The virtual work can then be reparametrized by the generalized coordinates : δ δ W = ∑ ∑ k = 1 f [ ( ∑ ∑ i = 1 m F i ⋅ ⋅ ∂ ∂ r i ∂ ∂ q k + ∑ ∑ j = 1 n M j ⋅ ⋅ ∂ ∂ ϕ ϕ j ∂ ∂ q k ) δ δ q k ] = ∑ ∑ k = 1 f Q k δ δ q k , where the generalized forces Q k are defined as Q k = ∑ ∑ i = 1 m F i ⋅ ⋅ ∂ ∂ r i ∂ ∂ q k + ∑ ∑ j = 1 n M j ⋅ ⋅ ∂ ∂ ϕ ϕ j ∂ ∂ q k , k = 1 , 2 , … … , f . Kane shows that these generalized forces can also be formulated in terms of the ratio of time derivatives. That is, Q k = ∑ ∑ i = 1 m F i ⋅ ⋅ ∂ ∂ v i ∂ ∂ q ˙ ˙ k + ∑ ∑ j = 1 n M j ⋅ ⋅ ∂ ∂ ω ω j ∂ ∂ q ˙ ˙ k , k = 1 , 2 , … … , f .