A note on the solution of the variational equations of a class of dynamical systems

Some properties are derived for the solutions of the variational equations of a class of dynamical systems. It is shown that under rather general conditions, the matrix of the linearized Lagrangian equations of motion have an important property for which the word 'skew-symplectic' has been introduced. It is also shown that the fundamental matrix of solutions is 'symplectic', the word symplectic being used here in a more general sense than in the classical literature. Two consequences of the symplectic property are that the fundamental matrix is easily invertible and that the eigenvalues appear in reciprocal pairs. The effect of coordinate transformations is also analyzed; in particular, the change from Lagrangian to canonical systems.