Scale invariance and constants of motion

Abstract

Scale invariance in the theory of classical mechanics can be induced from the scale invariance of background fields. In this paper we consider the relation between the scale invariance and the constants of particle motion in a self-similar spacetime, only in which the symmetry is well-defined and is generated by a homothetic vector. Relaxing the usual conservation condition by the Hamiltonian constraint in a particle system, we obtain a conservation law holding only on the constraint surface in the phase space. By the conservation law, we characterize constants of motion associated with the scale invariance not only for massless particles but for massive particles and classify the condition for the existence of the constants of motion. Furthermore, we find the explicit form of the constants of motion by solving the conservation equations.

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

“…(The argument can be extended to electromagnetic backgrounds, provided the latter are also preserved). Similar results hold for homothetic fields whose conformal factor is a constant [1][2][3][4], or for massless geodesics [5][6][7].…”

“…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.…”

Integrability of geodesic motions in curved manifolds through nonlocal conserved charges

“…Kuchař was the first to introduce what we call a conditional symmetry [49]. This is defined as a quantity, linear in the momenta, which is conserved due to the Hamiltonian constraint (for more recent applications of quantities which are conserved on the constrained surface see [18,50]). Assume that Q(x, p) is such a quantity; then if {Q, H} =ω(x)H, (3.17) whereω is some function of the configuration space variables, then this Q has the property of been conserved on the constrained surface, i.e.…”