Abstract
Linear evolution equations of mathematical physics admitting an invariant in the form of a positive quadratic form are considered. In particular, this includes the string vibration equation, the Liouville kinetic equation, the Maxwell system of equations and the Schrödinger equation. Conditions for the existence of an invariant Gaussian measure are indicated, which makes it possible to apply well-known results of ergodic theory (Poincaré’s recurrence theorem, Birkhoff–Khinchin ergodic theorem, etc.). We discuss the Hamiltonian property of such systems and conditions for their complete integrability. The ergodic properties of Kronecker flows on infinite-dimensional tori are studied. A general theorem on the averaging of quadratic forms is established.
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References
V. V. Kozlov, “Linear-Systems with a Quadratic Integral”, J. Appl. Math. Mech., 56:6 (1992), 803–809.
V. V. Kozlov, “Quadratic Conservation Laws for Equations of Mathematical Physics”, Russian Math. Surveys, 75:3 (2020), 445–494.
V. V. Kozlov, “Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46.
J. Williamson, “An Algebraic Problem Involving the Involutory Integrals of Linear Dynamical Systems”, Amer. J. Math., 62 (1940), 881–911.
H. Koçak, “Linear Hamiltonian Systems are Integrable with Quadratics”, J. Math. Phys., 23:12 (1982), 2375–2380.
A. B. Zheglov and D. V. Osipov, “On First Integrals of Linear Hamiltonian Systems”, Dokl. Math., 98:3 (2018), 616–618.
A. B. Zheglov and D. V. Osipov, “Lax Pairs for Linear Hamiltonian Systems”, Siberian Math. J., 60:4 (2019), 592–604.
D. V. Treschev and A. A. Shkalikov, “On the Hamiltonian Property of Linear Dynamical Systems in Hilbert Space”, Math. Notes, 101:6 (2017), 1033–1039.
V. Volterra, “Sopra una classe di equazioni dinamiche”, Atti Accad. Sci. Torino, 33 (1898), 451–475.
I. A. Bizyaev and V. V. Kozlov,, “Homogeneous Systems with Quadratic Integrals, Lie-Poisson Quasibrackets, and Kovalevskaya’s Method”, Sb. Math., 206:12 (2015), 1682–1706.
N. Bourbaki, “Functions of a Real Variable”, Elements of Mathematics, Springer-Verlag, Berlin, 2004.
A. M. Vershik, “Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?”, Proc. Steklov Inst. Math., 259 (2007), 248–272.
V. I. Bogachev, Gaussian Measures, Nauka, Fizmatlit, Moscow, 1997 (Russian).
P. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950.
L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, N.Y., London, Sydney, Toronto, 1974.
H. Queffélec and M. Queffélec, Diophantine Approximation and Dirichlet Series, Hindustan Book Agency, New Delhi, 2013.
J. von Neumann, “Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik”, Zischr. Phys., 57 (1929), 30–70.
S. Klimek and A. Leśniewski, “Ergodic Theorems for Quantum Kronecker Flows”, Perspectives of quantization, Amer. Math. Soc., Providence, RI, 1998, 71–80.
J. C. Oxtoby, “Ergodic Sets”, Bull. Amer. Math. Soc., 58:2 (1952), 116–136.
V. V. Kozlov, “Weighted Means, Strict Ergodicity, and Uniform Distributions”, Math. Notes, 78:3 (2005), 329–337.
J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932.
F. A. Berezin, Lectures on Statistical Mechanics, In-t Komp. Issled., Moscow, Izhevsk, 2002 (Russian).
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Kozlov, V.V. On the Ergodic Theory of Equations of Mathematical Physics. Russ. J. Math. Phys. 28, 73–83 (2021). https://doi.org/10.1134/S1061920821010088
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DOI: https://doi.org/10.1134/S1061920821010088