Integrability of geodesic motions in curved manifolds through nonlocal conserved charges

Abstract

In this work we study the general system of geodesic equations for the case of a massive particle moving on an arbitrary curved manifold. The investigation is carried out from the symmetry perspective. By exploiting the parametrization invariance property of the system we define nonlocal conserved charges that are independent from the typical integrals of motion constructed out of possible Killing vectors/tensors of the background metric. We show that with their help every two dimensional surface can -at least in principle -be characterized as integrable. Due to the nonlocal nature of these quantities not more than two can be used at the same time unless the solution of the system is known. We demonstrate that even so, the two dimensional geodesic problem can always be reduced to a single first order ordinary differential equation; we also provide several examples of this process.

Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields

“…The theory of ref. [11] is indeed the staring point for our considerations here. First we extend the result above to conformal fields Y (L Y g = 2ω Y g) which preserve also some electromagnetic background given by a potential A, i.e.,…”

“…In sec. II we generalize the results obtained in [1,2,11] to proper conformal fields and electromagnetic backgrounds preserved by them; we present both Lagrangian and Hamiltonian approaches. The explicit form of integrals of the motion for pp-waves is spelled out in sec.…”

“…The charge may become local using a special parametrisation. We can, for example, perform the transformation [11]…”

Scaling and conformal symmetries for plane gravitational waves

“…The paradigm of a Chrono-Projective transformation is provided by Kepler's third law [31,32]. 2 Symmetries of timelike geodesics were considered recently using non-local conservation laws [38].…”

“Kepler Harmonies” and conformal symmetries

“…Unconventional conserved quantities for geodesic motion in a curved space were studied recently in[14].…”