American Institute of Mathematical Sciences

Advanced
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  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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Dürr and S. Teufel, Bohmian Mechanics: The Physics and Mathematics of Quantum Theory, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/b99978.  Google Scholar

[5]

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 3$^nd$ edition, Addison Wesley, 2001. Google Scholar

[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.  Google Scholar

[7]

H. A. Kramers, Quantum Mechanics, North-Holland Publishing Company, Amsterdam, 1957.  Google Scholar

[8]

L. D. Landau and E. M. Lifshitz, Mechanics, Vol. 1, 3$^nd$ edition, Butterworth-Heinemann, 1976. Google Scholar

[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.  Google Scholar

[10]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Vol. 6, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959.  Google Scholar

[11]

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. 2, 4$^nd$ edition, Butterworth-Heinemann, 1975.  Google Scholar

[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.  Google Scholar

[13]

E. Madelung, Quantentheorie in hydrodynamischer form (Quantum theory in hydrodynamic form), Z Phys, 40 (1926), 322–326. In German. Google Scholar

[14]

A. Messiah, Quantum Mechanics, Dover Publications Inc. New York, 1961.  Google Scholar

[15] R. Omnès, The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, N.J., 1994.   Google Scholar
[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.  Google Scholar

[17]

R. Schiller, Quasi-classical theory of the nonspinning electron, Physical Review, 125 (1962), 1100-1108.  doi: 10.1103/PhysRev.125.1100.  Google Scholar

[18]

R. Schiller, Quasi-classical theory of the spinning electron, Physical Review, 125 (1962), 1116-1123.  doi: 10.1103/PhysRev.125.1116.  Google Scholar

[19]

A. M. Scarfone, Canonical quantization of nonlinear many-body systems, Physical Review E, 71 (2005), 051103, 15pp. doi: 10.1103/PhysRevE.71.051103.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. Auletta, Foundations and Interpretation of Quantum Mechanics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789810248215.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Dürr and S. Teufel, Bohmian Mechanics: The Physics and Mathematics of Quantum Theory, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/b99978.  Google Scholar

[5]

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, 3$^nd$ edition, Addison Wesley, 2001. Google Scholar

[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.  Google Scholar

[7]

H. A. Kramers, Quantum Mechanics, North-Holland Publishing Company, Amsterdam, 1957.  Google Scholar

[8]

L. D. Landau and E. M. Lifshitz, Mechanics, Vol. 1, 3$^nd$ edition, Butterworth-Heinemann, 1976. Google Scholar

[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.  Google Scholar

[10]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Vol. 6, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959.  Google Scholar

[11]

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Vol. 2, 4$^nd$ edition, Butterworth-Heinemann, 1975.  Google Scholar

[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.  Google Scholar

[13]

E. Madelung, Quantentheorie in hydrodynamischer form (Quantum theory in hydrodynamic form), Z Phys, 40 (1926), 322–326. In German. Google Scholar

[14]

A. Messiah, Quantum Mechanics, Dover Publications Inc. New York, 1961.  Google Scholar

[15] R. Omnès, The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, N.J., 1994.   Google Scholar
[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.  Google Scholar

[17]

R. Schiller, Quasi-classical theory of the nonspinning electron, Physical Review, 125 (1962), 1100-1108.  doi: 10.1103/PhysRev.125.1100.  Google Scholar

[18]

R. Schiller, Quasi-classical theory of the spinning electron, Physical Review, 125 (1962), 1116-1123.  doi: 10.1103/PhysRev.125.1116.  Google Scholar

[19]

A. M. Scarfone, Canonical quantization of nonlinear many-body systems, Physical Review E, 71 (2005), 051103, 15pp. doi: 10.1103/PhysRevE.71.051103.  Google Scholar

[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.  Google Scholar

[1]

Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317
[2]

Håkon Hoel, Anders Szepessy. Classical Langevin dynamics derived from quantum mechanics. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 4001-4038. doi: 10.3934/dcdsb.2020135
[3]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[4]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421
[5]

Sandra Lucente, Eugenio Montefusco. Non-hamiltonian Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 761-770. doi: 10.3934/dcdss.2013.6.761
[6]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
[7]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
[8]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295
[9]

Cesare Tronci. Momentum maps for mixed states in quantum and classical mechanics. Journal of Geometric Mechanics, 2019, 11 (4) : 639-656. doi: 10.3934/jgm.2019032
[10]

Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461
[11]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513
[12]

María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207
[13]

Eliot Fried. New insights into the classical mechanics of particle systems. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469
[14]

Luis C. García-Naranjo, Mats Vermeeren. Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021011
[15]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017
[16]

Jean-Claude Zambrini. Stochastic deformation of classical mechanics. Conference Publications, 2013, 2013 (special) : 807-813. doi: 10.3934/proc.2013.2013.807
[17]

Jamie Cruz, Miguel Gutiérrez. Spiral motion in classical mechanics. Conference Publications, 2009, 2009 (Special) : 191-197. doi: 10.3934/proc.2009.2009.191
[18]

Cristina Stoica. An approximation theorem in classical mechanics. Journal of Geometric Mechanics, 2016, 8 (3) : 359-374. doi: 10.3934/jgm.2016011
[19]

Weinan E, Jianfeng Lu. Mathematical theory of solids: From quantum mechanics to continuum models. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5085-5097. doi: 10.3934/dcds.2014.34.5085
[20]

Federica Masiero. Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 223-263. doi: 10.3934/dcds.2012.32.223

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (115)
  • HTML views (251)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences