Geodesic
Noether theorem and first integrals of constrained Lagrangean systems
Mathematica Bohemica, Tome 122 (1997) no. 3, pp. 257-265
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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.
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.
DOI : 10.21136/MB.1997.126152
Classification : 37B99, 58F05, 58F35, 70H03, 70H33, 70H35
Keywords: Lagrangian system; Lepagean two-form; Euler-Lagrange form; singular Lagrangian; constrained system; Noether theorem; symmetry; constants of motion; first integrals 
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Krupková, Olga. Noether theorem and first integrals of constrained Lagrangean systems. Mathematica Bohemica, Tome 122 (1997) no. 3, pp. 257-265. doi: 10.21136/MB.1997.126152

[1] J. F. Cariñena M. F. Rañada: Noether's theorem for singular Lagrangians. Lett. Math. Phys. 15 (1988), 305-311. | DOI | MR

[2] C Ferrario A. Passerini: Symmetries and constants of motion for constrained Lagrangian systems: a presymplectic version of the Noether theorem. J. Phys. A 23 (1990), 5061-5081. | DOI | MR

[3] C. Ferrario A. Passerini: Dynamical symmetries in constrained systems: a Lagrangian analysis. J. Geom. Phys. 9 (1992), 121-148. | DOI | MR

[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.)

[5] D. Krupka: Some geometric aspects of variational problems in fibered manifolds. Folia Fac. Sci. Nat. UJEP Brunensis 14 (1973), 1-65.

[6] D. Krupka: A geometric theory of ordinary first order variational problems in fibered manifolds. I. Critical sections, II. Invariance. J. Math. Anal. Appl. 49 (1975), 180-206; 469-476. | DOI | MR

[7] D. Krupka: Geometry of Lagrangean structures 2. Arch. Math. (Brno) 22 (1986), 211-228. | MR

[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. | MR

[9] O. Krupková: Variational analysis on fibered manifolds over one-dimensional bases. PhD Thesis, Dept. of Math., Silesian University, Opava, 1992, pp. 67.

[10] O. Krupková: Symmetries and first integrals of time-dependent higher-order constrained systems. J. Geom. Phys. 18 (1996), 38-58. | DOI | MR

[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. | MR

[12] E. Noether: Invariante Variationsprobleme. Nachr. Kgl. Ges. Wiss. Göttingen, Math. Phys. Kl. (1918), 235-257.

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