GENERALIZED NATURAL MECHANICAL SYSTEMS OF 2-DEGREES OF FREEDOM WITH QUADRATIC INTEGRALS

This paper is devoted to the problem of constructing integrable mechanical systems with two degrees of freedom whose Lagrangians contain terms linear in the velocities and whose second integrals have the form of polynomials of the second degree in the velocity variables in which the coefficients depend only on the coordinates. The solution of this problem is known only in the case of reversible systems, for which the linear terms do not affect Lagrange's equations of motion. Using the method developed in a previous paper we classify all the possible irreversible systems into three types according to a certain normal form of the line element on the configuration space. The most general systems of the first two types are constructed. Several many-parameter systems of the third type are also found. Some of the new cases are found to be generalizations, by the introduction of some additional parameters, to well known integrable problems in particle and rigid body dynamics. Mechanical interpretation is also given for some other cases.