterdam, 1957.
[8]	L. D. Landau and E. M. Lifshitz, Mechanics, Vol. 1, 3$^nd$ edition, Butterworth-Heinemann, 1976.
[9]	L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Vol. 3. Translated from the Russian by J. B. Sykes and J. S. Bell. Addison-Wesley Series in Advanced Physics. Pergamon Press Ltd., London-Paris; for U.S.A. and Canada: Addison-Wesley Publishing Co., Inc., Reading, Mass; 1958.
[10]	L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Vol. 6, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959.
[11]	L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. 2, 4$^nd$ edition, Butterworth-Heinemann, 1975.
[12]	A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion, Springer-Verlag New York Inc., 1983. doi: 10.1007/978-1-4757-4257-2.
[13]	E. Madelung, Quantentheorie in hydrodynamischer form (Quantum theory in hydrodynamic form), Z Phys, 40 (1926), 322–326. In German.
[14]	A. Messiah, Quantum Mechanics, Dover Publications Inc. New York, 1961.
[15]	R. Omnès, The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, N.J., 1994.
[16]	S. A. Rashkovskiy, Eulerian and Newtonian dynamics of quantum particles, Progress of Theoretical and Experimental Physics, 2013 (2013), 063A02. doi: 10.1093/ptep/ptt036.
[17]	R. Schiller, Quasi-classical theory of the nonspinning electron, Physical Review, 125 (1962), 1100-1108. doi: 10.1103/PhysRev.125.1100.
[18]	R. Schiller, Quasi-classical theory of the spinning electron, Physical Review, 125 (1962), 1116-1123. doi: 10.1103/PhysRev.125.1116.
[19]	A. M. Scarfone, Canonical quantization of nonlinear many-body systems, Physical Review E, 71 (2005), 051103, 15pp. doi: 10.1103/PhysRevE.71.051103.
[20]	V. V. Vedenyapin and N. N. Fimin, Eulerian and Newtonian dynamics of quantum particles, Doklady Mathematics, 91 (2015), 154-157. doi: 10.1134/s1064562415020131.

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References:

[1]	G. Auletta, Foundations and Interpretation of Quantum Mechanics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789810248215.
[2]	D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden variables", Ⅰ, Physical Review, 85 (1952), 166-179. doi: 10.1103/PhysRev.85.166.
[3]	D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden variables", Ⅱ, Physical Review, 85 (1952), 180-193. doi: 10.1103/PhysRev.85.180.
[4]	D. Dürr and S. Teufel, Bohmian Mechanics: The Physics and Mathematics of Quantum Theory, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/b99978.
[5]	H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 3$^nd$ edition, Addison Wesley, 2001.
[6]	P. Holland, Computing the wavefunction from trajectories: Particle and wave pictures in quantum mechanics and their relation, Annals of Physics, 315 (2005), 505-531. doi: 10.1016/j.aop.2004.09.008.
[7]	H. A. Kramers, Quantum Mechanics, North-Holland Publishing Company, Amsterdam, 1957.
[8]	L. D. Landau and E. M. Lifshitz, Mechanics, Vol. 1, 3$^nd$ edition, Butterworth-Heinemann, 1976.
[9]	L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Vol. 3. Translated from the Russian by J. B. Sykes and J. S. Bell. Addison-Wesley Series in Advanced Physics. Pergamon Press Ltd., London-Paris; for U.S.A. and Canada: Addison-Wesley Publishing Co., Inc., Reading, Mass; 1958.
[10]	L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Vol. 6, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959.
[11]	L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. 2, 4$^nd$ edition, Butterworth-Heinemann, 1975.
[12]	A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion, Springer-Verlag New York Inc., 1983. doi: 10.1007/978-1-4757-4257-2.
[13]	E. Madelung, Quantentheorie in hydrodynamischer form (Quantum theory in hydrodynamic form), Z Phys, 40 (1926), 322–326. In German.
[14]	A. Messiah, Quantum Mechanics, Dover Publications Inc. New York, 1961.
[15]	R. Omnès, The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, N.J., 1994.
[16]	S. A. Rashkovskiy, Eulerian and Newtonian dynamics of quantum particles, Progress of Theoretical and Experimental Physics, 2013 (2013), 063A02. doi: 10.1093/ptep/ptt036.
[17]	R. Schiller, Quasi-classical theory of the nonspinning electron, Physical Review, 125 (1962), 1100-1108. doi: 10.1103/PhysRev.125.1100.
[18]	R. Schiller, Quasi-classical theory of the spinning electron, Physical Review, 125 (1962), 1116-1123. doi: 10.1103/PhysRev.125.1116.
[19]	A. M. Scarfone, Canonical quantization of nonlinear many-body systems, Physical Review E, 71 (2005), 051103, 15pp. doi: 10.1103/PhysRevE.71.051103.
[20]	V. V. Vedenyapin and N. N. Fimin, Eulerian and Newtonian dynamics of quantum particles, Doklady Mathematics, 91 (2015), 154-157. doi: 10.1134/s1064562415020131.

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