Point symmetries of the Euler-Lagrange equations

G.F. Torres del Castillo

Departamento de Física Matemática, Instituto de Ciencias, Universidad Autónoma de Puebla, 72570 Puebla, Pue., México.

Received 6 August 2013. ]]> Accepted 7 January 2014.

Abstract

We give an elementary derivation of the equations for the point symmetries of the Euler-Lagrange equations for a Lagrangian of a system with a finite number of degrees of freedom. We show that given a divergence symmetry of a Lagrangian, there exists an equivalent Lagrangian that is strictly invariant under that transformation. The corresponding description in the Hamiltonian formalism is also investigated.

Keywords: Lagrangians; symmetries; equivalent Lagrangians; constants of motion; Hamiltonian formalism.

Resumen

Damos una derivación elemental de las ecuaciones para las simetrías puntuales de las ecuaciones de Euler-Lagrange para una lagrangiana de un sistema con un número finito de grados de libertad. Mostramos que dada una simetría hasta una divergencia de una lagrangiana, existe una lagrangiana equivalente que es estrictamente invariante bajo esa transformación. También se investiga la descripción correspondiente en el formalismo hamiltoniano.

Descriptores: Lagrangianas; simetrías; lagrangianas equivalentes; constantes de movimiento; formalismo hamiltoniano.

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Acknowledgment

The author is grateful to Dr. Jose Luis Lopez Bonilla for bringing Ref. [10] to his attention.

References

1. H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations (Van Nostrand, London, 1966). Chap. 2.         [ Links ]

2. H. Stephani, Differential Equations: Their Solution Using Symmetries (Cambridge University Press, Cambridge, 1990).         [ Links ]

3. P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer-Verlag, New York, 2000).         [ Links ]

4. P.E. Hydon, Symmetry Methods for Differential Equations: A Beginner's Guide (Cambridge University Press, Cambridge, 2000).         [ Links ]

5. B. van Brunt, The Calculus of Variations (Springer-Verlag, New York, 2004).         [ Links ]

6. G.F. Torres del Castillo, C. Andrade Mirón, and R.I. Bravo Rojas, Rev. Mex. Fís. E 59 (2013) 140.         [ Links ]

7. Y. Kosmann-Schwarzbach, The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century (Springer, New York, 2011). Chap. 4.         [ Links ]

8. O. Krupková, The Geometry of Ordinary Variational Equations (Springer-Verlag, Berlin, 1997).         [ Links ]

9. S. Weinberg, Lectures on Quantum Mechanics (Cambridge University Press, Cambridge, 2013).         [ Links ]

10. M. Havelková, Communications in Mathematics 20 (2012) 23.         [ Links ]

11. M.G. Calkin, Lagrangian and Hamiltonian Mechanics (World Scientific, Singapore, 1996).         [ Links ]

12. G.F. Torres del Castillo, Rev. Mex. Fis. E 57 (2011) 158.         [ Links ]