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Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems
Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Vernadskogo Ave., 101/1, Moscow, 119526, Russia |
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.
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. |
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. |
[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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