Inverse problems in Classical Field Theories through Tulczyjew?s triples

CAPRIOTTI, SANTIAGO; JUAN CARLOS MARRERO

La Laguna

Jornada; Jornada iberoamericana de álgebra y geometría, Campus América; 2017

Universidad de La Laguna

The search of conditions ensuring that a given system of ordinary or partial dieren-tial equations coincides with the equations of motion of a Lagrangian or a Hamiltonian density,constitutes what is commonly called the inverse problem in mechanics and classical eld the-ory [2, 57, 1012].Tulczyjew´s triples [4], on the other hand, provide a framework in which both Lagrangianand Hamiltonian viewpoints are treated as equivalent. A crucial fact of this approach is therepresentation of the underlying equations of motion as submanifolds of a suitable jet bundle.Now, there exists a way to encode in a geometrical way dierential equations through the notionof exterior dierential systems [3,8,9]. In this context a procedure called prolongation can be usedin order to to represent an exterior dierential system as a submanifold of a jet bundle.This fact gives rise to a formulation of the inverse problem in Tulczyjew´s triple terms: It isenough to represent the give system of dierential equations as a submanifold of a jet bundle, andto nd conditions under which the submanifold so determined is the submanifold correspondingto equations of motion of a Lagrangian or a Hamiltonian in the Tulczyjew approach.In the present talk an approach to the inverse problem from the Tulczyjew´s triple viewpointalong these lines will be discussed, extending to the eld theory realm previous work by De Diegoet al. [1].References[1] María Barbero-Liñán, Marta Farré Puiggalí, and David Martín de Diego. Isotropic submanifolds and the inverseproblem for mechanical constrained systems. Journal of Physics A: Mathematical and Theoretical , 48(4):045210,2015.[2] A.M. Bloch, O.E. Fernandez, and T. Mestdag. Hamiltonization of nonholonomic systems and the inverse problemof the calculus of variations. Reports on Mathematical Physics , 63(2):225 249, 2009.[3] R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, and P.A. Griths. Exterior dierential systems .Springer-Verlag, 1991.[4] C. M. Campos, E. Guzmán, and J. C. Marrero. Classical eld theories of rst order and Lagrangian submanifoldsof premultisymplectic manifolds. J. Geom. Mech. , 4(1):126, 2012.[5] G F Torres del Castillo. The Hamiltonian description of a second-order ODE. Journal of Physics A: Mathematicaland Theoretical , 42(26):265202, 2009.[6] Partha Guha and A. Ghose Choudhury. Hamiltonization of higher-order nonlinear ordinary dierential equationsand the Jacobi last multiplier. Acta Appl. Math. , 116(2):179197, November 2011.[7] Sergio Hojman and Luis F. Urrutia. On the inverse problem of the calculus of variations. Journal of MathematicalPhysics , 22(9):18961903, 1981.[8] T. A. Ivey and J. M. Landsberg. Cartan for beginners: dierential geometry via moving frames and exteriordierential systems . Graduate Texts in Mathematics. American Mathematical Society, 2003.[9] N. Kamran. An elementary introduction to exterior dierential systems. In Geometric approaches to dierentialequations (Canberra, 1995) , volume 15 of Austral. Math. Soc. Lect. Ser. , pages 100115. Cambridge Univ. Press,Cambridge, 2000.[10] Volker Perlick. The Hamiltonization problem from a global viewpoint. Journal of Mathematical Physics ,33(2):599606, 1992.[11] W Sarlet. The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics.Journal of Physics A: Mathematical and General , 15(5):1503, 1982.[12] D.J. Saunders. Thirty years of the inverse problem in the calculus of variations. Reports on Mathematical Physics ,66(1):43 53, 2010.