Intrinsic variational equations for conservative dynamical systems with N degrees of freedom

For a conservative dynamical system with n deg. of freedom we show that the equations of variation along an orbit may be written with respect to an orthonormal moving frame (a generalized Frenet frame) in which the tangential variation is given by a quadrature and the normal and n-2 binormal variations are solutions of n-1 coupled second order equations of the form of Hill's equation.