Nonlinear constraints in nonholonomic mechanics

In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic constraints and according to a lagrangian and some Chetaev-like conditions. Using some natural regular conditions, a simple form of these equations is given. In the particular cases of linear and affine constraints, one recovers the classical equations in the forms known previously, for example, by Bloch and all \cite {Bl, BKMM}. The case of time-dependent constraints is also considered. Examples of linear constraints, time independent and time depenndent nonlinear constraints are considered, as well as their dynamics given by suitable lagrangians. All examples are based on classical ones, such as those given by Appell's machine.

The possibility to involve a nonlinear constraint and a Lagrangian that rules the dynamics is usually associated with Chetaev or generalized Chetaev principles. A criticism of Chetaev principle is performed, for example, in [21,20], where some situations (as Appell machine) are presented as examples when Chetaev principle fails to a real situation. Other authors use Chetaev principle in some special conditions, as for example in [19], as a generalized Chetaev principle, when the constraint is homogeneous in the relative velocities and the constraints are time dependent. Our goal in the paper is not to study the workability of Chetaev or generalized Chetaev principle, but the possibility to put in an unitary form the dynamics equations coming from linear, affine and regular nonlinear constraints (Theorem 4.1).
The Chetaev principle, generally accepted in nonlinear constraint case, comes from the following principle: taking the variation before imposing the constraints, that is, not imposing the constraints on the family of curves defining the variation. In this case, one follow similar arguments as in the linear and affine constraints in [4,5] and we give a new form expressed in Theorem 4.1. Adapting these results in the case of time dependent nonlinear constraints, we obtain a similar general result that applies in the cases of generalized Chetaev case [19,Section 2] or the example in [19,Section 3].
Some short preliminaries on foliations are given in the second section. Nonlinear constraints, including linear and affine ones, are considered for Lagrangians in the next section using foliations, but following the classical bundle setting as in [4] for linear and affine constraints. The implicit forms of constraints and a link with the Lagrange multipliers form of Euler-Lagrange equation are also considered. For a nonlinear constraint C, a C-semispray S and an S-curvature R of C are defined using Proposition 3.3 and Proposition 3.4 respectively. Notice that in the cases of linear and affine constraints, the curvature need no semispray to be defined. A short form of nonholonomic Lagrangian dynamics, subject of linear and affine constraints, are presented in the third section, following [4] and it is extended in the case of dynamics generated by Cregular Lagrangians, having nonlinear constraint systems. The main result is Theorem 4.1, where a synthetic form of linear and regular-nonlinear cases is given. This result can be adapted to other situations; for example, in the case of time dependent constraints, but a time independent lagrangian (as studied in [19], see Example 3. in the last section). In order to illustrate the constructions performed in the paper, the cases of five well-known examples are discussed in the section five.

Preliminaries on foliations
Let us consider M an (n + m)-dimensional manifold which will be assumed to be connected and orientable.
A codimension n foliation F on M is defined by a foliated cocycle {U i , ϕ i , f i,j } such that: Every fibre of ϕ i is called a plaque of the foliation. Condition (2.1) says that, on the intersection U i ∩U j the plaques defined respectively by ϕ i and ϕ j coincides. The manifold M is decomposed into a family of disjoint immersed connected submanifolds of dimension m; each of these submanifolds is called a leaf of F . If U ⊂ M is an open subset, that a foliation F on M induces a foliation F U on U , called an induced foliation.
By T F we denote the tangent bundle to F and Γ(F ) is the space of its global sections i.e. vector fields tangent to F .
A particular example of a foliation is a locally trivial fibration. There are elementary examples of foliations that are not locallly trivial fibrations and the spaces of leaves are not Hausdorff separated. For example, considering the natural projection π 1 : IR 2 → IR, (x, y) π1 → x one obtain a foliation F on IR 2 ; but on U = IR 2 \{(x, 0)|x ≥ 0} ⊂ IR 2 the induced foliation F U is not a locally trivial fibration and the space of leaves is not Hausdorff separated (even if the leaves are fibers of a surjective submersion). According to the above conventions, the coordinates are denoted by x = x 1 and y = x1.

Linear, affine and nonlinear constraints and Lagrangians
A linear constraint system of a foliation F is a left splitting of the inclusion T F I0 → T M . Since there is a short exact sequence of vector bundle morphisms it follows that the existence a left splitting C of I 0 is equivalent with the existence a right splitting D of the projection Π 0 , thus a inclusion of N F in T M , via the injective morphism D, that gives a decomposition One say that a sectionX ∈ Γ(N F ) is transverse field if for every vector fields X, Y ∈ X (M ) such thatX = Π 0 (X) and Y ∈ Γ(T F ), then [X, Y ] f s ∈ Γ(T F ); we say that D(X) ∈ Γ(N F ) is the horizontal lift ofX. Thus ifX,Ȳ ∈ Γ(N F ) are transverse, the curvature has the form Using local coordinates, a linear constraint C has the local form and the corresponding D is The curvature B of C has the local form As an example, we consider the linear Appell constraints (see, for example, [21]). The manifold is M = IR 3 × T 2 and the foliation is the simple foliation defined by the fibers of the canonical projection IR 3 × T 2 → T 2 . Consider the coordinates (x 1 , x 2 , x 3 ) on IR 3 and (x1, x2) on T 2 . The linear Appell constraints are given by the formulas (3.8) Using formulas (3.7), its curvature B has the coefficients An affine constraint system of a foliation F is a fibered map D ′ : N F → T M affine on fibers. One can decompose D ′ as where D comes from a linear constraint C : T M → T F and b ∈ Γ(T F ) is a tangent vector field to F . We can define also a map C ′ : T M → T F , by C ′ (X) = C(X) + b. In the affine case, giving C and b is equivalent giving D and b, as easily can be seen.
In the similar way one can extend the definition of an adapted Lagrangian L, asking that L has the form L (X) = L 0 (C ′ (X)) +L (Π 0 (X)) , X ∈ X ( T M ), (3.10) where C ′ is an affine constraint and T M = T M − {zero section}. According to [4,Ch. 5], a covariant derivative of b, along a horizontal vector fieldX ∈ Γ(D(N F )), can be considered as a vector field ∇X b ∈ X (M ) that projects by Π 0 onX. Using local coordinates, if a linear constraint D has the local form (3.6), then C (corresponding to D) and D ′ have the forms (3.5) and We deal now with nonlinear constraints. Let us consider the endomorphismJ ∈ End (T N F ), induced by the projection of the canonical almost tangent structure J ∈ End (T T M ). Using local coordinates, it is given by : We say that a map C : N F → T M , viewed also as a section C ∈ Γ(π * N F T M ), is a nonlinear constraint ifJ (C) = Γ 0 . Using local coordinates, Proposition 3.1. A nonlinear constraint give rise to a left splitting C ′′ or, equivalently, a right splitting D ′′ of the exact sequence of vector bundle morphisms Proof. Using local coordinates, it can be proved that the map gives a left splitting of 16) where N ′′ F = D ′′ (π * N F N F ). Let us deal now with an implicit realization of nonlinear constraints. Let F : T M → T F be a fibered manifold map. Using local coordinates, F has the form Let us notice that the property of a point z ∈ T M , of coordinates (x v , xv, y v , yv), to have F u (x v , xv, y v , yv) = 0, does not depend on coordinates; we say that a such point z is a constraint point.
We also say that F is a contravariant implicit constraint (or a con-constraint for short) if 1. for every x ∈ M and any transverse vectorX 2. the local matrices ∂F u ∂y v (z) are non-singular in all constraint points z.
Using the implicit mapping theorem, and local coordinates, these conditions read that the local equations F u (x v , xv, y v , yv) = 0 can be solved with respect to y v , giving local functions (x u , xū, yū) → C u (x u , xū, yū) in a neighborhood of any point in N F , such that F u (x v , xv, C v , yv) = 0. Finally, we obtain local nonlinear constraints C U : Let us consider now the covariant case. Let G : T M → T * F be a fibered manifold map. Using local coordinates, G has the form As in the contravariant case, the property of a point z ∈ T M , called also a constraint point, of coordinates (x v , xv, y v , yv), to have G u (x v , xv, y v , yv) = 0, does not depend on coordinates. We say that G is a covariant implicit constraint (or a cov-constraint for short) if 1. for every x ∈ M and any transverse vectorX 2. the local matrices ∂G u ∂y v (z) are non-singular in all constraint points z.
These conditions read that the local equations G u (x v , xv, y v , yv) = 0 can be solved with respect to y u , giving local functions (x u , xū, yū) → C u (x u , xū, yū) in a neighborhood of any point in N F , such that G u (x v , xv, C v , yv) = 0. Finally, as in the contravariant case, we obtain local nonlinear constraints The implicit form of constraints can be used to give an invariant form to the condition that a covector type be a combination of partial derivatives of functions that give the constraints; for example, in nonholonomic mechanics, the Chetaev condition reads that the covector giving the Euler-Lagrange derivative is such a combination.
for cov-constraints F , or for con-constraints F , then one have Proof. We consider the con-constraints case, since the cov-constraints case is analogous. Differentiating the implicit equation thus using the hypothesis, the conclusion follows. ✷ Nonlinear constraints lifts to linear constraints of the natural lifted foliation F N F on N F , as follows. On an intersection of two adapted charts, the rule is Using this formula, by a direct computation, one can check that the formulas C ū u = ∂C u ∂yū , C u v = 0 gives rise a linear constraint on F N F , i.e. a splitting (left C or right D) of the exact sequence it is a nonlinear one as well. Indeed, the right splitting An affine constraint gives rise also to a nonlinear one, in a similar way. Indeed, an affine constraint is given by a linear constraint C and a vector field b ∈ Γ (T F ). The vector field gives a nonlinear constraint, where D is the right splitting of (3.2) corresponding to C.
We see below that linear and affine constraint have in common a curvature that is a tensor. An almost transverse semi-spray is a (non-necessarily foliated) section S : N F → N N F that is a section for the both structures of vector bundle of N N F over N F (one of usual vector bundle, the other one induced by the transversal component of the differential of the canonical projection N F → M , as a foliated map). In the case of the trivial foliation by the points of M , we recover the definition of a semi-spray on M . Using local coordinates, an almost transverse semi-spray S has the local form x u , xū, yū S → x u , xū, yū, yū, Sū(x u , xū, yū) . If we ask that S be a foliate section, we say that S is a transverse semi-spray. The only difference in formula (3.19) is that Sū = Sū(xū, yū).
In order to lift of an (almost) transverse semi-spray one need a nonlinear constraint (in particular it can be a linear or an affine one).
By a straightforward verification of chain rules on the intersection domains, one can check that S is a global vector field. ✷ Notice that considering coordinates (x u , xū, yū) and A vector field S ∈ X (N F ) given by Proposition 3.3 will be called a C-semispray. Let us notice that C V and C have the same formulas, but they are different as vector fields; C : N F → T M , but C V ∈ X (V ) = X (N F U ) is a local vector field. As a representative nonlinear example, we consider the Appell's example of nonlinear constraint. Take the foliation of IR 3 0 = IR 3 \{0} generated by ∂ ∂z . Denote x = x1, y = x2 and z = x 1 and consider the nonlinear constraint given by the implicit equation We have C 3 y1, y2 = ±α y1 2 + y2 2 , but we take C 3 y1, y2 = α y1 2 + y2 2 .
Formula (3.15) gives We can consider time dependent constraints, as follows. Let us consider N T F = N F × IR or N T F = N F × S 1 and the foliation F T on N T F is induced by the foliation F = F N F on N F , such that the canonical projection N T F → N F is a diffeomorphism of leaves, thus the new parameter is transverse.
A time dependent nonlinear constraint on M is a map C : N T F → T M , viewed also as a section C ∈ Γ(π * N T F T M ), such thatJ (C) = Γ 0 . Using local coordinates, There is an exact sequence, induced by (3.14): As in the time independent case, a time dependent nonlinear constraint on M gives also rise to a left splitting C ′′ or, equivalently, a right splitting D ′′ of the exact sequence (3.24); using local coordinates, then analogous formulas (3.15) and (3.16) holds. A more general approach of time dependent constraints can be considered taking M ′ = M × IR instead M and the parameter from IR being transverse. Then transverse coordinates get xū, wherē u = 0, n and x0 = t ∈ IR. The case considered above is when y0 = 1, corresponding to (t, 1) ≡ ∂ ∂t , the tangent vector to curve t → t in IR. We do not use this general situation in the paper.
A classical example of time dependent nonlinear constraint is the Appell-Hammel dynamic system in an elevator considered in [19], having the time dependent constraints It is easy to see that the above Appell example corresponds to the particular case when v 0 (t) = 0.
Formula (3.15) gives Notice that formula (3.17) on T (N F ) shows that ∂xū This tensor vanishes only for linear or affine constraint.
In both nonlinear Appell's examples, the matrix of C is (3.26) Proposition 3.4. If C : N F → T M is a nonlinear constraint and S ∈ X (N F ) is a C-semispray, then the local formula Proof. Let us consider two coordinates systems on V and V ′ , V ∩ V ′ = ∅, on N F , as in the proof of Proposition 3.3. One can check that We have By a long and straightforward computation, one obtain the formula Using also formula (3.21), we obtain the conclusion. ✷ We call R given by Proposition 3.4 as the S-curvature of C; this R is free on S only in the case when C = 0, i.e. when C is a linear or affine constraint, as in [5,4], (see the formulas (4.34) and (4.35) below). In general, the formula gives only a local linear map L(V N F U , T F N FU ) that does not extends to L(V N F , T F N F ). We say that R V is the pseudo-curvature of C; it is not a tensor. In the case of Appell's nonlinear constraint, one have R V = 0, only using the euclidean coordinates.

The Lagrangian dynamics for linear, affine and nonlinear constraint systems
In this section we look closer to properties that involve together Lagrangians and linear constraint on foliations, following [5,4]. Specifically, the dynamics of the system is ruled by a master Lagrangian L : T M → IR and a linear or affine constraint C : T M → T F , or D : N F → T M , considered in the previous section. Let L : T M → IR be a Lagrangian on the total space of a foliated manifold endowed with a system of a nonlinear (possible linear or affine) constraint. We study the case of nonlinear constraints, thus we consider one given by a left splitting C of I ′ 0 , or by a right splitting D of the projection Π ′ 0 in the exact sequence (3.18 Notice that δ is subject to t = const. Substituting δxū in the Lagrange equations, one obtain the induced constrained Lagrange equations: As we see below for implicit nonlinear constraints, these equations are concordant to Chetaev conditions.
A nonlinear constraint C ∈ Γ(π * (4.30) According to [4,Sect. 5.2], in the cases when i.e. of linear and affine constraints respectively, the constrained Lagrange equations (4.29) can be written in terms of the constrained Lagrangian as are both tensors, B and γ (see [5,4] for more details).
In the linear constraint case (i.e. b u = 0), the formula (4.32) gives, according to formula (3.28), that the curvature R of C is R ū u = yvB ū vū . (4.34) In the affine constraint case, the formulas (4.32), (4.33) and (3.28) give the curvature of C by In the sequel we extend formulas (4.31) to the case of nonlinear constraints. The equation of motion of the extended nonholonomic system is In the case of the induced Lagrangian L c , one have Thus, using (3.28), one have On the other hand, where Comparing the relations (4.36) and (4.37), we obtain ∂L ∂y u It is easy to see that h = (hūv) gives a global bilinear form in the fibers of V N F =π * N F N F . We have, by a straightforward computation, Using the splitting (C ′′ at left or D ′′ at right) of exact the sequence (3.14) given by Proposition 3.1, one can easy deduce an interpretation of h. Recall that the Hessian of L is a bilinear form in the fibers of V T M = π * T M T M . Proof. We use adapted coordinates. The conclusion follows using the form (3.15) and the identity We say that the Lagrangian L is C-regular if the bilinear form h is nondegenerted on the fibers of V N F . If it is the case, denoting hūv = (hūv) −1 , the equation (4.39) gives By a straightforward computation, based on the equality (4.39), one can prove that the local functions (Sū) verify the rule (3.22) on the intersection of compatible domains, giving by formula (3.19) an almost transverse semi-spray S, called canonically associated with C and L. Using Proposition 3.3 and the above constructions, one have the following result. Notice that in the particular case of linear and affine constraints, using formulas (3.28), then (4.31) can be deduced from (4.41).
The Legendre map of L, L : T M → T * M , or L ∈ Γ(π * T M T * M ), is given in local coordinates by The statement below gives the equations of motion in the same form the C-regular case and the linear and affine cases of constraints, studied in [5,4].
where R is the S-curvature of C and S is the almost transverse semi-spray S canonically associated, in the first case, or R is the curvature of C, in the last two cases.
In the case of time dependent constraints (as in [19], see the Example 3 in the next section), but a time independent lagrangian, the equations of motion are obtained in the same way as equations (4.36), taking into account the fact that the constraints are time dependent, but the lagrangian is not. One obtain that equations (4.36) are replaced by and the equations (4.39) are valid in the same form, but with
We have C 3 x 1 , x1, y1 = f (x 1 , x1, y1). Formula (3.15) gives If one consider on V = IR 2 \{(x 1 , 0)|x 1 ≥ 0} the induced foliation (thus generated by ∂ ∂x 1 ), this is not a locally trivial one and the space of leaves is not Hausdorff separated, thus the use of a foliation is justified in this case.