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Revisiting Noether’s theorem on constants of motion. (English) Zbl 1420.34059

Summary: In this paper we revisit Noether’s theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, with some new perspectives on both the theoretical and the applied side. We make full use of invariance up to a divergence, or, as we call it here, Bessel-Hagen (BH) invariance. By recognizing that the Bessel-Hagen (BH) function need not be a total time derivative, we can easily deduce nonlocal constants of motion. We prove that we can always trivialize either the time change or the BH-function, so that, in particular, BH-invariance turns out not to be more general than Noether’s original invariance. We also propose a version of time change that simplifies some key formulas. Applications include Lane-Emden equation, dissipative systems, homogeneous potentials and superintegrable systems. Most notably, we give two derivations of the Laplace-Runge-Lenz vector for Kepler’s problem that require space and time change only, without BH invariance, one with and one without use of the Lagrange equation.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
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[1] Arnold, V. I., Mathematical Methods of Classical Mechanics (1989), Springer-Verlag · Zbl 0692.70003
[2] Bessel-Hagen, E., Über die Erhaltungssätze der Elektrodynamik, Math. Ann, 84, 258-276 (1921) · JFM 48.0877.02
[3] Bluman, G. W.; Kumei, S., Symmetries and Differential Equations (1989), Springer-Verlag · Zbl 0698.35001
[4] Bobillo-Ares, N., Noether’s theorem in discrete classical mechanics, Am. J. Phys, 56, 174-177 (1988)
[5] Boyer, T. H., Continuous symmetries and conserved currents, Annals of Physics, 42, 445-466 (1967) · Zbl 0149.21602
[6] Chandrasekhar, S., An introduction to the study of stellar structure (1939), Chicago: The University of Chicago Press, Chicago · Zbl 0022.19207
[7] Danby, J. M.A., Fundamentals of Celestial Mechanics (1988), Richmond, VA: Willmann-Bell, Inc., Richmond, VA
[8] Desloge, E. A.; Karch, R. I., Noether’s theorem in classical mechanics, Am. J. Phys, 45, 4 (1977)
[9] Djukic, D. J. S., A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians. Internat, J. Non-Linear Mech, 8, 479-488 (1973) · Zbl 0269.70012
[10] Giaquinta, M.; Hildebrandt, S., Calculus of Variations I. Grundlehren der mathematischen Wissenschaften, 310 (1996), Springer-Verlag · Zbl 0853.49001
[11] Gorni, G.; Zampieri, G., A class of integrable Hamiltonian systems including scattering of particles on the line with repulsive interactions, Differential and Integral Equations, 4, 2, 305-329 (1991) · Zbl 0722.34046
[12] Gorni, G.; Zampieri, G., Variational aspects of analytical mechanics, São Paulo J. Math. Sci, 5, 2, 203-231 (2011)
[13] Govinder, K. S.; Leach, P. G.L., Noetherian integrals via nonlocal transformation, Physics Letters A, 201, 91-94 (1995) · Zbl 1020.70504
[14] Kobussen, J. A., On a systematic search for integrals of motion, Helv. Phys. Acta, 53, 183-200 (1980)
[15] Kosmann-Schwarzbach, Y., The Noether theorems (2011), Springer-Verlag · Zbl 1216.01011
[16] Leach, P. G.L.; Flessas, G. P., Noetherian first integrals, Journal of Nonlinear Mathematical Physics, 15, 1, 9-21 (2008) · Zbl 1160.70008
[17] Leach, P. G.L., Lie symmetries and Noether symmetries, Applicable Analysis and Discrete Mathematics, 6, 238-246 (2012) · Zbl 1289.70027
[18] Lévy-Leblond, J., Conservation laws for gauge-variant Lagrangians in classical mechanics, Am. J. Phys, 39, 502-506 (1971)
[19] Logan, J. D., Invariant Variational Principles (1977), Academic Press
[20] Mach, P., All solutions of the n = 5 Lane-Emden equation, J. Math. Phys, 53, 6, 6 (2012) · Zbl 1282.34017
[21] Moser, J., Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Mathematics, 16, 197-220 (1975) · Zbl 0303.34019
[22] Moser, J., Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), 38, Finitely many mass points on the line under the influence of an exponential potential—an integrable system, 467-497 (1975), Springer: Springer, Berlin · Zbl 0323.70012
[23] Neuenschwander, D. E., Emmy Noether’s wonderful theorem (2011), John Hopkins University Press · Zbl 1239.00018
[24] Noether, E., Nachr. d. Kö nig. Gesellsch. d. Wiss. zu Göttingen, Mathphys. Klasse, Invariante Variationsprobleme, 235-257 (1918) · JFM 46.0770.01
[25] Olver, P. J., Graduate Texts in Mathematics, 107, Applications of Lie groups to differential equations. Second Edition (1975), Springer-Verlag
[26] Rund, H., A direct approach to Noether’s theorem in the calculus of variations, Util. Math, 2, 205-214 (1972) · Zbl 0256.49019
[27] Sarlet, W.; Cantrijn, F., Generalizations of Noether’s theorem in classical mechanics, SIAM Review, 23, 4, 467-494 (1981) · Zbl 0474.70014
[28] Trautman, A., Noether’s equations and conservation laws, Commun. Math. Phys, 6, 248-261 (1967) · Zbl 0172.27803
[29] Whittaker, E. T., A treatise on the analytical dynamics of particles and rigid bodies (1961), Cambridge University Press
[30] Zampieri, G., Completely integrable Hamiltonian systems with weak Lyapunov instability or isochrony, Comm. Math. Phys, 303, 73-87 (2011) · Zbl 1220.37063
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