Consider the kinetic energy for a system of N point masses. If it is assumed that T ( q , q ˙ ˙ , t ) = T ( q , q ˙ ˙ ) , then it can be shown that r ˙ ˙ k ( q , q ˙ ˙ , t ) = r ˙ ˙ k ( q , q ˙ ˙ ) (See Scleronomous § Application). Therefore, the kinetic energy is T ( q , q ˙ ˙ ) = 1 2 ∑ ∑ k = 1 N ( m k r ˙ ˙ k ( q , q ˙ ˙ ) ⋅ ⋅ r ˙ ˙ k ( q , q ˙ ˙ ) )