On Generalized Non-holonomic Systems

. Generalized non-holonomic mechanical systems are analyzed from a geometric point of view. The existence and uniqueness of solutions, D’Alembert principle, Gauss principle of minimal constraint, the non-holonomic momentum and Gibbs–Appell equations for such systems are studied in an invariant Lagrangian framework.


Introduction
Restricted mechanical systems have been traditionally studied assuming the vanishing of the work made by the constraining force on the virtual displacements. Nevertheless, in many interesting problems, for instance when the restriction is realized by the action of a servo mechanism [13], that assumption is not fulfilled: the constraining force yields a null work on vectors that non-necessarily coincide with the virtual displacements.
In order to encompass this kind of systems, Marle [13] developed, some years ago, the theory of generalized non-holonomic systems in a series of remarkable articles.
The analysis in [13], carried out in the Hamiltonian context, includes the proof of existence and uniqueness of solutions, the theory of reduction when symmetries are present, and a clever framework for dealing with forces in hamiltonian mechanics.
The aim of this letter is to consider some properties of these systems from the Lagrangian point of view, by using a simple geometrical setting.
An expression for the generator of the restricted dynamics in terms of the generator of the unrestricted one, furnishing a geometric proof of the existence and uniqueness of solutions, will be presented.
Furthermore, by means of suitable inner products defined on the tangent spaces of the configuration manifold, we will give a version of the Gauss principle of minimal constraint covering the generalized non-holonomic systems. As a particular case, we will recover the classical Chetaev's result about virtual displacements [6,16]. Besides, the D'Alembert principle as stated in [19,20], the Gibbs-Appell equations [1,10] and the formula for non-holonomic momentum [2] will also be extended to the case of generalized non-holonomic systems.
We expect that the point of view adopted in this letter can contribute to enlighten some aspects of the remarkable Marle's contributions.
It is worth to note that the geometric setting presented below involves neither covariant derivatives, nor metrics on the phase space of velocities, nor jets.
The development of this letter is as follows: In Section 2, a summary of the invariant setting introduced in [19,20] for the Lagrangian mechanics is presented.
In Section 3, we will give a geometric proof of the existence and uniqueness of solutions for generalized non-holonomic systems, and its expression in terms of the generator of the unrestricted system.
In Section 4, a version of the Gauss principle of minimal constraint encompassing the generalized non-holonomic systems is introduced.
In Section 5, two possible generalizations of the Gibbs-Appell equations as presented in [12] are analyzed.
The last section aims at extending the formula of the non-holonomic momentum [2] to the generalized case.

Lagrangian Mechanics in Invariant Form
In this section, we will make a summary of the framework introduced in [19,20] for the Lagrangian mechanics.
Given a differential manifold Q, we say that a vector w ∈ T (q,v) (T Q) is vertical if π * (w) = 0, with π : T Q → Q the canonical projection π(q, v) = q. That is, w is vertical when it is tangent to the fiber.
For each (q, v) ∈ T Q, we will denote When the point (q, v) is understood, we will write τ instead of τ (q,v) . The space of tangent vector fields on a manifold M will be denoted by X (M). The space of vertical vector fields on T Q, i.e. vector fields taking values in V (q,v) at each (q, v) ∈ T Q, will be denoted by V(T Q).
For T (Q) := {X : T Q → T Q s.t. X (q, v) ∈ T q Q}, we will also denote by τ the isomorphism from T (Q) onto V(T Q) given by Forces are represented by horizontal differential 1-forms on T Q, that is 1-forms vanishing on vertical vectors. That is, if F is a horizontal 1-form, for every w tangent to T Q, F(w) only depends on π * w.
Let us denote The vector fields on T Q relevant for mechanics are the special ones [19,20]. A vector field X ∈ X (T Q) is special if, at each (q, v) ∈ T Q, π * (X (q, v)) = v. In fact, only special vectors fields can be tangent to the lifting to T Q of curves in Q.
The space of special vector fields on T Q will be denoted by S(T Q).

Remark 1. It is clear that S(T Q)
is an affine subspace of X (T Q) and V(T Q) is its associated vector subspace.
Let us now consider a mechanical system described by the smooth Lagrangian function where Q, its configuration space, is a n-dimensional manifold.
The equations of motion of such a system are given by the Euler-Lagrange equations. In any coordinate patch, they read d dt If an external force F e acts on the system, the equations of motion become d dt The symmetric matrix with is assumed positive definite at every (q, v) ∈ T Q. Thus, for each (q, v) ∈ T Q, it gives rise to a inner product , M(q,v) on T q Q. Notice that, since L is C ∞ , , M(q,v) depends on (q, v)in a smooth way. The Lagrangian form associated to L is the 2-form ω L on T Q locally defined as Remark 2. Note that ω L = d ∂L ∂v j dq j . This definition of the Lagrangian form coincides with the one given in [20]. In [2] and [14] the definition is Since M is assumed positive definite at each point, ω L turns out to be symplectic and then, it gives rise to an isomorphism from the space of 1-forms on T Q onto X (T Q), with X α the unique vector field on T Q such that Remark 3. It is proved in [19] that Straightforward computations show that, for w 1 ∈ V (q,v) , The energy function associated to the Lagrangian L is For simplicity, we will write That is It is shown in [19,20] that X L is the generator of the dynamics for the (unconstrained) system, i.e. X L ∈ S(T Q) and a curve q(t) on Q is a solution of the Euler-Lagrange equations (2) if and only if (q(t),q(t)) is an integral curve of X L on T Q.
It follows from Remark 3 and Equation (8) that X L + X F ∈ S(T Q) for any force F.
The D'Alembert principle in the sense of [19] or [20] (see also [2] or [14]) asserts that, if an external force F e acts on the system, the dynamics is generated by the unique special vector field X U satisfying

That is, a curve q(t) on Q is a solution of the Euler-Lagrange equations (3) if and only if (q(t),q(t)) is an integral curve of
on T Q. Now, let us consider the mechanical system restricted by the constraint with C a submanifold of T Q locally defined as the zeros of k smooth functions such that their differential are independent at each (q, v) ∈ C.
Notice that the set of generators of dynamics compatible with the restrictions is the set S(C) of special vector fields tangent to C: It is easy to show that, for admissible constraints, S(C) is non empty (see, for instance [19]).
In any coordinate patch, the equations of motion are now given by d dt with F c the force exerted by the physical realization of the constraint in order the restriction to be satisfied. Consequently, the generator of the dynamics is the unique vector field X R on C satisfying It is equivalent to say Remark 4. We consider Lagrangians and constraints independent of time just for simplicity: the analysis for time-dependent ones is completely analogous.
The space of virtual displacements is, at each (q, v) ∈ C (see, for instance [18]), If, at each (q, v) ∈ C, the force F c does a null work on the virtual displacements, the constraint is usually called ideal.
It is proved in [20] that, for ideal and admissible constraints, there exists a unique F c such that X R = X U + X F c ∈ S(C). This X R is the unique vector field on C generating the dynamics of the restricted system.
In this case, the equations of motion can be written as (D'Alembert principle for non-holonomic systems) [2,14,20]: X R is the unique special vector field on C such that X R ∈ S(C) and

Generalized Non-holonomic Systems
As quoted in Section 1, for the generalized non-holonomic systems, at each (q, v)∈ C the constraining force F c is known to vanish on a subspace of T q Q, which can be different from D (q,v) . We will call W (q,v) that space and assume that it depends smoothly on (q, v). Henceforth, all the maps between manifolds are assumed to be smooth. Let us denote We will assume that W (q,v) depends smoothly on (q, v) and It is obvious that, in particular, this property holds in the classical nonholonomic case W (q,v) = D (q,v) . In fact, decompositions similar to (19) have already been considered in this case (see, for instance [7] or [12]).
The projections from T q Q onto D (q,v) and onto W ⊥ (q,v) associated to this direct sum will be denoted by D and W ⊥ , respectively.
Let us define In order the dynamics to satisfy (q(t),q(t)) ∈ C or, equivalently X R ∈ S(C), the force F c ∈ W 0 must satisfy Our proof of the existence and uniqueness of a force F c ∈ W 0 with this property will be based on the following two simple lemmas: LEMMA 1. For admissible constraints, the space S(C) is a non empty affine subspace of X (C). Its associated vector subspace is τ (D).
Proof. As quoted above, S(C) is non-empty for admissible constraints.
On the other hand, if X, Z ∈ S(C), then X − Z is vertical and So, according to definition (17), Proof. Due to Remark 3, X F is vertical. Then, for each w ∈ T (q,v) T Q such that π * (w) = u, we have from Equation (9) The existence and uniqueness of F c ∈ W 0 such that X R = X U + X F c ∈ S(C) and its expression in terms of X U is a direct consequence of the following geometric result: PROPOSITION 1. If condition (19) holds and the constraint is admissible, then If X 0 is any fixed element of S(C), Proof. Let us choose any fixed vector field X 0 ∈ S(C). (Recall that S(C) is non empty for admissible constraints).
It is obvious that, for X F X as in Equation (22), τ −1 (X F X ) ∈ W ⊥ . So, the previous lemma implies F X ∈ W 0 . (Note that we can take τ −1 (X − X 0 ) because X − X 0 is vertical.) Now we have Thus, from Lemma 1 we have X + X F X ∈ S(C).
On the other hand, ifF ∈ W 0 and X + XF ∈ S(C) then, again from Lemma 1 we obtain τ −1 (X F X − XF ) ∈ D. But, from the previous lemma, since F X −F ∈ W 0 , we Notice that the characterization (22) of F X is clearly independent of the choice of X 0 ∈ S(C).

Remark 5.
On the hamiltonian side, via the Legendre transform, condition (19) becomes the condition under which the existence and uniqueness of solutions for generalized non-holonomic systems is shown in [13]: the constraining force must belong to the symplectic complement in T ξ (T * Q), with respect to the canonical form on T * Q, of (π * ) −1 (D), whereπ : T * Q → Q is the canonical projection.
Thus, the previous proposition can be seen as a geometric version, in the Lagrangian context, of that Marle's result.
It is worth to notice that we are only considering first order constraints (i.e when just q(t) andq(t) are involved in them). Extensions of Marle's results to higher order constraints have been carried out in [3][4][5] and applied to pneumatic tyres and other models. Remark 6. Lemma 2 and Proposition 1 enlighten the physical sense of condition (19): in order to determine the acceleration of the restricted system, we need n independent equations; F c ∈ W 0 implies τ −1 (X F c ) ∈ W ⊥ and condition (19) guarantees that this amounts for n − k equations, independent of the k equations derived from the constraint X U + X F c ∈ S(C).

COROLLARY 1. Under the assumptions of the previous proposition, X R is the unique special vector field on C such that
Proof. It is a direct consequence of the previous proposition and Lemma 2.
The previous result can be rewritten in the following way: D'Alembert principle for generalized non-holonomic systems: For admissible constraints, if condition (19) holds and F c ∈ W 0 , then the generator of the dynamics for the restricted system is the unique X R ∈ S(C) such that, at each (q, v) ∈ C, ∀w ∈ In fact, since ω L (X U , w) = F e (w), ∀w ∈ T (q,v) (T Q), Equation (24) yields ω L (X R − X U , w) = 0, ∀w ∈ T (q,v) (T Q) s.t. π * w∈W (q,v) . So, from Remark 3, EXAMPLE. Two particles with masses m 1 and m 2 , and the same electrical charge e, are jointed by a spring and move on a straight line l. At time t, their positions on l are given by q 1 (t) and q 2 (t) with q 1 < q 2 . Under the action of a time dependent electric field E(t), parallel to l and with (unknown) strength E(t) constant on l, their velocities satisfẏ with A a function such that In order to apply the previous results for writing down the equations of motion of the system, we think about (25) as a constraint, and consider the force exerted by the electric field E(t) on the particles as the constraining force F c (t).
By doing so, we have: and The Lagrangian of the unrestricted system (i.e., disregarding the field E(t)) is where whit k the constant of Coulomb, a the equilibrium length of the spring and κ its elasticity constant; Since E(t) is independent of q and the particles are identically charged, then Given that It is easy to see that condition (19) is equivalent to assumption (26). We now calculate X F c . A straightforward computation shows that the projections D and W ⊥ are given by According to Equation (27), we can take X 0 (q, v) = (v 1 , v 2 ; 0, b(q, v)) with b as in (28). If

Proposition 1 and Equation (36) yield
The equations of motion of the system are given by Consequently,

The Gauss Principle of Minimal Constraint
The classical Gauss principle of minimal constraint (see, for instance [11]) asserts that F c vanishing on the virtual displacements is equivalent to characterize the generator X R of the restricted dynamics as being the only vector field in S(C) such that, ∀(q, v) ∈ C, with . M(q,v) the norm associated to ., . M(q,v) . In order to establish a generalized Gauss principle of minimal constraints encompassing the generalized non-holonomic systems, we will consider for each (q, v)∈ C a inner product [., .] (q,v) on T q Q, depending smoothly on (q, v), such that the subspaces D (q,v) and W ⊥ (q,v) are [., .] (q,v) -orthogonal. It is clear that, if condition (19) holds, then there exist inner products [., .] (q,v) with these properties. For instance, we could take, for u 1 , u 2 ∈ T q Q, Obviously, D and W ⊥ turn out to be orthogonal projections for [., .] (q,v) . We will write [., .] instead of [., .] (q,v) when no confusion can arise. The norm on T q Q associated to [., .] (q,v) will be denoted by . [,](q,v) . We now introduce a generalized version of Gauss principle: PROPOSITION 2. (Generalized Gauss principle of minimal constraint) If condition (19) holds and the constraint is admissible, then X R is the unique special vector field on C such that Proof. At each (q, v) ∈ C, On the other hand, from Corollary 1 we have It is clear that the first term in the r.h.s in this equation vanishes only for X = X R .

Remark 7.
It is obvious that we can take [., .] (q,v) = , M(q,v) if and only if W (q,v) = D (q,v) at each (q, v) ∈ C and that we would obtain the classical Gauss principle of minimal constraint if we had [., .] = , M(q,v) in the equality (40). Then, from the previous proposition we can directly recover the well known Chetaev's result: for the (classical) Gauss principle of minimal constraint to hold, the constraining force has to make a null work along each u ∈ D (q,v) , ∀(q, v) ∈ C, even if the restrictions are non-linear in the velocity variables.
EXAMPLE. In the example of the previous section, we can also determine X R by means of the Gauss Principle: If we take the inner product then X R is the unique vector field belonging to S(C) such that But we have seen that the minimum of is reached when the first term in the r.h.s vanishes. Thus, from (27) and (36), we have

Gibbs-Appell Equations
The Gibbs-Appell equations were introduced independently by Appell [1] and Gibbs [10] in the nineteenth century. A renewed attention has been payed to them in the last years (see, for instance [8,12] or [15]), in particular by their relation with Kane's method [9]. With our notations, the Gibbs-Appell function for classical non-holonomic systems [12] can be written as where, as above, X U is the generator of the unrestricted dynamics.
The vector field X R ∈ S(C) generating the restricted dynamics is the solution of the Gibbs-Appell equation where For a generalized non-holonomic system with F c ∈ W 0 , we can consider the same function G, or we can define another Gibbs-Appell function G W besides the one defined by Equation (42): Both functions allow us for writing down equivalent Gibbs-Appell equations determining X R , the generator of the restricted dynamics.
PROPOSITION 3. Let us assume that condition (19) holds and that the constraint is admissible. The vector field X R is the unique in S(C) satisfying or the equivalent requirement Proof. At each (q, v) ∈ C and ∀u ∈ T q Q, Thus, as a consequence of Corollary 1, X R is the generator of the restricted dynamics if and only if it belongs to S(C) and satisfies (44).
On the other hand, But [., .] (q,v) was chosen so as to have W ⊥ (q,v) = D ⊥ [,] (q,v) . Then, requirements (44) and (45) are equivalent. EXAMPLE. We now write down the Gibbs-Appell equations associated to G and G W for the example considered in the previous sections. If the extended Gibbs-Appell functions read and . Remark 8. The results presented hitherto could be rewritten in terms of any metric If we can define [., .] (q,v) only depending on q, for instance when W = D and the restrictions are linear in v, these inner product give rise to a metric g on Q.
In this case, we could take on T Q the Sasaki's extension g S [17] of g.

The Generalized Non-holonomic Momentum Map
We now consider the action of a Lie group G on Q. The Lagrangian L is assumed to be invariant by the lifting of the action to T Q. We will see that an equation for the non-holonomic momentum map, similar to the one obtained in [2] for classical non-holonomic systems, can be deduced for the generalized case.
We will denote g the Lie algebra of G and, for each (q, v) ∈ C, where ξ Q ∈ X (Q) is the infinitesimal generator of the action on Q associated to ξ . Let g W be the bundle over C whose fiber at (q, v) is given by g W (q,v) and [g W ] * its dual bundle.
The non-holonomic momentum map J nh is the section of [g W ] * defined by J nh (q, v)(ξ ) := D v L(ξ Q (q)).
For classical non-holonomic constraints (i.e., W = D), it is shown in [2] that, if ξ is a section of the bundle g D and q(t) is a trajectory of the constrained system, the following momentum equation holds: For the generalized non-holonomic systems we have Proof. Just take g W (q(t),q(t)) instead of g D (q(t),q(t)) in the proof of Theorem 4.5 of [2].
The main applications of the non-holonomic momentum formula appear in the theory of reduction (see, for instance [2]). Although that theory has not been considered in this letter, the previous proposition was included just to present one more equation where the space W plays, in the generalized non-holonomic case, the role played by the space D in the classical non-holonomic systems.