Covariant differentiation

In general, the derivation of the Euler-Lagrange equations representing, e. g., the dynamics of multi-body systems requires to differentiate a vector field on a differentiable manifold M. The process of differentiation on a manifold is a generalization of the ordinary process of differentiation inRⁿ. The main difference becomes clear by considering (for a moment) the example of a vector fieldv(q) inRⁿdefined along a parameterized curveq=q(t)∈Rⁿ with timet[Fra03]. The ordinary derivative of this vector field along the curve is defined by

dv(t)

dt = lim

∆t→0

v(t+ ∆t)−v(t)

∆t , (2.16)

8This implies that the metric tensor is only regular, i. e., det(M)6= 0 (cf. (2.14)).

2.2. Tensors on manifolds

where it can be seen that vectors at the different points q(t+ ∆t) and q(t) are related.

This makes only sense in an affine space, where a vector can be translated parallel. If one is concerned with a vector fieldv(q) on a general manifoldM, then the vectors in (2.16) are even in the different tangent spacesv(t+ ∆t)∈ Mq(t+∆t) respectively v(t) ∈ Mq(t). An important implication of this state of affairs is that the (time) derivatives

˙

v^h= ∂v^h

∂q^kq˙^k (2.17)

of a vector field v^h(q(t)) on a manifold M do not represent the components of a con-travariant vector. Since ˙q^k are of type (1,0), the partial derivatives ∂v^h/∂q^k would need to be of type (1,1). However considering the transformation of the vector field

w^j(x) = ∂x^j

∂q^hv^h(q), (2.18)

it can be seen, by partial differentiation,

∂w^j

∂x^k = ∂²x^j

∂q^l∂q^h

∂q^l

∂x^kv^h +∂x^j

∂q^h

∂q^l

∂x^k

∂v^h

∂q^l (2.19)

that∂v^h/∂q^l is even not tensorial (cf. Definition 2.3).

The circumstance that the ordinary derivative of a vector field on a general manifold is not tensorial motivates the introduction of a more general concept of differentiation.

Definition 2.9 (Covariant derivative of a vector field). Let v^h(q^k) be a contravariant vector field on a differentiable manifold M. Then, the partial covariant derivative w. r. t.

q^k can be expressed as

∇kv^j = ∂v^h

∂q^k + Γ_h^j_kv^h, (2.20)

where the three-index symbols Γ_h^j_k are the connection coefficients on the differentiable manifoldM.

The notion of covariant differentiation involves the components of an affine connection Γ_h^j_k. Thereby, it is assumed that the manifold is equipped with such a connection. In accordance with the example of ordinary differentiation (2.16), the affine connection (to be defined later) specifies how to relate vectors of distinct (neighboring) tangent spaces of a general manifold. This process of translation is sometimes referred to as transportation by parallel displacement.

Definition 2.10 (Local parallelism of vectors). Let v^j(q^k) be an arbitrary contravari-ant vector at the point r(q^k) ∈ M, a unique vector v^j + dv^j at the neighboring point r(q^k+ dq^k)∈ M is defined such that the covariant differential of the vector field is zero, i. e.,

Dv^j =∇kv^jdq^k= dv^j + Γ_h^j_kv^hdq^k= 0, (2.21) where Γ_h^j_k is to be evaluated at r(q^k), while dq^k refers to the displacement fromr(q^k) to r(q^k+ dq^k). The vector v^j + dv^j constructed at r(q^k+ dq^k) is said to be parallel to the vector v^j atr(q^k) if dv^j satisfies the condition (2.21).

The process of covariant differentiation—leading to the concept of parallelism—is de-fined for a general tensor field on an affinely connected space⁹:

Definition 2.11 (Covariant derivative of a tensor field). The partial covariant derivative of a general type(r, s) tensor field Q^j1···jr_l1···ls,

2. The covariant derivative is a linear operator, i. e., given the real numbersa and b,

∇^m

aX^j1···jr_l1···ls+bY^h1···hr_k1···ks

=a∇^mX^j1···jr_l1···ls+b∇^mY^h1···hr_k1···ks. 3. The covariant derivative of the product of two tensor fields follows a rule formally

identically to the product rule of ordinary partial differentiation, i. e.,

∇^m

X^j1···jr_l1···lsY^h1···hr_k1···ks

=

∇mX^j1···jr_l1···lsY^h1···hr_k1···ks+X^j1···jr_l1···ls∇mY^h1···hr_k1···ks.

Note that the above properties do not completely specify the connection coefficients.

Definition 2.12 (Connection coefficients [LR89]). Any set of three-index symbolsΓ_h^l_k is said to constitute the components of an affine connection on the differentiable manifold Mif they transform under a change of coordinates (2.1) by

⋆Γ_m^j_p(x) = ∂x^j

In case of a Riemannian manifold of Definition 2.8, there is a particular connection that relates parallel displacement with the Riemannian metric of Definition 2.7.

Definition 2.13(Riemannian connection). The metric tensor M_hj(r), det(M_hj(r))6= 0 for all r∈ M, defines a connection on the Riemannian manifold as introduced by Defini-tion 2.8. In local coordinatesqof the coordinate neighborhoodQ, its connection coefficients are given by the Christoffel symbols of the first kind and second kind

Γ_hlk = 1

respectively, where the contravariant tensor of rank two in (2.25), M^jl represents the inverse of the metric tensor M_jl.

9An affinely connected space is a manifold equipped with an affine connection.

2.2. Tensors on manifolds

Especially the motion of our Euler-Lagrange systems considered here, takes place on a differentiable manifold equipped with a Riemannian connection. Such a so-called Levi-Civita connection has some important implications. A property of paramount importance for the stability analysis of nonlinear Euler-Lagrange systems relies on the following lemma:

Lemma 2.1(Ricci’s lemma [LR89]). The covariant derivative of the metric tensor w. r. t.

the connection (2.25) vanishes identically:

∇kM_jh= 0, ∇kM^jh = 0. (2.26)

Proof. Permuting the indicesh and lin (2.24) and adding the result Γ_lhk = 1

2

∂M_hk

∂q^l +∂M_lh

∂q^k − ∂M_lk

∂q^h

to (2.24) and exploiting the symmetry property of Corollary 2.1, yields the identity Γ_hlk+ Γ_lhk = ∂M_hl

∂q^k . (2.27)

Furthermore, it follows from (2.22) of Definition 2.11 that the covariant derivative of the tensor fieldM_hl(q) w. r. t. the Christoffel symbols (2.25) is given by

∇kM_hl= ∂M_hl

∂q^k −Γ_hlk−Γ_lhk, (2.28)

where the inverse of the relation (2.25) has already been used. By considering the identity (2.27) in (2.28) it follows directly that∇kM_hl= 0. The result∇kM^hl = 0 can be obtained analogously.

An additional important implication of the Riemannian connection is the property that parallel displacement preserves scalar products.

Remark 2.4. Let v and w be vectors at the same point r of the Riemannian mani-fold equipped with a metric connection as introduced by Definition 2.13. Then, the scalar product hv,wi is preserved under transportation by parallel displacement, i. e., for a dis-placement from r(q^k) to r(q^k+ dq^k),

dhv,wi= 0. (2.29)

Proof. According to statement 2 of Definition 2.6, the scalar product of the vectorsv and wis an invariant. Therefore, it can be written

dhv,wi= Dhv,wi= D v^TM w

= Dv^T

M w+v^T (DM)w+v^TM(Dw) = 0.

The first and the third term in the three-term expression of the above equation are zero due to Definition 2.10. The second term is zero as a consequence of Lemma 2.1, since DM_jh =∇kM_jhdq^k= 0.

The concept of covariant differentiation of tensors on affinely connected spaces of the Riemannian manifold will be extensively exploited for the derivation of the Euler-Lagrange equation, describing the dynamics of the class of systems considered here.

Im Dokument Multi-Dimensional Nonlinear Oscillation Control of Compliantly Actuated Robots  (Seite 32-36)