Quasi-Lagrangian systems of Newton equations

Systems of Newton equations of the form q = -1/2 A(-1)(q)del k with an inte gral of motion quadratic in velocities are studied. These equations general ize the potential case (when A=I, the identity matrix) and they admit a cur ious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of moti on is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by e mbedding into five-dimensional integrable systems. They are characterized b y a linear, second-order partial differential equation PDE which we call th e fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of nonconfocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamic al systems of the Henon-Heiles type. (C) 1999 American Institute of Physics . [S0022-2488(99)00912-3].