# Noether theorem and first integrals of constrained Lagrangean systems

Mathematica Bohemica (1997)

• Volume: 122, Issue: 3, page 257-265
• ISSN: 0862-7959

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## Abstract

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The dynamics of singular Lagrangean systems is described by a distribution the rank of which is greater than one and may be non-constant. Consequently, these systems possess two kinds of conserved functions, namely, functions which are constant along extremals (constants of the motion), and functions which are constant on integral manifolds of the corresponding distribution (first integrals). It is known that with the help of the (First) Noether theorem one gets constants of the motion. In this paper it is shown that every constant of the motion obtained from the Noether theorem is a first integral; thus, Noether theorem can be used for an effective integration of the corresponding distribution.

## How to cite

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Krupková, Olga. "Noether theorem and first integrals of constrained Lagrangean systems." Mathematica Bohemica 122.3 (1997): 257-265. <http://eudml.org/doc/248128>.

@article{Krupková1997,
abstract = {The dynamics of singular Lagrangean systems is described by a distribution the rank of which is greater than one and may be non-constant. Consequently, these systems possess two kinds of conserved functions, namely, functions which are constant along extremals (constants of the motion), and functions which are constant on integral manifolds of the corresponding distribution (first integrals). It is known that with the help of the (First) Noether theorem one gets constants of the motion. In this paper it is shown that every constant of the motion obtained from the Noether theorem is a first integral; thus, Noether theorem can be used for an effective integration of the corresponding distribution.},
author = {Krupková, Olga},
journal = {Mathematica Bohemica},
keywords = {Lagrangian system; Lepagean two-form; Euler-Lagrange form; singular Lagrangian; constrained system; Noether theorem; symmetry; constants of motion; first integrals; Lagrangian system; Lepagean two-form; Euler-Lagrange form; singular Lagrangian; constrained system; Noether theorem; symmetry; constants of motion; first integrals},
language = {eng},
number = {3},
pages = {257-265},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Noether theorem and first integrals of constrained Lagrangean systems},
url = {http://eudml.org/doc/248128},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Krupková, Olga
TI - Noether theorem and first integrals of constrained Lagrangean systems
JO - Mathematica Bohemica
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 122
IS - 3
SP - 257
EP - 265
AB - The dynamics of singular Lagrangean systems is described by a distribution the rank of which is greater than one and may be non-constant. Consequently, these systems possess two kinds of conserved functions, namely, functions which are constant along extremals (constants of the motion), and functions which are constant on integral manifolds of the corresponding distribution (first integrals). It is known that with the help of the (First) Noether theorem one gets constants of the motion. In this paper it is shown that every constant of the motion obtained from the Noether theorem is a first integral; thus, Noether theorem can be used for an effective integration of the corresponding distribution.
LA - eng
KW - Lagrangian system; Lepagean two-form; Euler-Lagrange form; singular Lagrangian; constrained system; Noether theorem; symmetry; constants of motion; first integrals; Lagrangian system; Lepagean two-form; Euler-Lagrange form; singular Lagrangian; constrained system; Noether theorem; symmetry; constants of motion; first integrals
UR - http://eudml.org/doc/248128
ER -

## References

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1. J. F. Cariñena M. F. Rañada, 10.1007/BF00419588, Lett. Math. Phys. 15 (1988), 305-311. (1988) MR0952453DOI10.1007/BF00419588
2. C Ferrario A. Passerini, 10.1088/0305-4470/23/21/040, J. Phys. A 23 (1990), 5061-5081. (1990) MR1083892DOI10.1088/0305-4470/23/21/040
3. C. Ferrario A. Passerini, 10.1016/0393-0440(92)90016-T, J. Geom. Phys. 9 (1992), 121-148. (1992) MR1166718DOI10.1016/0393-0440(92)90016-T
4. J. Hrivňák, Symmetries and first integrals of equations of motion in higher-order mechanics, Thesis, Dept. of Math., Silesian University, Opava, 1995, pp. 59. (In Czech.) (1995)
5. D. Krupka, Some geometric aspects of variational problems in fibered manifolds, Folia Fac. Sci. Nat. UJEP Brunensis 14 (1973), 1-65. (1973)
6. D. Krupka, 10.1016/0022-247X(75)90169-9, J. Math. Anal. Appl. 49 (1975), 180-206; 469-476. (1975) MR0362397DOI10.1016/0022-247X(75)90169-9
7. D. Krupka, Geometry of Lagrangean structures 2, Arch. Math. (Brno) 22 (1986), 211-228. (1986) MR0868536
8. O. Krupková, Lepagean 2-forms in higher order Hamiltonian mechanics, I. Regularity, II. Inverse problem, Arch. Math. (Brno) 22 (1986), 97-120; 23 (1987), 155-170. (1986) MR0868124
9. O. Krupková, Variational analysis on fibered manifolds over one-dimensional bases, PhD Thesis, Dept. of Math., Silesian University, Opava, 1992, pp. 67. (1992)
10. O. Krupková, 10.1016/0393-0440(95)00002-X, J. Geom. Phys. 18 (1996), 38-58. (1996) MR1370828DOI10.1016/0393-0440(95)00002-X
11. G. Marmo G. Mendella W. M. Tulczyjew, Symmetries and constants of the motion for dynamics in implicit form, Ann. Inst. Henri Poincaré, Phys. Theor. 57(1992), 147-166. (1992) MR1184887
12. E. Noether, Invariante Variationsprobleme, Nachr. Kgl. Ges. Wiss. Göttingen, Math. Phys. Kl. (1918), 235-257. (1918)

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