Adjoint symmetries, separability, and volume forms (Q2737975)

The authors discuss two aspects of symmetries of Lagrangian systems: characterization of general dynamical symmetries which belong to the class of Noether symmetries and separation of variables for the Hamilton-Jacobi system which can be related to the calculations of symmetry operators. They deal with rather particular Lagrangians \(L= \frac{1}{2} g_{ij} \dot x^i\dot x^j-V(x)\) and symmetries \(Y= \xi^i \partial/\partial x^i+ \Gamma(\xi^i) \partial/\partial\dot x^i\) (where \(\xi^i= -g^{ij} \partial F/\partial x^i\) are assumed to be polynomial functions in velocities), the orthogonal separation of variables is ensured by the existence of \(n-1\) additional quadratic first integrals. The final results are efficient which is illustrated with a number of examples. The article involves valuable references and comments of recent literature together with large outlook for further investigations.