A single particle freely moving in R 3 has 3 degrees of freedom. The configuration space is M = R 3 , and P ( M ) = C ∞ ∞ ( [ t 0 , t 1 ] , M ) . For every path γ γ ∈ ∈ P ( M ) and a variation Γ Γ ( t , ϵ ϵ ) of γ γ , there exists a unique σ σ ∈ ∈ T 0 R 3 such that Γ Γ ( t , ϵ ϵ ) = γ γ ( t ) + σ σ ( t ) ϵ ϵ + o ( ϵ ϵ ) , as ϵ ϵ → → 0. By the definition,