ZealOS has an advanced algorithm for integrating ordinary differential equations suitable for use in video games.  (Not 
scientific work.)  It also has some support for systems of masses and springs, to save you some work.

See CMathODE and ODEsUpdate for an overview.
See ODECallDerivative to see what support there is for masses and springs.

See ::/Demo/Games/Whap.CC, ::/Demo/Games/Rocket.CC, ::/Demo/Games/MassSpring.CC, ::/Apps/Span/SpanMain.CC or 
::/Apps/X-Caliber/X-Caliber.CC.