Abstract
The basics of Hamiltonian mechanics and its generalizations are analyzed to find the most general laws of motion. The specific features of the variational principle in Hamiltonian mechanics (the problems of covariant formulation and boundary conditions) and the Maupertuis principle are discussed. The connection between Hamiltonian mechanics and statistical physics (Hamiltonian equations of motion preserve the Gibbs distribution and the evolution of nonequilibrium states of the harmonic oscillator in the thermal bath is described by the probability amplitudes) is underlined. The most well known generalizations, the Birkhoff and Nambu mechanics, are considered from this point of view. The Ostrogradsky mechanics, in which the Lagrangian depends on higher derivatives, theories not on symplectic manifolds, theories not on manifolds, and theories with complex variables are discussed. The simplest generalization of Poisson brackets for description of the evolution of nonequilibrium states results in the cosmological constant that arises in gravitational equations.
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References
L. V. Prokhorov and S. V. Shabanov, “Hamiltonian Mechanics of Gauge Systems” (KomKniga, Moscow, 2006) [in Russian].
Y. Nambu, Phys. Rev. D: Part. Fields 7, 2405 (1973).
L. V. Prokhorov, quant-ph/0406079.
L. V. Prokhorov, gr-qc/0602023.
L. V. Prokhorov, Fiz. Elem. Chastits At. Yadra 38, 696 (2007) [Phys. Part. Nucl. 38, 364 (2007)].
V. I. Arnold, Mathematical Methods of Classical Mechanics (2nd Eng. ed., Springer, Berlin, 1982; Nauka, Moscow, 1979).
A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems, Vol. 1 (Udmurdsky University, Izhevsk, 1999; CRC, London, 2004).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 1: Mechanics (Fizmatgiz, Moscow, 1958; 3rd Eng. ed., Butterworth-Heinemann, Oxford, 1982).
M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].
P. A. M. Dirac, The Principles of Quantum Mechanics (4th Eng. ed., Oxford University, Oxford, 1982; Fizmatlit, Moscow, 1960).
K. Jakobi, Lectures on Dynamics (Glavn. Red. Obshetekhnich. Literatury, Moscow, Leningrad, 1936) [in Russian].
Ya. G. Sinai, Topics in Ergodic Theory (Princeton University, Princeton, 1993; FAZIS, Moscow, 1996).
L. V. Prokhorov, Vestn. SPbGU, Ser. 4., No. 4, 3 (2005).
V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961).
B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, in Current Problems in Mathematics. Fundamental Directions (Moscow, 1985), Vol. 4, p. 179 [in Russian].
S. V. Shabanov, J. Phys. A 25, L1245 (1992).
A. A. Martinez-Merino and M. Montesinos, Ann. Phys. (N.Y.) 321, 318 (2006).
R. Finkelstein and M. Villasante, Phys. Rev. D: Part. Fields 33, 1666 (1986).
G. D. Birkhoff, Dynamical Systems (AMS Colloq. Publ. V. IX, Hamburg, 1927).
M. Santilli, Foundations of Theoretical Mechanics, Vols. I, II (Springer, Berlin, 1978, 1983).
J. McEwan, Found. Phys. 23, 313 (1993).
Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Math. Phys. K1, p. 966 (1921).
O. Klein, Z. Phys. 37, 895 (1926).
H. Mandel, Z. Phys. 39, 136 (1926).
V. Fock, Z. Phys. 39, 226 (1926).
M. Green, J. Schwarz, and E. Witten, Superstring Theory (Cambridge University, Cambridge, 1987; Mir, Moscow, 1990), Vols. 1, 2.
L. V. Prokhorov, Space As a Network (NIIKh St. Petersburg State University, St. Petersburg, 2004) [in Russian].
G. M. Zaslavskii, Physics of Chaos in Hamiltonian Systems (Izhevsk, Moscow, 2004; Imperial College, London, 1998).
N. I. Gerasimenko and B. S. Pavlov, Teor. Mat. Fiz. 74, 345 (1988).
Yu. V. Pokornyi, et al., Differential Equations on Networks (Geometric Graphs) (Fizmatlit, Moscow, 2004) [in Russian].
M. Ostrogradsky, Mem. Acad. St. Petersburg 4, 385 (1850).
V. V. Nesterenko, J. Phys. A 22, 1673 (1989).
L. V. Prokhorov, Yad. Fiz. 35, 229 (1982) [Sov. J. Nucl. Phys. 35, 129 (1982)].
M. Mondragon and M. Montesinos, Int. J. Mod. Phys. A 19, 2473 (2004).
A. Sergi, cond-mat/0508193.
M. Milgrom, Astrophys. J. 270, 365 (1983); Astrophys. J. 270, 371 (1983); Astrophys. J. 270, 384 (1983).
J. D. Bekenstein, Phys. Rev. D: Part. Fields 70, 083509 (2004).
J. H. Gundlach, et al., Phys. Rev. Lett. 98, 150801 (2007).
S. Dodelson and M. Liguori, Phys. Rev. Lett. 97, 231301 (2006).
H.-C. Lee, Am. J. Math. 65, 433 (1943).
V. G. Lemlein, Dokl. Akad. Nauk SSSR 115, 655 (1957).
Ph. Tondeur, Comm. Math. Helvetici 36, 234 (1961).
B. V. Fedosov, J. Diff. Geom. 40, 213 (1994).
I. Gelfand, V. Retakh, and M. Shubin, Advan. Math. 136, 104 (1998).
O. I. Mokhov, Symplectic and Poisson Geometry on Loop Spaces of Smooth Manifolds and Integrable Equations (Izhevsk, Moscow, 2004) [in Russian].
D. I. Blokhintsev, Usp. Fiz. Nauk 122, 745 (1977) [Sov. Phys. Usp. 20, 683 (1977)].
A. G. Lisi, “Quantum Mechanics from a Universal Action Reservoir,” physics/0605068.
C. Lanczos, Analytical Dynamics (Mir, Moscow, 1965) [in Russian].
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry—Methods and Applications, Parts I–III (Nauka, Moscow, 1979; Springer, New York, 1984, 1985, 1990).
V. V. Nesterenko, Phys. Rev. D 75, 087703 (2007).
E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (4th Eng. ed., Cambridge University, Cambridge, 1989; Glavn. Red. Tekhniko-Teoretich. Literatury, Moscow, 1937).
Variation Principles of Mechanics, Ed. By L. S. Polak (Fizmatgiz, Moscow, 1959) [in Russian].
M. V. Ostrogradsky, Complete Collected Works (Izd-vo Akad. nauk Ukrainskoi SSR, Kiev, 1961), Vol. 2 [in Russian].
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Original Russian Text © L.V. Prokhorov, 2008, published in Fizika Elementarnykh Chastits i Atomnogo Yadra, 2008, Vol. 39, No. 5.