For each path γ γ ∈ ∈ P ( M ) and ϵ ϵ 0 > 0 , a variation of γ γ is a function Γ Γ : [ t 0 , t 1 ] × × [ − − ϵ ϵ 0 , ϵ ϵ 0 ] → → M such that, for every ϵ ϵ ∈ ∈ [ − − ϵ ϵ 0 , ϵ ϵ 0 ] , Γ Γ ( ⋅ ⋅ , ϵ ϵ ) ∈ ∈ P ( M ) and Γ Γ ( t , 0 ) = γ γ ( t ) . The virtual displacement δ δ γ γ : [ t 0 , t 1 ] → → T M ( T M being the tangent bundle of M ) corresponding to the variation Γ Γ assigns to every t ∈ ∈ [ t 0 , t 1 ] the tangent vector