The principle of stationary action in the calculus of variations

López, Emanuel   ; Molgado, Alberto   ; Vallejo, José A.

Communications in Mathematics, Tome 20 (2012) no. 2, pp. 89-116

Abstract (VO)

We review some techniques from non-linear 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 clear the ideas here exposed.

Détail

Classification : 34K11, 49K15, 49S05
Keywords: stationary action; functional extrema; conjugate points; oscillatory solutions; Lane-Emden equations

Bibliographie

[1] Adams, R. A.: Sobolev spaces. 1978, Academic Press, | Zbl

[2] Agarwal, R. P., O'Regan, D.: An introduction to ordinary differential equations. 2008, Springer Verlag, | MR | Zbl

[3] Baez, J. C.: Spin foam models. Class. Quant. Grav., 15, 1998, 1827-1858, | DOI | MR | Zbl

[4] Baker, L. M., Fairlie, D. B.: Hamilton-Jacobi equations and Brane associated Lagrangians. Nucl. Phys. B, 596, 2001, 348-364, | DOI | MR | Zbl

[5] Basdevant, J. L.: Variational principles in Physics. 2010, Springer, | MR

[6] Berezin, V.: Square-root quantization: application to quantum black holes. Nucl. Phys. Proc. Suppl., 57, 1997, 181-183, | DOI | MR | Zbl

[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, | MR | Zbl

[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, | DOI | MR

[11] Carlini, A., Greensite, J.: Square Root Actions, Metric Signature, and the Path-Integral of Quantum Gravity. Phys. Rev. D, 52, 1995, 6947-6964, | DOI | MR

[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. | DOI | MR

[13] Chandrasekhar, S.: An Introduction to the Study of Stellar Structure. 1967, Dover publications, | MR

[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, | DOI | MR | Zbl

[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, | MR | Zbl

[17] Emden, R.: Gaskugeln, Anwendungen der mechanischen Warmen-theorie auf Kosmologie und meteorologische Probleme. 1907, B. G. Teubner,

[18] Fiziev, P. P.: Relativistic Hamiltonian with square root in the path integral formalism. Theor. Math. Phys., 62, 2, 1985, 123-130, | DOI | MR

[19] Flett, T. M.: Differential analysis. 1980, Cambridge University Press, | MR | Zbl

[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, | MR

[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,

[22] Fowler, R. H.: The solution of Emden's and similar differential equations. Monthly Notices Roy. Astro. Soc., 91, 1930, 63-91,

[23] Fowler, R. H.: Further studies of Emden's and similar differential equations. Quart. J. Math., 2, 1931, 259-288, | DOI | Zbl

[24] Fox, C.: An introduction to the calculus of variations. 1963, Cambridge University Press, Reprinted by Dover (1987). | MR

[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, | DOI | MR

[26] García, P. L.: The Poincaré-Cartan invariant in the calculus of variations. Symposia Mathematica, 14, 1974, 219-246, | MR | Zbl

[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, | DOI

[28] Gelfand, I. M., Fomin, S. V.: Calculus of variations. 2000, Dover, | Zbl

[29] Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Advanced classical field theory. 2009, World Scientific, | MR | Zbl

[30] Giaquinta, M., Hildebrandt, S.: Calculus of variations I: The Lagrangian formalism. 1996, Springer Verlag, | MR

[31] Goenner, H., Havas, P.: Exact solutions of the generalized Lane--Emden equation. J. Math. Phys., 41, 10, 2000, 7029-7043, | DOI | MR | Zbl

[32] Goldschmidt, H., Sternberg, S.: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier (Grenoble), 23, 1973, 203-267, | DOI | MR | Zbl

[33] Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd. ed. 2001, Addison-Wesley, | MR

[34] González, G.: Relativistic motion with linear dissipation. Int. J. Theor. Phys., 46, 3, 2007, 417-423, | DOI | MR | Zbl

[35] Gray, C. G., Karl, G., Novikov, V. A.: Progress in classical and quantum variational principles. Rep. Prog. Phys., 67, 2004, 159-208, | DOI | MR

[36] Gray, C. G., Poisson, E.: When action is not least for orbits in general relativity. Am. J. Phys., 79, 1, 2011, 43-56, | DOI

[37] Gray, C. G., Taylor, E. F.: When action is not least. Am. J. Phys., 75, 5, 2007, 434-458, | DOI

[38] Hamber, H. W., Williams, R. M.: Discrete gravity in one dimension. Nuclear Physics B, 451, 1995, 305-324, | DOI | MR | Zbl

[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, | DOI | MR

[41] Hermes, H., Lasalle, J. P.: Functional analysis and time optimal control. 1969, Academic Press, | MR | Zbl

[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, | DOI | MR | Zbl

[44] José, J. V., Saletan, E. J.: Classical Dynamics, a contemporary approach. 1998, Cambridge University Press, | MR | Zbl

[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, | DOI

[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, | MR

[48] Krupková, O., (eds.), D. J. Saunders: Variations, geometry and physics. 2009, Nova Science Publishers, | MR | Zbl

[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, | MR | Zbl

[51] Lucha, W., Schöberl, F. F.: Relativistic Coulomb Problem: Energy Levels at the Critical Coupling Constant Analytically. Phys. Lett. B, 387, 1996, 573-576, | DOI

[52] Martyushev, L. M., Seleznev, V. D.: Maximum entropy production principle in physics, chemistry and biology. Physics Reports, 426, 2006, 1-45, | DOI | MR

[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, | MR | Zbl

[54] Moore, T. A.: Getting the Most Action from the Least Action: A proposal. Am. J. Phys., 72, 4, 2004, 522-527, | DOI

[55] Nersesyan, A. P.: Hamiltonian formalism for particles with a generalized rigidity. Theor. Math. Phys., 117, 1, 1998, 1214-1222, | DOI | MR | Zbl

[56] Pars, L. A.: An introduction to the calculus of variations. 1962, Heinemann, Reprinted by Dover (2010).. | MR | Zbl

[57] Puzio, R.: On the square root of the Laplace--Beltrami operator as a Hamiltonian. Class. Quantum Grav., 11, 1994, 609-620, | DOI | MR | Zbl

[58] Rajaraman, R.: Solitons and Instantons. 1988, North--Holland Publishing, | MR

[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, | MR

[61] Rojas, E.: Higher order curvature terms in Born-Infeld type brane theories. Int. J. Mod. Phys. D, 20, 2011, 59-75, | DOI | MR | Zbl

[62] Saunders, D. J.: An alternative approach to the Cartan form in Lagrangian field theories. J. Phys. A, 20, 1987, 339-349, | DOI | MR | Zbl

[63] Sagan, H.: Introduction to the calculus of variations. 1992, Dover, | MR

[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

[66] Stephani, H., Kramer, D., MacCallum, M. A. H., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein's Field Equations. 2003, Cambridge University Press, | MR | Zbl

[67] Stirzaker, D.: Elementary probability. 2003, Cambridge University Press, | MR | Zbl

[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, | DOI

[69] Taylor, E. F.: Guest Editorial: A Call to Action. Am. J. Phys., 71, 5, 2003, 423-425, | DOI

[70] Taylor, J. R.: Classical mechanics. 2005, University Science Books, | Zbl

[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, | MR | Zbl

[73] Brunt, B. Van: The calculus of variations. 2004, Springer Verlag, | MR

[74] Wang, Q. A.: Maximum entropy change and least action principle for non equilibrium systems. Astrophysics and Space Sciences, 305, 2006, 273-281, | DOI

[75] Whittaker, E. T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th. ed. 1937, Dover, | MR

[76] Williams, R. M.: Recent Progress in Regge Calculus. Nucl. Phys. Proc. Suppl., 57, 1997, 73-81, | DOI | MR | Zbl

[77] Wong, J. S. W.: On the generalized Emden-Fowler equation. SIAM Rev., 17, 2, 1975, 339-360, | DOI | MR | Zbl

[78] Zaslavski, A. J.: Turnpike properties in the calculus of variations and optimal control. 2006, Springer Verlag, | MR | Zbl

[79] Zeidler, E.: Nonlinear functional analysis and its applications, Vol. III: Variational methods and optimization. 1986, Springer Verlag,

[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, | DOI | Zbl

[81] Zwiebach, B.: A First Course in String Theory. 2004, Cambridge University Press, | MR | Zbl