Integrable lagrangian systems and symmetries

Resumen

This thesis is written by student-candidate Agustín Caparrós Quintero for the PhD degree Doctor por la Universidad Complutense de Madrid y la Universidad Politécnica de Madrid, after completion of all academic activities under IMEIO doctoral program (http://www.mat.ucm.es/imeio/). It is based on the personal research of the student under guidance of supervisor Rafael Hernández Heredero on the subject of Lagrangian systems integrability, a topic in Mathematical Physics. After introducing the symmetry approach to integrability, the extended formal symmetry approach is used to obtain new results in second order integrable Lagrangian systems. As an application, a classification of such systems is done and described in full detail. All the computational techniques used are detailed as well as the exact functional expressions of the integrable classes representatives. This thesis focuses on the extension to higher orders of the symmetry approach for the integrability of Partial Differential Equation (PDE)s. The extended symmetry approach is applied to non-evolutionary equations of order (4, 1) arising as Euler-Lagrange equations of Lagrangian systems, whose integrability conditions are explicitly computed and applied to the classification of integrable second order Lagrangian densities. The theory underlying the symmetry approach and its extension, as described in part I, is that becoming from Lie groups, differential invariance and symmetries applied to systems of differential equations. Although a similar theoretical framework exists for the determination of symmetries of variational problems, the foundations of the computational techniques used in this thesis are not those derived from variational symmetries methods, but the ones derived from the extended symmetry approach to integrability for non-evolutionary equations which is introduced in chapter 3. In the second part of this thesis we detail the calculus techniques employed to compute all cases expressions resulting in the integrable Lagrangian systems classification. In the third part, classification results are presented and some of these are checked to comply the theoretical formulation of the extended symmetry approach for equations of generic order (n,m). For this, some recursion operators are computed along with explicit canonical densities for well-known Lagrangian systems like Bussinesq or Non Lineal Schrodinger (NLS).