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The principle of stationary action in the calculus of variations. (English) Zbl 1272.49100
Summary: We review some techniques from nonlinear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Our main intention in this regard is to expose precise mathematical conditions for critical paths to be minimum solutions in a variety of situations of interest in physics. Our claim is that, with a few elementary techniques, a systematic analysis (including the domain for which critical points are genuine minima) of non-trivial models is possible. We present specific models arising in modern physical theories in order to make the ideas exposed here clear.
MSC:
49S05 Variational principles of physics
49K15 Optimality conditions for problems involving ordinary differential equations
34K11 Oscillation theory of functional-differential equations
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References:
[1] Adams, R. A.: Sobolev spaces. 1978, Academic Press, · Zbl 0347.46040
[2] Agarwal, R. P., O’Regan, D.: An introduction to ordinary differential equations. 2008, Springer Verlag, · Zbl 1158.34001
[3] Baez, J. C.: Spin foam models. Class. Quant. Grav., 15, 1998, 1827-1858, · Zbl 0932.83014
[4] Baker, L. M., Fairlie, D. B.: Hamilton-Jacobi equations and Brane associated Lagrangians. Nucl. Phys. B, 596, 2001, 348-364, · Zbl 0972.81146
[5] Basdevant, J. L.: Variational principles in Physics. 2010, Springer, · Zbl 1149.49037
[6] Berezin, V.: Square-root quantization: application to quantum black holes. Nucl. Phys. Proc. Suppl., 57, 1997, 181-183, · Zbl 0976.83531
[7] Binney, J., Tremain, S.: Galactic dynamics. 1994, Princeton University Press,
[8] Buck, B., (eds), V. A. Macaulay: Maximum Entropy in Action: A Collection of Expository Essays. 1991, Oxford University Press,
[9] Burghes, D. N., Graham, A.: Introduction to Control Theory, Including Optimal Control. 1980, Wiley, · Zbl 0428.93001
[10] Carlini, A., Frolov, V. P., Mensky, M. B., Novikov, I. D., Soleng, H. H.: Time machines: the Principle of Self-Consistency as a consequence of the Principle of Minimal Action. Int. J. Mod. Phys. D, 4, 1995, 557-580,
[11] Carlini, A., Greensite, J.: Square Root Actions, Metric Signature, and the Path-Integral of Quantum Gravity. Phys. Rev. D, 52, 1995, 6947-6964,
[12] Carlip, S.: \((2+1)\)-Dimensional Chern-Simons Gravity as a Dirac Square Root. Phys. Rev. D, 45, 1992, 3584-3590, Erratum-ibid D47 (1993) 1729.
[13] Chandrasekhar, S.: An Introduction to the Study of Stellar Structure. 1967, Dover publications, · Zbl 0022.19207
[14] Coffman, C. V., Wong, J. S. V.: Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations. Transactions of the AMS, 167, 1972, 399-434, · Zbl 0278.34026
[15] Collins, G. W.: The fundamentals of stellar astrophysics. 1989, Freeman,
[16] Curtain, R. F., Pritchard, A. J.: Functional analysis in modern applied mathematics. 1977, Academic Press, · Zbl 0448.46002
[17] Emden, R.: Gaskugeln, Anwendungen der mechanischen Warmen-theorie auf Kosmologie und meteorologische Probleme. 1907, B. G. Teubner, · JFM 38.0952.02
[18] Fiziev, P. P.: Relativistic Hamiltonian with square root in the path integral formalism. Theor. Math. Phys., 62, 2, 1985, 123-130,
[19] Flett, T. M.: Differential analysis. 1980, Cambridge University Press, · Zbl 0442.34002
[20] Fowler, R. H.: Some results on the form near infinity of real continuous solutions of a certain type of second order differential equations. Proc. London Math. Soc., 13, 1914, 341-371,
[21] Fowler, R. H.: The form near infinity of real, continuous solutions of a certain differential equation of the second order. Quart. J. Math., 45, 1914, 289-350, · JFM 45.0479.01
[22] Fowler, R. H.: The solution of Emden’s and similar differential equations. Monthly Notices Roy. Astro. Soc., 91, 1930, 63-91, · JFM 56.0389.02
[23] Fowler, R. H.: Further studies of Emden’s and similar differential equations. Quart. J. Math., 2, 1931, 259-288, · Zbl 0003.23502
[24] Fox, C.: An introduction to the calculus of variations. 1963, Cambridge University Press, Reprinted by Dover (1987). · Zbl 0041.42801
[25] Friedman, J. L., Louko, J., Winters-Hilt, S. N.: Reduced phase space formalism for spherically symmetric geometry with a massive dust shell. Phys. Rev. D, 56, 1997, 7674-7691,
[26] García, P. L.: The Poincaré-Cartan invariant in the calculus of variations. Symposia Mathematica, 14, 1974, 219-246, · Zbl 0303.53040
[27] Garrett, B. C., Abusalbi, N., Kouri, D. J., Truhlar, D. G.: Test of variational transition state theory and the least action approximation for multidimensional tunneling probabilities against accurate quantal rate constants for a collinear reaction involving tunneling into an excited state. J. Chem. Phys., 83, 1985, 2252-2258,
[28] Gelfand, I. M., Fomin, S. V.: Calculus of variations. 2000, Dover, · Zbl 0964.49001
[29] Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Advanced classical field theory. 2009, World Scientific, · Zbl 1179.81002
[30] Giaquinta, M., Hildebrandt, S.: Calculus of variations I: The Lagrangian formalism. 1996, Springer Verlag, · Zbl 0853.49001
[31] Goenner, H., Havas, P.: Exact solutions of the generalized Lane–Emden equation. J. Math. Phys., 41, 10, 2000, 7029-7043, · Zbl 1009.34002
[32] Goldschmidt, H., Sternberg, S.: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier (Grenoble), 23, 1973, 203-267, · Zbl 0243.49011
[33] Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd. ed. 2001, Addison-Wesley, · Zbl 1132.70001
[34] González, G.: Relativistic motion with linear dissipation. Int. J. Theor. Phys., 46, 3, 2007, 417-423, · Zbl 1115.83002
[35] Gray, C. G., Karl, G., Novikov, V. A.: Progress in classical and quantum variational principles. Rep. Prog. Phys., 67, 2004, 159-208,
[36] Gray, C. G., Poisson, E.: When action is not least for orbits in general relativity. Am. J. Phys., 79, 1, 2011, 43-56,
[37] Gray, C. G., Taylor, E. F.: When action is not least. Am. J. Phys., 75, 5, 2007, 434-458,
[38] Hamber, H. W., Williams, R. M.: Discrete gravity in one dimension. Nuclear Physics B, 451, 1995, 305-324, · Zbl 0925.83004
[39] Hand, L. N., Finch, J. D.: Analytical Mechanics. 1998, Cambridge University Press,
[40] Harremoës, P., Topsøe, F.: Maximum Entropy Fundamentals. Entropy, 3, 2001, 191-226, · Zbl 1006.94007
[41] Hermes, H., Lasalle, J. P.: Functional analysis and time optimal control. 1969, Academic Press, · Zbl 0203.47504
[42] Horedt, G. P.: Polytropes: Applications in astrophysics and related fields. 2004, Kluwer,
[43] Jaynes, E. T.: Information theory and statistical mechanics. Phys. Rev., 106, 1957, 620-630, · Zbl 0084.43701
[44] José, J. V., Saletan, E. J.: Classical Dynamics, a contemporary approach. 1998, Cambridge University Press, · Zbl 0918.70001
[45] Kapustnikov, A. A., Pashnev, A., Pichugin, A.: The canonical quantization of the kink–model beyond the static solution. Phys. Rev. D, 55, 1997, 2257-2264,
[46] Klein, J. F.: Physical significance of entropy or of the second law. 2009, Cornell University Library,
[47] Krupková, O.: The geometry of ordinary variational equations, Lecture Notes in Mathematics 1678. Springer Verlag, 1997, · Zbl 0936.70001
[48] Krupková, O., (eds.), D. J. Saunders: Variations, geometry and physics. 2009, Nova Science Publishers, · Zbl 1209.58002
[49] Lane, I. J. H.: On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestial experiment. Amer. J. Sci. and Arts, 4, 1870, 57-74,
[50] Lebedev, L. P., Cloud, M. J.: The calculus of variations and functional analysis (with optimal control and applications in mechanics). 2003, World Scientific, · Zbl 1042.49001
[51] Lucha, W., Schöberl, F. F.: Relativistic Coulomb Problem: Energy Levels at the Critical Coupling Constant Analytically. Phys. Lett. B, 387, 1996, 573-576,
[52] Martyushev, L. M., Seleznev, V. D.: Maximum entropy production principle in physics, chemistry and biology. Physics Reports, 426, 2006, 1-45,
[53] Menotti, P.: Hamiltonian structure of 2+1 dimensional gravity. Recent developments in general relativity, 14th SIGRAV Conference on General Relativity and Gravitational Physics, Genova, Italy (2000), 2002, 165-177, Springer, · Zbl 1202.83095
[54] Moore, T. A.: Getting the Most Action from the Least Action: A proposal. Am. J. Phys., 72, 4, 2004, 522-527,
[55] Nersesyan, A. P.: Hamiltonian formalism for particles with a generalized rigidity. Theor. Math. Phys., 117, 1, 1998, 1214-1222, · Zbl 1086.37523
[56] Pars, L. A.: An introduction to the calculus of variations. 1962, Heinemann, Reprinted by Dover (2010).. · Zbl 0108.10303
[57] Puzio, R.: On the square root of the Laplace–Beltrami operator as a Hamiltonian. Class. Quantum Grav., 11, 1994, 609-620, · Zbl 0812.32009
[58] Rajaraman, R.: Solitons and Instantons. 1988, North–Holland Publishing, · Zbl 0493.35074
[59] Ramond, P.: Field theory: A modern primer (Frontiers in Physics series Vol. 74). 2001, Westview Press,
[60] Razavy, M.: Classical And Quantum Dissipative Systems. 2006, Imperial College Press, · Zbl 1105.82014
[61] Rojas, E.: Higher order curvature terms in Born-Infeld type brane theories. Int. J. Mod. Phys. D, 20, 2011, 59-75, · Zbl 1213.83122
[62] Saunders, D. J.: An alternative approach to the Cartan form in Lagrangian field theories. J. Phys. A, 20, 1987, 339-349, · Zbl 0652.58002
[63] Sagan, H.: Introduction to the calculus of variations. 1992, Dover,
[64] Simmons, G. F., Krantz, S. G.: Differential Equations: Theory, Technique, and Practice. 2006, McGraw-Hill,
[65] Smith, D.: Variational methods in optimization. 1998, Dover, · Zbl 0918.49001
[66] Stephani, H., Kramer, D., MacCallum, M. A. H., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. 2003, Cambridge University Press, · Zbl 1057.83004
[67] Stirzaker, D.: Elementary probability. 2003, Cambridge University Press, · Zbl 1071.60002
[68] Sussmann, H. J., Willems, J. C.: \(300\) years of optimal control: from the brachystochrone problem to the maximum principle. IEEE Control Systems, 17, 1997, 32-44,
[69] Taylor, E. F.: Guest Editorial: A Call to Action. Am. J. Phys., 71, 5, 2003, 423-425,
[70] Taylor, J. R.: Classical mechanics. 2005, University Science Books, · Zbl 1075.70002
[71] Thornton, S. T., Marion, J. B.: Classical Dynamics of Particles and Systems. 2004, Brooks/Cole,
[72] Troutman, J. L.: Variational Calculus and Optimal Control: Optimization With Elementary Convexity. 1996, Springer Verlag, · Zbl 0865.49001
[73] Brunt, B. Van: The calculus of variations. 2004, Springer Verlag, · Zbl 1039.49001
[74] Wang, Q. A.: Maximum entropy change and least action principle for non equilibrium systems. Astrophysics and Space Sciences, 305, 2006, 273-281, · Zbl 1104.82038
[75] Whittaker, E. T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th. ed. 1937, Dover, · Zbl 0061.41806
[76] Williams, R. M.: Recent Progress in Regge Calculus. Nucl. Phys. Proc. Suppl., 57, 1997, 73-81, · Zbl 0976.83506
[77] Wong, J. S. W.: On the generalized Emden-Fowler equation. SIAM Rev., 17, 2, 1975, 339-360, · Zbl 0295.34026
[78] Zaslavski, A. J.: Turnpike properties in the calculus of variations and optimal control. 2006, Springer Verlag, · Zbl 1100.49003
[79] Zeidler, E.: Nonlinear functional analysis and its applications, Vol. III: Variational methods and optimization. 1986, Springer Verlag, · Zbl 0583.47050
[80] Zloshchastiev, K. G.: Quantum kink model and \(SU(2)\) symmetry: Spin interpretation and T-violation. J. Phys. A: Math. Gen., 31, 1998, 6081-6085, · Zbl 0954.81046
[81] Zwiebach, B.: A First Course in String Theory. 2004, Cambridge University Press, · Zbl 1072.81001
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