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.

9 pages; v2:published version in PTEP

References in corpus (8)

• Covariant hamiltonian dynamics
• Critical Exponents of Extremal Kerr Perturbations
• Constants of Motion for Constrained Hamiltonian Systems: A Particle around a Charged Rotating Black Hole
• Conformal Killing Tensors and covariant Hamiltonian Dynamics
• Energy extraction from Kerr black holes by rigidly rotating strings
• The essence of the Blandford-Znajek process
• Membranes with a symmetry of cohomogeneity one
• Self-similar motion of a Nambu-Goto string

Cited by in corpus (3)

• Scaling and conformal symmetries for plane gravitational waves
• Integrability of geodesic motions in curved manifolds through non-local conserved charges
• Chaotic particle motion around a homogeneous circular ring