Contact symmetries of constrained quadratic Lagrangians

23rd International Conference on Integrable Systems and Quantum Symmetries

Dimakis, N.^1 ; Terzis, Petros A^2 ; Christodoulakis, T.^2

Instituto de Ciencias Fisicas y Matematicas, Universidad Austral de Chile, Valdivia, Chile^1

Nuclear and Particle Physics Section, Physics Department, University of Athens, Athens

GR, 157-71, Greece^2

关键词: Conformal factors;    Constrained systems;    Degree of freedom;    Gauge conditions;    Infinitesimal generator;    Integrals of motion;    Loss of symmetry;    Relativistic particles;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/670/1/012021/pdf
DOI  :  10.1088/1742-6596/670/1/012021

来源: IOP

【 摘 要 】

The conditions for the existence of (polynomial in the velocities) contact symmetries of constrained systems that are described by quadratic Lagrangians is presented. These Lagrangians mainly appear in mini-superspace reductions of gravitational plus matter actions. In the literature, one usually adopts a gauge condition (mostly for the lapse N) prior to searching for symmetries. This, however, is an unnecessary restriction which may lead to a loss of symmetries and consequently to the respective integrals of motion. A generalization of the usual procedure rests in the identification of the lapse function N as an equivalent degree of freedom and the according extension of the infinitesimal generator. As a result, conformal Killing tensors (with appropriate conformal factors) can define integrals of motion (instead of just Killing tensors used in the regular gauge fixed case). Additionally, rheonomic integrals of motion - whose existence is unique in this type of singular systems - of various orders in the momenta can be constructed. An example of a relativistic particle in a pp-wave space-time and under the influence of a quadratic potential is illustrated.