TempleOS 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.HC, ::/Demo/Games/Rocket.HC, ::/Demo/Games/MassSpring.HC, 
::/Apps/Span/SpanMain.HC or ::/Apps/X-Caliber/X-Caliber.HC.