Educación

 

Variational symmetries of Lagrangians

 

G.F. Torres del Castillo*, C. Andrade Mirón**, and R.I. Bravo Rojas**

 

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

* Facultad de Ciencias Físico Matemáticas Universidad Autónoma de Puebla, Apartado postal 165, 72001 Puebla, Pue., México.

 

Received 25 June 2013
Accepted 8 November 2013

 

Abstract

We present an elementary derivation of the equation for the infinitesimal generators of variational symmetries of a Lagrangian for a system with a finite number of degrees of freedom. We also give a simple proof of the existence of an infinite number of Lagrangians for a given second-order ordinary differential equation.

Keywords: Lagrangians; symmetries; constants of motion; ordinary differential equations.

 

Resumen

Presentamos una derivación elemental de la ecuación para los generadores infinitesimales de simetrías variacionales de una lagrangiana para un sistema con un número finito de grados de libertad. Damos tambien una prueba simple de la existencia de un número infinito de lagrangianas para una ecuación diferencial ordinaria de segundo orden dada.

Descriptores: Lagrangianas; simetrías; constantes de movimiento; ecuaciones diferenciales ordinarias.

PACS: 45.20.Jj; 02.30.Hq; 02.20.Sv

 

DESCARGAR ARTÍCULO EN FORMATO PDF

 

References

1. V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer-Verlag, New York, 1989).         [ Links ]

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

3. H. Goldstein, C.P. Poole, Jr. and J.L. Safko, Classical Mechanics, 3rd ed. (Addison-Wesley, San Francisco, 2002).         [ Links ]

4. E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. (Cambridge University Press, Cambridge, 1993). Chap. X.         [ Links ]

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

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

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

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

9. G.W. Blumanand S. Kumei, Symmetries and Differential Equations (Springer-Verlag, New York, 1989).         [ Links ]

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

11 . G.F. Torres del Castillo, Differentiable Manifolds: A Theoretical Physics Approach (Birkhäuser Science, New York, 2012).         [ Links ]

12. M.C. Nucci and P.G.L. Leach, J. Nonlinear Math. Phys. 16 (2009) 431.         [ Links ]

13. P. Guha and A. Ghose Choudhury, Acta Appl. Math. 116 (2011) 179.         [ Links ]

14. M.C. Nucci, J. Phys.: Conf. Ser. 380 (2012) 012008.         [ Links ]

15. O.P. Bhutani and K. Vijayakumar, J. Austral. Math. Soc. B 32 (1991) 457.         [ Links ]

16. S.C. Anco and G. Bluman, Euro. Jnl of Applied Mathematics 9 (1998) 245.         [ Links ]

17. M. Marcelli and M.C. Nucci, J. Math. Phys. 44 (2003) 2111.         [ Links ]

18. W. Sarlet and F. Cantrijn, SlAM Rev. 23 (1981) 467.         [ Links ]