Mathematica Bohemica, Vol. 143, No. 3, pp. 291-328, 2018
The symmetry reduction of variational integrals
Václav Tryhuk, Veronika Chrastinová
Received January 19, 2017. First published December 14, 2017.
Abstract: 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.
References: [1] L. Adamec: A route to Routh - the classical setting. J. Nonlinear Math. Phys. 18 (2011), 87-107. DOI 10.1142/S1402925111001180 | MR 2786936 | Zbl 1217.49040
[2] L. Adamec: A route to Routh - the parametric problem. Acta Appl. Math. 117 (2012), 115-134. DOI 10.1007/s10440-011-9654-2 | MR 2886172 | Zbl 1231.49038
[3] S. L. Bazański: The Jacobi variational principle revisited. Classical and Quantum Integrability (Warsaw, 2001) (J. Grabowski et al.). Banach Cent. Publ. 59. Polish Academy of Sciences, Institute of Mathematics, Warsaw (2003), 99-111. DOI 10.4064/bc59-0-4 | MR 2003718 | Zbl 1082.70008
[4] S. Capriotti: Routh reduction and Cartan mechanics. J. Geom. Phys. 114 (2017), 23-64. DOI 10.1016/j.geomphys.2016.11.015 | MR 3610032 | Zbl 06688396
[5] É. Cartan: Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes. S. M. F. Bull 42 (1914), 12-48. DOI 10.24033/bsmf.938 | MR 1504722 | JFM 45.1294.04
[6] É. Cartan: Leçons sur les invariants intégraux. Hermann, Paris (1971). MR 0355764 | Zbl 0212.12501
[7] J. Chrastina: The Formal Theory of Differential Equations. Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis. Mathematica 6. Masaryk University, Brno (1998). MR 1656843 | Zbl 0906.35002
[8] V. Chrastinová, V. Tryhuk: On the internal approach to differential equations 1. The involutiveness and standard basis. Math. Slovaca 66 (2016), 999-1018. DOI 10.1515/ms-2015-0198 | MR 3567912 | Zbl 06662105
[9] V. Chrastinová, V. Tryhuk: On the internal approach to differential equations 3. Infinitesimal symmetries. Math. Slovaca 66 (2016), 1459-1474. DOI 10.1515/ms-2016-0236 | MR 3619165
[10] V. Chrastinová, V. Tryhuk: On the internal approach to differential equations 2. The controllability structure. Math. Slovaca 67 (2017), 1011-1030. DOI 10.1515/ms-2017-0029 | MR 3674124 | Zbl 06760210
[11] M. Crampin, T. Mestdag: Routh's procedure for non-abelian symmetry groups. J. Math. Phys. 49 (2008), Article ID 032901, 28 pages. DOI 10.1063/1.2885077 | MR 2406795 | Zbl 1153.37396
[12] A. T. Fuller: Stability of Motion. A collection of early scientific papers by Routh, Clifford, Sturm and Bocher. Reprint. Taylor & Francis, London (1975). Zbl 0312.34033
[13] P. A. Griffiths: Exterior Differential Systems and The Calculus of Variations. Progress in Mathematics 25. Birkhäuser, Boston (1983). DOI 10.1007/978-1-4615-8166-6 | MR 0684663 | Zbl 0512.49003
[14] R. Hermann: Differential form methods in the theory of variational systems and Lagrangian field theories. Acta Appl. Math. 12 (1988), 35-78. DOI 10.1007/BF00047568 | MR 0962880 | Zbl 0664.49018
[15] D. Hilbert: Über den Begriff der Klasse von Differentialgleichungen. Math. Ann. 73 (1912), 95-108. DOI 10.1007/BF01456663 | MR 1511723 | JFM 43.0378.01
[16] I. S. Krasil'shchik, V. V. Lychagin, A. M. Vinogradov: Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Advanced Studies in Contemporary Mathematics 1. Gordon and Breach Science Publishers, New York (1986). MR 0861121 | Zbl 0722.35001
[17] P. Libermann, C.-M. Marle: Symplectic Geometry and Analytical Mechanics. Mathematics and Its Applications 35. D. Reidel Publishing, Dordrecht (1987). DOI 10.1007/978-94-009-3807-6 | MR 0882548 | Zbl 0643.53002
[18] S. Lie: Vorlesungen über Differentialgleichungen mit bekanten infinitesimale Transformationen. Teubner, Leipzig (1891). JFM 23.0351.01
[19] J. E. Marsden, T. S. Ratiu, J. Scheurle: Reduction theory and the Lagrange-Routh equations. J. Math. Phys. 41 (2000), 3379-3429. DOI 10.1063/1.533317 | MR 1768627 | Zbl 1044.37043
[20] T. Mestdag: Finsler geodesics of Lagrangian systems through Routh reduction. Mediterr. J. Math. 13 (2016), 825-839. DOI 10.1007/s00009-014-0505-z | MR 3483865 | Zbl 1338.53103
[21] V. Tryhuk, V. Chrastinová: On the mapping of jet spaces. J. Nonlinear Math. Phys. 17 (2010), 293-310. DOI 10.1142/S140292511000091X | MR 2733205 | Zbl 1207.58004
[22] V. Tryhuk, V. Chrastinová: Automorphisms of ordinary differential equations. Abstr. Appl. Anal. (2014), Article ID 482963, 32 pages. DOI 10.1155/2014/482963 | MR 3166619
[23] V. Tryhuk, V. Chrastinová, O. Dlouhý: The Lie group in infinite dimension. Abstr. Appl. Anal. 2011 (2011), Article ID 919538, 35 pages. DOI 10.1155/2011/919538 | MR 2771243 | Zbl 1223.22018
[24] E. Vessiot: Sur l'intégration des systèmes différentiels qui admettent des groupes continus de transformations. Acta Math. 28 (1904), 307-349. DOI 10.1007/BF02418390 | MR 1555005 | Zbl 0
[25] A. M. Vinogradov: Cohomological Analysis of Partial Differential Equations and Secondary Calculus. Translations of Mathematical Monographs 204. AMS, Providence (2001). MR 1857908 | Zbl 1152.58308
Affiliations: Václav Tryhuk, Brno University of Technology, Faculty of Civil Engineering, AdMaS centre, Purkyňova 139, 612 00 Brno, Czech Republic, e-mail: tryhuk.v@fce.vutbr.cz, tryhuk.v@outlook.com; Veronika Chrastinová, Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics, Veveří 331/95, 602 00 Brno, Czech Republic, e-mail: chrastinova.v@fce.vutbr.cz
