Non-conservative extension of Keplerian integrals and a new class of integrable system

Abstract

The invariance of the Lagrangian under time translations and rotations in Kepler's problem yields the conservation laws related to the energy and angular momentum. Noether's theorem reveals that these same symmetries furnish generalized forms of the first integrals in a special nonconservative case, which approximates various physical models. The system is perturbed by a biparametric acceleration with components along the tangential and normal directions. A similarity transformation reduces the biparametric disturbance to a simpler uniparametric forcing along the velocity vector. The solvability conditions of this new problem are discussed, and closed-form solutions for the integrable cases are provided. Thanks to the conservation of a generalized energy, the orbits are classified as elliptic, parabolic, and hyperbolic. Keplerian orbits appear naturally as particular solutions to the problem. After characterizing the orbits independently, a unified form of the solution is built based on the Weierstrass elliptic functions. The new trajectories involve fundamental curves such as cardioids and logarithmic, sinusoidal, and Cotes' spirals. These orbits can represent the motion of particles perturbed by solar radiation pressure, of spacecraft with continuous thrust propulsion, and some instances of Schwarzschild geodesics. Finally, the problem is connected with other known integrable systems in celestial mechanics.