A rigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom. The configuration space here is M = S O ( 3 ) , the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and P ( M ) = C ∞ ∞ ( [ t 0 , t 1 ] , M ) . We use the standard notation s o ( 3 ) to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map exp : s o ( 3 ) → → S O ( 3 ) guarantees the existence of ϵ ϵ 0 > 0 such that, for every path γ γ ∈ ∈ P ( M ) , its variation Γ Γ ( t , ϵ ϵ ) , and t ∈ ∈ [ t 0 , t 1 ] , there is a unique path Θ Θ t ∈ ∈ C ∞ ∞ ( [ − − ϵ ϵ 0 , ϵ ϵ 0 ] , s o ( 3 ) ) such that Θ Θ t ( 0 ) = 0 and, for every ϵ ϵ ∈ ∈ [ − − ϵ ϵ 0 , ϵ ϵ 0 ] , Γ Γ ( t , ϵ ϵ ) = γ γ ( t ) exp ⁡ ⁡ ( Θ Θ t ( ϵ ϵ ) ) . By the definition,