Atlas of two-dimensional irreversible conservative lagrangian mechanical systems with a second quadratic integral

TL;DR

This paper systematically constructs and catalogs 41 new and existing two-dimensional mechanical systems with quadratic invariants, covering various geometries and including applications to rigid body dynamics and motion in different surfaces.

Contribution

It provides a comprehensive classification of 2D irreversible conservative systems with quadratic invariants, introducing many new integrable cases and unifying previous known systems.

Findings

Constructed 41 multi-parameter integrable systems mostly on Riemannian manifolds.

Included all known planar and rigid body motion cases as special instances.

Discovered new integrable cases related to Steklov's problem and motions on various surfaces.

Abstract

This paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant, quadratic in velocities, other than the Hamiltonian. The configuration space is in general a 2D Riemannian or pseudo-Riemannian manifold and the determination of its geometry is a part of the process of solution. Forces acting on the system include a part derived from a scalar potential and a part derived from a vector potential, associated with terms linear in velocities in the Lagrangian function of the system. The last cause time-irreversibility of the system. We construct 41 multi-parameter integrable systems of the type described in the title mostly on Riemannian manifolds. They are mostly new and cover all previously known systems as special cases, corresponding to special values of the parameters. Those include all known cases of motion of a particle in the plane and all known cases in the dynamics of rigid body. In the last field we introduce a new integrable case related to Steklov's case of motion of a body in a liquid. Several new cases of motion in the plane, on the sphere and on the pseudo-sphere or in the hyperbolic plane are found as special cases. Prospective applications in mathematics and physics are also pointed out.