On linear equations of variation in dynamical problems

The linear equations of variation, associated with a motion of a particle moving in a plane under a field of force which admits a first integral of the motion of any form, are drawn up in terms of the tangential and normal displacements. The existence of the first integral implies that the normal displacement satisfies a single second-order differential equation, the tangential displacement being given from the solution of this by a single quadrature. The special cases are examined in which the integral is one of energy, and in which it is one of angular momentum. The extension is made to the motion of two particles moving in a plane under a conservative force-field depending on their positions, which admits also an integral of angular momentum. (The study of the relative motion in the gravitational problem of three bodies in the plane may be put into this form by Jacobi's formulation). An equation is given for finding the non-zero characteristic exponents of a periodic solution of this second problem.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.