The symmetry reduction of variational integrals

Václav Tryhuk; Veronika Chrastinová

Mathematica Bohemica (2018)

  • Volume: 143, Issue: 3, page 291-328
  • ISSN: 0862-7959

Abstract

top
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.

How to cite

top

Tryhuk, Václav, and Chrastinová, Veronika. "The symmetry reduction of variational integrals." Mathematica Bohemica 143.3 (2018): 291-328. <http://eudml.org/doc/294419>.

@article{Tryhuk2018,
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.},
author = {Tryhuk, Václav, Chrastinová, Veronika},
journal = {Mathematica Bohemica},
keywords = {Routh reduction; Lagrange variational problem; Poincaré-Cartan form; diffiety; standard basis; controllability; variation},
language = {eng},
number = {3},
pages = {291-328},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The symmetry reduction of variational integrals},
url = {http://eudml.org/doc/294419},
volume = {143},
year = {2018},
}

TY - JOUR
AU - Tryhuk, Václav
AU - Chrastinová, Veronika
TI - The symmetry reduction of variational integrals
JO - Mathematica Bohemica
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 143
IS - 3
SP - 291
EP - 328
AB - 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.
LA - eng
KW - Routh reduction; Lagrange variational problem; Poincaré-Cartan form; diffiety; standard basis; controllability; variation
UR - http://eudml.org/doc/294419
ER -

References

top
  1. Adamec, L., 10.1142/S1402925111001180, J. Nonlinear Math. Phys. 18 (2011), 87-107. (2011) Zbl1217.49040MR2786936DOI10.1142/S1402925111001180
  2. Adamec, L., 10.1007/s10440-011-9654-2, Acta Appl. Math. 117 (2012), 115-134. (2012) Zbl1231.49038MR2886172DOI10.1007/s10440-011-9654-2
  3. Bażański, S. L., 10.4064/bc59-0-4, 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. (2003) Zbl1082.70008MR2003718DOI10.4064/bc59-0-4
  4. Capriotti, S., 10.1016/j.geomphys.2016.11.015, J. Geom. Phys. 114 (2017), 23-64. (2017) Zbl06688396MR3610032DOI10.1016/j.geomphys.2016.11.015
  5. Cartan, É., 10.24033/bsmf.938, S. M. F. Bull 42 (1914), 12-48 9999JFM99999 45.1294.04. (1914) MR1504722DOI10.24033/bsmf.938
  6. Cartan, É., Leçons sur les invariants intégraux, Hermann, Paris (1971). (1971) Zbl0212.12501MR0355764
  7. Chrastina, J., The Formal Theory of Differential Equations, Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis. Mathematica 6. Masaryk University, Brno (1998). (1998) Zbl0906.35002MR1656843
  8. Chrastinová, V., Tryhuk, V., 10.1515/ms-2015-0198, Math. Slovaca 66 (2016), 999-1018. (2016) Zbl06662105MR3567912DOI10.1515/ms-2015-0198
  9. Chrastinová, V., Tryhuk, V., 10.1515/ms-2016-0236, Math. Slovaca 66 (2016), 1459-1474. (2016) MR3619165DOI10.1515/ms-2016-0236
  10. Chrastinová, V., Tryhuk, V., 10.1515/ms-2017-0029, Math. Slovaca 67 (2017), 1011-1030. (2017) Zbl06760210MR3674124DOI10.1515/ms-2017-0029
  11. Crampin, M., Mestdag, T., 10.1063/1.2885077, J. Math. Phys. 49 (2008), Article ID 032901, 28 pages. (2008) Zbl1153.37396MR2406795DOI10.1063/1.2885077
  12. Fuller, A. T., Stability of Motion. A collection of early scientific papers by Routh, Clifford, Sturm and Bocher, Reprint. Taylor & Francis, London (1975). (1975) Zbl0312.34033MR0371578
  13. Griffiths, P. A., 10.1007/978-1-4615-8166-6, Progress in Mathematics 25. Birkhäuser, Boston (1983). (1983) Zbl0512.49003MR0684663DOI10.1007/978-1-4615-8166-6
  14. Hermann, R., 10.1007/BF00047568, Acta Appl. Math. 12 (1988), 35-78. (1988) Zbl0664.49018MR0962880DOI10.1007/BF00047568
  15. Hilbert, D., 10.1007/BF01456663, Math. Ann. 73 (1912), 95-108 9999JFM99999 43.0378.01. (1912) MR1511723DOI10.1007/BF01456663
  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). (1986) Zbl0722.35001MR0861121
  17. Libermann, P., Marle, C.-M., 10.1007/978-94-009-3807-6, Mathematics and Its Applications 35. D. Reidel Publishing, Dordrecht (1987). (1987) Zbl0643.53002MR0882548DOI10.1007/978-94-009-3807-6
  18. Lie, S., Vorlesungen über Differentialgleichungen mit bekanten infinitesimale Transformationen, Teubner, Leipzig (1891),9999JFM99999 23.0351.01. (1891) 
  19. Marsden, J. E., Ratiu, T. S., Scheurle, J., 10.1063/1.533317, J. Math. Phys. 41 (2000), 3379-3429. (2000) Zbl1044.37043MR1768627DOI10.1063/1.533317
  20. Mestdag, T., 10.1007/s00009-014-0505-z, Mediterr. J. Math. 13 (2016), 825-839. (2016) Zbl1338.53103MR3483865DOI10.1007/s00009-014-0505-z
  21. Tryhuk, V., Chrastinová, V., 10.1142/S140292511000091X, J. Nonlinear Math. Phys. 17 (2010), 293-310. (2010) Zbl1207.58004MR2733205DOI10.1142/S140292511000091X
  22. Tryhuk, V., Chrastinová, V., 10.1155/2014/482963, Abstr. Appl. Anal. (2014), Article ID 482963, 32 pages. (2014) MR3166619DOI10.1155/2014/482963
  23. Tryhuk, V., Chrastinová, V., Dlouhý, O., 10.1155/2011/919538, Abstr. Appl. Anal. 2011 (2011), Article ID 919538, 35 pages. (2011) Zbl1223.22018MR2771243DOI10.1155/2011/919538
  24. Vessiot, E., 10.1007/BF02418390, Acta Math. 28 (1904), 307-349. (1904) MR1555005DOI10.1007/BF02418390
  25. Vinogradov, A. M., 10.1090/mmono/204/01, Translations of Mathematical Monographs 204. AMS, Providence (2001). (2001) Zbl1152.58308MR1857908DOI10.1090/mmono/204/01

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.