Geodesic
The symmetry reduction of variational integrals
Mathematica Bohemica, Tome 143 (2018) no. 3, pp. 291-328
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The Routh reduction of cyclic variables in the Lagrange function and the Jacobi-Maupertuis principle of constant energy systems are generalized. The article deals with one-dimensional variational integral subject to differential constraints, the Lagrange variational problem, that admits the Lie group of symmetries. Reduction to the orbit space is investigated in the absolute sense relieved of all accidental structures. In particular, the widest possible coordinate-free approach to the underdetermined systems of ordinary differential equations, Poincaré-Cartan forms, variations and extremals is involved for the preparation of the main task. The self-contained exposition differs from the common actual theories and rests only on the most fundamental tools of classical mathematical analysis, however, they are applied in infinite-dimensional spaces. The article may be of a certain interest for nonspecialists since all concepts of the calculus of variations undergo a deep reconstruction.
The Routh reduction of cyclic variables in the Lagrange function and the Jacobi-Maupertuis principle of constant energy systems are generalized. The article deals with one-dimensional variational integral subject to differential constraints, the Lagrange variational problem, that admits the Lie group of symmetries. Reduction to the orbit space is investigated in the absolute sense relieved of all accidental structures. In particular, the widest possible coordinate-free approach to the underdetermined systems of ordinary differential equations, Poincaré-Cartan forms, variations and extremals is involved for the preparation of the main task. The self-contained exposition differs from the common actual theories and rests only on the most fundamental tools of classical mathematical analysis, however, they are applied in infinite-dimensional spaces. The article may be of a certain interest for nonspecialists since all concepts of the calculus of variations undergo a deep reconstruction.
DOI : 10.21136/MB.2017.0008-17
Classification : 49N99, 49S05, 70H03
Keywords: Routh reduction; Lagrange variational problem; Poincaré-Cartan form; diffiety; standard basis; controllability; variation 
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Tryhuk, Václav; Chrastinová, Veronika. The symmetry reduction of variational integrals. Mathematica Bohemica, Tome 143 (2018) no. 3, pp. 291-328. doi: 10.21136/MB.2017.0008-17

[1] Adamec, L.: A route to Routh---the classical setting. J. Nonlinear Math. Phys. 18 (2011), 87-107. | DOI | MR | JFM

[2] Adamec, L.: A route to Routh---the parametric problem. Acta Appl. Math. 117 (2012), 115-134. | DOI | MR | JFM

[3] Bażański, S. L.: The Jacobi variational principle revisited. Classical and Quantum Integrability ({W}arsaw, 2001) J. Grabowski et al. Banach Cent. Publ. 59. Polish Academy of Sciences, Institute of Mathematics, Warsaw (2003), 99-111. | DOI | MR | JFM

[4] Capriotti, S.: Routh reduction and Cartan mechanics. J. Geom. Phys. 114 (2017), 23-64. | DOI | MR | JFM

[5] Cartan, É.: Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes. S. M. F. Bull 42 (1914), 12-48 \99999JFM99999 45.1294.04. | DOI | MR

[6] Cartan, É.: Leçons sur les invariants intégraux. Hermann, Paris (1971). | MR | JFM

[7] Chrastina, J.: The Formal Theory of Differential Equations. Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis. Mathematica 6. Masaryk University, Brno (1998). | MR | JFM

[8] Chrastinová, V., Tryhuk, V.: On the internal approach to differential equations 1. The involutiveness and standard basis. Math. Slovaca 66 (2016), 999-1018. | DOI | MR | JFM

[9] Chrastinová, V., Tryhuk, V.: On the internal approach to differential equations 3. Infinitesimal symmetries. Math. Slovaca 66 (2016), 1459-1474. | DOI | MR

[10] Chrastinová, V., Tryhuk, V.: On the internal approach to differential equations 2. The controllability structure. Math. Slovaca 67 (2017), 1011-1030. | DOI | MR | JFM

[11] Crampin, M., Mestdag, T.: Routh's procedure for non-abelian symmetry groups. J. Math. Phys. 49 (2008), Article ID 032901, 28 pages. | DOI | MR | JFM

[12] Fuller, A. T.: Stability of Motion. A collection of early scientific papers by Routh, Clifford, Sturm and Bocher. Reprint. Taylor & Francis, London (1975). | MR | JFM

[13] Griffiths, P. A.: Exterior Differential Systems and The Calculus of Variations. Progress in Mathematics 25. Birkhäuser, Boston (1983). | DOI | MR | JFM

[14] Hermann, R.: Differential form methods in the theory of variational systems and Lagrangian field theories. Acta Appl. Math. 12 (1988), 35-78. | DOI | MR | JFM

[15] Hilbert, D.: Über den Begriff der Klasse von Differentialgleichungen. Math. Ann. 73 (1912), 95-108 \99999JFM99999 43.0378.01. | DOI | MR

[16] Krasil'shchik, I. S., Lychagin, V. V., Vinogradov, A. M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Advanced Studies in Contemporary Mathematics 1. Gordon and Breach Science Publishers, New York (1986). | MR | JFM

[17] Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Mathematics and Its Applications 35. D. Reidel Publishing, Dordrecht (1987). | DOI | MR | JFM

[18] Lie, S.: Vorlesungen über Differentialgleichungen mit bekanten infinitesimale Transformationen. Teubner, Leipzig (1891),\99999JFM99999 23.0351.01.

[19] Marsden, J. E., Ratiu, T. S., Scheurle, J.: Reduction theory and the Lagrange-Routh equations. J. Math. Phys. 41 (2000), 3379-3429. | DOI | MR | JFM

[20] Mestdag, T.: Finsler geodesics of Lagrangian systems through Routh reduction. Mediterr. J. Math. 13 (2016), 825-839. | DOI | MR | JFM

[21] Tryhuk, V., Chrastinová, V.: On the mapping of jet spaces. J. Nonlinear Math. Phys. 17 (2010), 293-310. | DOI | MR | JFM

[22] Tryhuk, V., Chrastinová, V.: Automorphisms of ordinary differential equations. Abstr. Appl. Anal. (2014), Article ID 482963, 32 pages. | DOI | MR

[23] Tryhuk, V., Chrastinová, V., Dlouhý, O.: The Lie group in infinite dimension. Abstr. Appl. Anal. 2011 (2011), Article ID 919538, 35 pages. | DOI | MR | JFM

[24] Vessiot, E.: Sur l'intégration des systèmes différentiels qui admettent des groupes continus de transformations. Acta Math. 28 (1904), 307-349. | DOI | MR

[25] Vinogradov, A. M.: Cohomological Analysis of Partial Differential Equations and Secondary Calculus. Translations of Mathematical Monographs 204. AMS, Providence (2001). | DOI | MR | JFM

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