Reduction and reconstruction aspects of second-order dynamical systems   with symmetry
  We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results. 
M. Crampin and T. Mestdag