Kinematic reduction and the Hamilton-Jacobi equation

A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces.


Introduction
The reduction of mechanical control systems to kinematic systems is very interesting and useful for solving control problems such as optimal control problems [3] and for designing suitable control laws (see [21], Chapter 8 in [4] and references therein). For instance, the planning motion for the associated kinematic system determines trajectories of the mechanical control system. Thus the methodologies to find these trajectories have been simplified because the kinematic reduction gives rise to a first-order control-linear system defined on the configuration manifold. Hence in general it is easier to solve or to analyze the kinematic system. If the mechanical control system is reducible to a kinematic one, then the controlled trajectories of this kinematic system under reparametrization define solutions of the original second-order problem on the phase space. An interesting particular case is the one defined by kinematic reductions of order 1. This kind of reductions define a decoupling vector field. Unfortunately, there is not a systematic procedure for finding such kinematic reductions.
The philosophy of kinematic reductions of order 1 seems, in a first approach, quite similar to the standard Hamilton-Jacobi theory. This theory, that appeared with the dawn of analytical mechanics, is a valuable tool for the exact integration of Hamilton's equations, for instance using the technique of separation of variables (see [1] and references therein). In many cases, the Hamilton-Jacobi theory allows us to simplify the integration of Hamilton's equations or, at least, to find some particular solutions. To be more precise, consider a configuration manifold Q and a hamiltonian function H : T * Q → R. The Hamilton-Jacobi equation can be written as H q, ∂W ∂q = constant for some function W : Q → R. If we find such a function W , then the integration of the associated Hamilton's equations (with initial conditions along dW (Q)) is reduced to knowing the integral curves of a vector field X dW H on Q. This vector field is given by X dW H = T τ T * Q •X H •dW ∈ X(Q), where τ T * Q : T * Q → Q is the canonical projection and X H is the hamiltonian vector field associated to H. Hence, from the integration of a vector field on the configuration space it is possible to recover some of the solutions of the original hamiltonian system. Recent developments in Hamilton-Jacobi theory are described in [5,6,13,14,19,22]. Of course, the possible similarities with the theory of kinematic reductions are now clearer.
One of the main objectives in our paper is to carefully study the underlying geometry of the kinematic reduction theory by showing the close relation with the classical Hamilton-Jacobi theory. Moreover, advantage of recent developments in Hamilton-Jacobi theory for nonholonomic systems on skew-symmetric algebroids [2,14] even with external forces will be really useful to obtain a full novel theory of kinematic reduction for this type of systems. It is important to highlight this is not an arbitrary generalization since the mechanics on algebroids [8,9,11,17,20] is particularly relevant for the class of Lagrangian systems invariant under the action of a Lie group of symmetries including as a particular case nonholonomic dynamics (see [7] for a survey on the subject; see also [15,20,23]).
The main results of this paper can be summarized in the following points: • A description of nonholonomic mechanics in terms of the Levi-Civita connection associated to a fibered riemannian metric defined on the vector subbundle determined by the nonholonomic constraints. • A deduction of the Hamilton-Jacobi equation for nonholonomic systems in terms of the induced Levi-Civita connection. • An affine connection approach in presence of control forces.
• A description of Hamilton-Jacobi equation with controls.
• Relationship between Hamilton-Jacobi equation and kinematic reductions by decoupling vector fields.
It is interesting to observe that our approach allows us to extend the theory of kinematic reduction to controlled system with symmetries as for instance, nonholonomic Lagrange-Poincaré equations, etc.
In the sequel, all the manifolds are real, second countable and C ∞ . The maps are assumed to be also C ∞ . Sum over all repeated indices is understood.

Skew-symmetric algebroids
In this section we introduce the notion of a skew-symmetric algebroid on a vector bundle τ D : D → M . It is known that this geometric structure covers many interesting cases in mechanics, as for instance, nonholonomic mechanics (see [14]). Similarly to the intrinsic definition of the Euler-Lagrange equations for a Lagrangian function L : T M → R obtained by the canonical structures on it (standard Lie bracket, exterior differential...), it is possible to determine the motion equations for a Lagrangian L : D → R using the differential geometric structures naturally induced by the skew-symmetric algebroid structure. We will show that this generalization is quite useful in applications and clarifies the dynamics of systems with nonholonomic constraints. Let us first introduce the notion of a skew-symmetric algebroid.
of sections of τ D and a vector bundle morphism ρ D : D → T M , so-called anchor map, such that: (ii) If we also denote by ρ D : Γ(τ D ) → X(M ) the morphism of C ∞ (M )-modules induced by the anchor map then If the bracket [[·, ·]] D satisfies the Jacobi identity, we have that the pair ([[·, ·]] D , ρ D ) is a Lie algebroid structure on the vector bundle is a skew-symmetric algebroid structure on the vector bundle τ D : D → M , then an almost differential d D of sections of Λ k τ D * being τ D * : D * → M the vector bundle projection of the dual bundle D * is defined as follows is a Lie algebroid structure on the vector bundle τ D : D → M if and only if (d D ) 2 = 0 (see [17,18,20,25] for more details about the Lie algebroids).
Suppose that (x i ) are local coordinates on M and that {e A } is a local basis of the space of sections Γ(τ D ), then are called the local structure functions of the skew-symmetric algebroid on τ D : D → M .
If {e A } is the dual basis of {e A }, then Given X ∈ Γ(τ D ), the integral curves of the section X are those curves σ : That is, they are the integral curves of the associated vector field ρ D (X) ∈ X(M ). If σ is an integral curve of X, then X • σ is a ρ Dadmissible curve. Locally, the integral curves are characterized as the solutions of the following system of differential equationṡ Consider now the vector space over R If M is connected and D is a transitive skew-symmetric algebroid, that is, 1) It is important to stress that condition (2.1) holds if the skew-symmetric algebroid has a connected base space and is completely nonholonomic, that is, [14] for more details. Given this bundle metric we can construct a unique torsion-less connection ∇ G D on D which is metric with respect to G (see [9] and references therein, for the standard case of Lie algebroids). The following construction mimics the classical construction of the Levi-Civita connection for a riemannian metric on a differentiable manifold. We will denote by G D : D → D * the vector bundle isomorphism induced by G D and by # G D : D * → D the inverse morphism. The bundle metric can be locally written as The Levi-Civita connection ∇ G D : Γ(τ D ) × Γ(τ D ) → Γ(τ D ) associated to the bundle metric G D is defined by the formula: Alternatively, ∇ G D is determined by These two properties allow to determine Christoffel symbols of the connection ∇ G D that satisfy More details about how to compute the Christoffel symbols are given in Section 5.3 if a G D -orthogonal basis of Γ(τ D ) is taken.
Additionally, we have the notion of derivative along an admissible curve. If γ : I ⊆ R → D is a ρ D -admissible curve and is the set of sections along γ, then the induced covariant derivative ∇ γ : Γ(γ) → Γ(γ) can be defined as the mapping from X ∈ Γ(γ) to ∇ γ X ∈ Γ(γ) with local expression 3.2. Geodesics. Given the bundle metric G D , a ρ D -admissible curve γ : If the local expression of γ is then γ is a geodesic if and only if The geodesics are just the integral curves of a vector field on D, called the geodesic spray ξ G D , whose local expresion is The associated symmetric product is defined as follows: . The symmetric product on Riemannian manifolds is a fundamental tool in controllability results, kinematic reduction of mechanical systems and in the characterization of geodesic invariance of distributions (see [4]). These results can be extended to our setting of skew-symmetric algebroids.
Proof. This is proved locally, similarly to the proof of this result on Riemannian manifolds in Lemma B.3 in [4].
Locally, X = X A e A and Y = Y B e B , then Then, On the other hand, This proves the equality since the vertical lift of this section is equal to the local expression of [X v , [ξ G D , Y v ]] we just computed above. Now, we can extend the characterization of geodesically invariant distributions already known on Riemannian manifolds [4] to skew-symmetric algebroids. A subbundle D of a skew-symmetric algebroid (D, [[·, ·]] D , ρ D ) is geodesically invariant if for any geodesic γ : I ⊆ R → D with initial condition γ(0) ∈ D(σ(0)), then γ(t) ∈ D(σ(t)) for any t ∈ I, where σ = τ D • γ.
Proof. (i) ⇔ (ii) The integral curves of ξ G D are the geodesics. This proves the equivalence.

Newtonian systems. Given a bundle map
we define a newtonian system as the triple (D, G D , F). This newtonian system induces the system of differential equations: where the solutions are curves γ : I ⊆ R → D which are ρ D -admissible. Given local coordinates (x i , y A ) associated with the basis {e A } for sections of D, Equations (3.3) can be written aṡ Observe that Equations (3.4) are the equations of the integral curves of a vector field ξ G D ,F on D. Locally, this vector field is given by Remark 3.3. The map F could be given by a section F ∈ Γ(τ D ) such that F = F • τ D . An interesting particular case is when we have a potential function V : M → R and F is the section −grad G D V ∈ Γ(τ D ) given by In particular, the solutions of this newtonian system (D, G D , F = −grad G D V ) are equivalent to the solutions of the Euler-Lagrange equations on a skew-symmetric algebroid with Lagrangian L : D −→ R: Therefore, these solutions are ρ D -admissible curves γ : (3.6) Locally, those solutions satisfẏ   [2,14]. Denote also by ι D : D → T M the canonical inclusion. We induce by restriction a bundle metric G D : D × M D → R and an skew-symmetric algebroid structure on D as follows: For this particular skew-symmetric algebroid structure, Equations (3.6) correspond with the equations of the nonholonomic system determined by the constraints induced by the distribution D and the mechanical lagrangian (3.5). These equations are also called in the literature Lagrange-D'Alembert's equations.
Consider a basis of G D -orthogonal vector fields Observe that for the induced coordinates (x i , y A , y α ) on T M the nonholonomic constraints are rewritten as y α = 0, m + 1 ≤ α ≤ n. That is, the induced coordinates on D are given by (x i , y A ). Therefore, the skew-symmetric algebroid structure induced on the vector subbundle D → M is locally described by: Example 3.6. Our theory is not only restricted to lagrangian systems defined on the tangent bundle T M or nonholonomic systems determined by a regular distribution on T M . The techniques described in this paper by means of skew-symmetric algebroids are general enough to cover the most important cases of reduction of mechanical systems subjected or not to nonholonomic constraints. As a particular example, we include in our analysis the case of Lie algebras g of finite dimension (it is clear that g is a Lie algebroid over a single point). Now, suppose that (l, d) is a nonholonomic Lagrangian system on g, where l : g → R is a Lagrangian function defined by l(ξ) = 1 2 I Iξ, ξ , I I : g → g * is a symmetric positive definite inertia operator and d is a vector subspace of g. We have the orthogonal decomposition Take now an adapted basis to this decomposition {e A , e α } where d = span {e A } and d ⊥ = span {e α }. Then, the Euler-Poincaré-Suslov equations for (l, d) arė Example 3.7. Similarly, more involved situations can be recovered using our techniques. For instance, nonholonomic systems on Atiyah algebroids associated with principal G-bundles. For the sake of simplicity, we will consider the particular case when the principal G-bundle is trivial. In such a case, the Atiyah algebroid is a vector bundle of the form where g is the Lie algebra of the Lie group G and M is a smooth manifold. The Lie bracket of the space Γ(τ A ) is characterized by the following condition for ξ, ξ ∈ g and X, X ∈ X(M ). The anchor map ρ A is the canonical projection onto the second factor. Suppose now that D is a vector subbundle of A over M of constant rank (the constraint bundle) such that is a vector subbundle of A. Then we can choose a local basis {ξ a } 1≤a≤r of Γ(τ D V ), with ξ a : U ⊆ M −→ g smooth maps, and a local basis where (x i , y c , y α ) are the corresponding local coordinates on D.
Note that in the particular case when M is a single point, we recover the Euler-Poincaré Suslov equations.
Geometric interpretations of nonholonomic LR systems or nonholonomic systems with semidirect product symmetry may also be given using skew-symmetric algebroids deduced from Lie algebroids (see [8] for more details).

Hamilton-Jacobi equation
The next result is a direct consequence of Equations (3.6). See [2] for an extension of this result (in a hamiltonian context) for many different types of mechanical systems (nonholonomic dynamics, dissipative systems...).
be a skew-symmetric algebroid and consider a newtonian system determined by (D, G D , F). Take an arbitrary section X ∈ Γ(τ D ) then, the following conditions on X are equivalent: Then, the result follows immediately from the fact that This result is analogous to the Hamilton-Jacobi theory already described in a Lagrangian framework for a free mechanical system in [5] and for a nonholonomic one in [6], but adapted to skew-symmetric algebroid structures.  From now on, we only consider mechanical problems given by (D, G D , V ) as described in Remark 3.3. Specializing Proposition 4.1 to this kind of problems and to vector fields X verifying an extra condition i X d D ( G (X)) = 0 we obtain a new expression of this Proposition. Indeed, the following Theorem can be compared with the classical expression of the Hamilton-Jacobi equation proposed in [14] for the hamiltonian function h : D * −→ R: where κ ∈ D * and G D * : D * × M D * −→ R is the induced bundle metric on the dual bundle. that is,σ (t) = ρ D (X)(σ(t)), γ(t))) = 0. (ii) X satisfies the Hamilton-Jacobi differential equation If, additionally, the skew-symmetric algebroid (D, Proof. First, we study how to express the condition i X d D ( G D (X)) = 0 in terms of the Levi-Civita connection associated to G D . Let Y ∈ Γ(τ D ), . Therefore, the condition i X d D ( G D (X)) = 0 is alternatively written as We only need to check that both condition (ii) in Proposition 4.1 and Theorem 4.4 are equivalent. If we examine the Hamilton-Jacobi differential equation In the last equality we have used condition (4.3). Therefore Equation (4.2) is written as [2,14]).
From Theorem 4.4, it is clear that we need to find sections X satisfying i X d D ( G D (X)) = 0. The most simple candidate to be a solution is After some straightforward calculations, this condition is equivalent to: This condition is always true if the bracket [[·, ·]] D satisfies the Jacobi identity, that is, if the pair ([[·, ·]] D , ρ D ) is a Lie algebroid structure on the vector bundle τ D : D → M (see [14]).

Mechanical control systems and kinematic reductions
Assume that the newtonian system determined by (D, G D , F) also contains some input forces. We model this set of input forces by a vector subbundle D (c) of D * . Locally, The vector fields Y 1 , . . . , Y k are called the control sections or input sections.
The equations of motion for a newtonian system with input sections are as follows where γ : I → D is a ρ D -admissible curve.
In terms of the control sections, Equation (5.1) can be rewritten as follows: for some u : I ⊆ R → R k , playing the role of controls. The corresponding local equations arė Remark 5.1. It is also possible to study the more realistic case when U is a proper subset of R k . In this case, the controls u take value in a proper set of R k (i.e., not all the linear combinations of controls are allowed). Our procedure can be adapted to that particular control set. But for geometrical clarity in this paper we only consider the control distribution D (c) .

5.1.
Kinematic reduction of mechanical control systems on a skew-symmetric algebroid. Now we introduce the notion of a kinematic reduction (see [4]).
Given an skew-symmetric algebroid structure on the vector bundle τ D : D → M , we define a driftless system as the set (M, D, U ), where D is a vector subbundle of D locally spanned by {X 1 , . . . , X k }, with X α ∈ Γ(τ D ), 1 ≤ α ≤ k , and U ⊂ R k is the set of admissible controls. For a section X = k α=1 u α X α ∈ Γ(τ D ), remember that an integral curve of X is a curve σ : I ⊂ R → M such thaṫ where (u 1 (t), . . . , u k (t)) = (ū 1 (σ(t)), . . . ,ū k (σ(t))) ∈ U for all t. It can also be said that the pair (σ, u) is a solution to the driftless system. Observe that for each pair (σ(t), u(t)) we have the curve γ : I −→ D ⊆ D defined by In the sequel, we will denote by τ D the restriction (τ D ) D . Definition 5.4 (Kinematic reduction). Let (D, G D , F, D (c) ) be a mechanical control system. A driftless system (M, D, U ) is called a kinematic reduction of (D, G D , F, D (c) ) if for every solution (σ(t), u(t)) of (5.3) there exists a pair (γ(t), u(t)) solution of (5.2), where γ(t) = k α=1 u α (t)X α (σ(t)).
The rank of a kinematic reduction is the rank of the distribution D. Rank-one kinematic reductions are particularly interesting. A section X of Γ(τ D ) is called a decoupling section if the rank-one kinematic system induced by D = span {X} is a kinematic reduction. . . , X k } whose involutive closure has maximum rank.
When a system is kinematically controllable, motion planning is possible by using concatenations of integral curves of the decoupling vector fields. Those curves must be reparametrized in such a way that each segment begins and ends with zero velocity, see [4] for more details.
We have the following adaptation of the results in Section 4.
Under extra assumptions, we have an alternative way to write condition (ii) in Proposition 5.6. This provides us with a new characterization of kinematic reductions in terms of the affine connection of the given mechanical control system. (See also [4,21]).
Proof. (i) ⇒ (ii) Observe that for each α = 1, . . . , k we have that f X α ∈ Γ(τ D ) for all f ∈ C ∞ (M ). If (i) holds, we have for every f ∈ C ∞ (M ). Then, taking suitable functions f , we obtain Q (c) (X α ) = 0 and Q (c) (∇ G D Xα X α ) = 0. Now, using the polarization identity we have that As (ii) is true, then statement (ii) in Proposition 5.6 is satisfied for every X ∈ Γ(τ D ). Hence statement (i) in Proposition 5.6 is true and we can conclude that (M, D, U ) is a kinematic reduction of (D, G D , F, D (c) ) according to Definition 5.4.
is a rank-one kinematic reduction of (D, G D , F, D (c) ).
In the sequel assume that F comes from a potential function V : M → R, that is, F = −grad G D V • τ D . Then Theorem 4.4 can be adapted to control systems as follows. Take a section X ∈ Γ(τ D ) such that i X d D ( G D (X))(Y ) = 0 for all Y ∈ Γ(τ D ⊥ (c) ). Under this hypothesis, the following conditions are equivalent: (i) If σ : I −→ M is an integral curve of ρ D (X), that is, γ(t)))) = 0. (ii) X satisfies the Hamilton-Jacobi differential equation Proof. Following similar arguments that in the proof of Theorem 4.4 it is easy to prove that the condition

The last expression is equivalent to the equation
Hence, we have just extended the notion of decoupling sections for mechanical control systems with nonzero potential since Theorem 5.9 gives sufficient and necessary conditions to have a kinematic reduction of such a mechanical control system. Remark 5.10. Note that the condition i X d D ( G D (X))(Y ) = 0 in the hypothesis of Theorem 5.9 is C ∞ (M )-linear in Y ∈ D ⊥ (c) . Hence only for a basis of vector fields in D ⊥ (c) the condition must be checked. However that same condition is not C ∞ (M )-linear in X ∈ Γ(τ D ).
This theorem plays a key role to define in the future the geometric notion of motion planning for mechanical control systems with non-zero potentials, only known so far for zero potentials [4]. In that sense it will be useful to have some notion of reparametrization of integral curves of sections X ∈ Γ(τ D ).
Proof. Let us rewrite the condition i f X d D ( G D (f X))(Y ) = 0 as follows because X ∈ Γ(τ D (c) ). From here, the equivalence is straightforward.
As a consequence of Proposition 5.11, Theorem 5.9 can also be written for such a f X.
From Theorem 5.9, we establish a connection between decoupling sections in the sense defined in [4] and the solutions to Hamilton-Jacobi differential equation.
) is a decoupling section, then the following conditions are equivalent: ), (iii) X satisfies the Hamilton-Jacobi differential equation Proof. The condition i X d D ( G D (X))(Y ) = 0 for all Y ∈ Γ(τ D ⊥ (c) ) can be rewritten as . As X is a decoupling section, both X and ∇ G D X X ∈ Γ(τ D (c) ) by Corollary 5.8. Hence the left-hand side of the above equality is zero and (i) is equivalent to (ii).
Condition (iii) can be rewritten as follows: ). Hence the equivalence between (ii) and (iii) is clear because of the hypothesis for V .
Note that the property of being a decoupling section is preserved by C ∞ (M )-multiplication. However, the condition i X d D ( G D (X))(Y ) = 0 for all Y ∈ Γ(τ D ⊥ (c) ) is not C ∞ (M )-linear on X. Proposition 5.11 has already characterized those functions f such that ). It might also be the case that a decoupling section X ∈ Γ(τ D (c) ) does not satisfy (ii) in Corollary 5.12, but there might exist functions . (5.5) Condition (ii) in Corollary 5.12 would be satisfied if the following partial differential equation for f has solutions: ). The chances to find a solution depend on the particular examples under study, see Section 5.3.

5.2.
Maximally reducible systems. If (M, D, U ) is a kinematic reduction of (D, G D , F, D (c) ), then any solution of (5.3) can be followed by a solution of (5.1). In this section we consider when the converse is also possible in such a way that we can talk about "equivalence" of controlled trajectories as mentioned in the introduction. The characterization of maximally reducible mechanical control systems defined on skew-symmetric algebroids is given by the following result. This a a generalization of the notion of being maximally reducible systems proved in [4] on Riemannian manifolds.
Theorem 5.14. Let (D, G D , F, D (c) ) be a mechanical control system such that D (c) has locally constant rank and F = 0. This mechanical control system is maximally reducible to a driftless system (M, D, U ) if and only if the following two conditions hold: where Sym (∞) (D (c) ) is the smallest distribution containing D (c) and closed under the symmetric product · : · G D .
Proof. It follows the same lines as the proof of Theorem 8.27 in [4].
Suppose that the rank of D ⊆ D is k . Note that X ∈ Γ(τ D ) is a section of D if and only if the vertical lift of X restricted to D is tangent to D.
Assume that (D, G D , 0, D (c) ) is maximally reducible to a driftless system (M, D, U ).
If γ : I ⊆ R → D is a geodesic for the bundle metric G D with initial condition γ(0) ∈ D σ(0) , then γ is a solution of Equations (5.2) with zero controls. Thus, by hypothesis, there exist controlsū α : I →Ū such that

γ(I) ⊆ D and we can conclude that D is geodesically invariant.
It remains to prove that D = D (c) . Remember that D (c) = span{Y 1 , . . . , Y k }. Then it is sufficient to prove that Y s is a section of D for s = 1, . . . , k in order to obtain D (c) ⊂ D. Having in mind that ξ G D is the geodesic spray on D and Y v s is the vertical lift of Y s ∈ Γ(τ D (c) ) to D (c) , it is clear that the integral curves of ξ G D + Y v s are solutions of Equations (5.2). By assumption, On the other hand, D is geodesically invariant, what implies that ξ G D restricted to D is tangent to D. Hence, (Y v s ) |D is tangent to D and Y s is a section of D. Next, we will see that D ⊆ D (c) . Let a be a vector in D. We consider the geodesicγ :Ĩ ⊆ R → D with initial conditionγ(0) = a. Hence, γ(Ĩ) ⊆ D because D is geodesically invariant. Now, take the curve γ : I ⊆ R → D given by Note that τ D • γ = τ D •γ • τ , where τ : R → R is the map defined by τ (t) = t 2 /2, and γ(0) = 0. From (3.1) it is straightforward to prove that On the other hand, define a curveσ on M given byσ = τ D •γ such that Then, it is clear that what implies that (σ • τ,ū) is a solution of Equations (5.3). Therefore, there exist controls u such that (γ, u) is a solution of Equations (5.2). In particular, . The other implication is proved as follows: First, we prove that (M, D, U ) is a kinematic reduction of (D, G D , F, D (c) ). Let σ : Thus γ is ρ D -admissible.
Now we have to prove that ∇ G D γ(t) γ(t) ∈ D (c) (σ(t)). By hypothesis, D = D (c) and D (c) is geodesically invariant. Then by Theorem 3.2 we conclude that ∇ G D γ(t) γ(t) ∈ D (c) (σ(t)) for all t ∈ I. Hence, there exist controls such that γ is a solution of Equation (5.2).
It remains to prove condition (ii) in Definition 5.13. Let (γ, u) : By assumption γ(0) ∈ D(σ(0)) and γ is an integral curve of ξ G D +X v , where ξ G D is the geodesic spray associated with G D and X ∈ Γ(τ D ).
Note that X v | D is tangent to D and (ξ G D ) | D is also tangent to D because D is geodesically invariant. Hence, if γ(0) ∈ D(σ(0)), then the integral curve of ξ G D + X v with initial condition γ(0) is entirely contained in D, that is, γ(t) ∈ D(σ(t)) for all t ∈ I.
Let us provide some results to find decoupling sections for the mechanical control systems under consideration. Proof. This follows immediately from Theorem 5.14 and the definition of decoupling sections.
The converse is not necessarily true. But it is true when D (c) has locally constant rank equal to one. Proof. It is straightforward from Theorem 5.14.

5.3.
Examples. The first two examples are specific cases of Example 3.4. As is shown, the example in Section 5.3.1 is not maximally reducible but admits rank-one kinematic reductions. Particular solutions to Hamilton-Jacobi differential equation are found using Theorem 5.9 and Corollary 5.12. In Section 5.3.2 we compute particular solutions to Hamilton-Jacobi differential equation for a maximally reducible system. The snakeboard described in Section 5.3.3 has nonholonomic constraints. Hence the use of skew-symmetric algebroids to find solutions to Hamilton-Jacobi differential equations is very natural.

5.3.1.
Planar rigid body with a variable-direction thruster. We refer to [4,Section 7.4.2] for a detailed description of the system. The configuration space is M = R 2 × S 1 . Consider local coordinates (x, y, θ).
Here the distribution D is the entire tangent space T M and V = 0. We are in the case explained in Example 3.4.
The riemannian metric is The control vector fields in D (c) are In this example The non-zero Christoffel symbols for the associated Levi-Civita ∇ G T M connection are , Let us compute the symmetric products of the control vector fields By Corollary 5.8, Y 1 and Y 2 are decoupling vector fields. However, D (c) = span{Y 1 , Y 2 } is not geodesically invariant because Sym (1) D (c) D (c) . Then according to Theorem 5.14 the mechanical control system is not maximally reducible to a driftless control system. As the involutive closure of the decoupling vector fields Y 1 and Y 2 has maximum rank, the system is kinematic controllable, see Definition 5.5. Then the motion planning is feasible.
Let us see if Y 1 and Y 2 satisfy condition (ii) in Corollary 5.12, that Thus Y 1 and Y 2 are both solutions to Hamilton-Jacobi differential equation because of Corollary 5.12.
Let us try to find more vector fields solution to Hamilton-Jacobi differential equation. In order to do this we have to find functions f ∈ C ∞ (M ) such that f Y 1 and f Y 2 also satisfy condition (ii) in Corollary 5.12. According to Proposition 5.11 f must satisfy For instance, f (x, y, θ) = g(x − h cos θ, y − h sin θ) where g : R 2 → R satisfies the above partial differential equation. Hence, all vector fields f Y 1 and f Y 2 are solutions to Hamilton-Jacobi differential equation, but not necessarily their linear combinations. So far we have found some particular solutions of Hamilton-Jacobi differential equation. Let us consider now the most general section in Γ(τ D ), X = α 1 Y 1 + α 2 Y 2 + α 3 Y 3 . Let us find functions α 1 , α 2 , α 3 such that condition i X (d D ( G D (X)))(Y ) = 0 for all Y ∈ D ⊥ (c) in Theorem 5.9 is satisfied, that is, . It can be proved that this condition and Hamilton-Jacobi differential equation (5.4) are satisified, for instance, by

5.3.2.
Robotic leg. We refer to [4, Section 7.4.1] for a detailed description of this system. The configuration manifold is M = R + × S 1 × S 1 with local coordinates (r, θ, ψ). The riemannian metric for the system is where m is the mass of the particle on the end of the extensible leg and J is the moment of inertia of the base rigid body about the pivot point. The control vector fields that span D (c) are From here, it is easy to compute the following symmetric products: According to Theorem 5.14 the system is maximally reducible. Thus all the C ∞ (M )-linear combination of Y 1 and Y 2 are decoupling sections and we can apply Corollary 5.12 to identify those decoupling sections that are solutions to Hamilton-Jacobi differential equation: . It can be proved that if α i (r, θ, ψ) = f i (r, θ − ψ) either for i = 1 or i = 2 and smooth functions f i : R + × S 1 → R, then the section α 1 Y 1 + α 2 Y 2 is a solution to Hamilton-Jacobi differential equation.
By Theorem 5.9, we can also check that are solutions to Hamilton-Jacobi differential equation.

5.3.3.
Snakeboard. We refer to [4,Section 13.4] and [21] for a detailed description of this system. The configuration manifold is M = SE(2) × S 1 × S 1 with local coordinates (x, y, θ, ψ, φ). Consider the following physical parameters: the mass m c of coupler, mass m r of rotor, mass m w of each wheel assembly, inertia J c of coupler about center of mass, inertia J r of rotor about center of mass, inertia J w of wheel assembly about center of mass, distance l from coupler center of mass to wheel assembly.
In order to find solutions to Hamilton-Jacobi differential equation, let us first check if there exist any decoupling section of Γ(D (c) ) so that we can use Corollary 5.12.
Let us check if the decoupling sections satisfy condition (ii) in Corollary 5.12.

Future work
The results in this paper extend the notion of decoupling sections for mechanical systems with nonzero potential. A decoupling section for those mechanical systems is a section that satisfies the assumption and condition (ii) in Theorem 5.9. The future research line consists of taking advatange of this geometric description of decoupling sections to do motion planning for mechanical systems with nonzero potential. One of the key points to succeed in motion planning is that not any reparametrization of decoupling sections in the sense of Theorem 5.9 is again decoupling in the sense of Theorem 5.9. In order to define the suitable notion of reparametrization Proposition 5.11 seems to be useful.