sentences:  That is, for some "period" formula_13,
  
  The textbook example of a periodic motion is the undamped pendulum.If the phase space is periodic in one or more coordinates, say formula_15, with formula_16 a vector , then there is a second kind of periodic motions defined by
  
  for every integer formula_18.The first step in writing an implicit system for a periodic motion is to move the period formula_13 from the boundary conditions to the ODE