Fractional Dynamics of Relativistic Particle

TL;DR

This paper explores fractional calculus applied to relativistic particles, revealing how long-term memory effects induce non-Hamiltonian, dissipative dynamics in relativistic systems with non-potential forces.

Contribution

It introduces a fractional dynamics framework for relativistic particles, analyzing conditions under which such systems are Hamiltonian or dissipative.

Findings

Fractional derivatives model long-term memory effects in relativistic particles.

Relativistic fractional dynamics are generally non-Hamiltonian and dissipative.

Conditions for Hamiltonian fractional relativistic systems are identified.

Abstract

Fractional dynamics of relativistic particle is discussed. Derivatives of fractional orders with respect to proper time describe long-term memory effects that correspond to intrinsic dissipative processes. Relativistic particle subjected to a non-potential four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_{\mu} u^{\mu}+c^2=0, where c is a speed of light in vacuum. In the general case, the fractional dynamics of relativistic particle is described as non-Hamiltonian and dissipative. Conditions for fractional relativistic particle to be a Hamiltonian system are considered.