A new approach to the vakonomic mechanics

The aim of this paper was to show that the Lagrange–d’Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them we consider the generalization of the Hamiltonian principle for nonholonomic systems with non-zero transpositional relations. We apply this variational principle, which takes into the account transpositional relations different from the classical ones, and we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with the d’Alembert–Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular, the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems, the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian mechanics. We illustrate our results with examples.

The nonholonomic mechanic is a remarkable generalization of the classical Lagrangian and Hamiltonian mechanic. The birth of the theory of dynamics of nonholonomic systems occurred when Lagrangian-Euler formalism was found to be inapplicable for studying the simple mechanical problem of a rigid body rolling without slipping on a plane.
A long period of time has been needed for finding the correct equations of motion of the nonholonomic mechanical systems and the study of the deeper questions associated with the geometry and the analysis of these equations. In particular, the integration theory of equations of motion for nonholonomic mechanical systems is not so complete as in the case of holonomic systems. This is due to several reasons. First, the equations of motion of nonholonomic systems have more complex structure than the Lagrange ones, which describes the behavior of holonomic systems. Indeed, a holonomic system can be described by a unique function of its state and time, the Lagrangian function. For the nonholonomic systems, this is not possible. Second, in general, the equations of motion of nonholonomic systems have no invariant measures, by the contrary the equations of motion of holonomic systems always have an invariant measure (see [6,26,30,50]).
One of the most important directions in the development of the nonholonomic mechanics is the research connected with the general mathematical formalism to describe the behavior of such systems which differs from the Lagrangian and Hamiltonian formalism. The main problem with the equations of motion of the nonholonomic mechanics has been centered on whether or not these equations can be derived from the Hamiltonian principle in the usual sense, such as for the holonomic systems (see for instance [33]). But there is not doubt that the correct equations of motion for nonholonomic systems are given by the d'Alembert-Lagrange principle.
The general understanding of inapplicability of Lagrange equations and variational Hamiltonian principles to the nonholonomic systems is due to Hertz, who expressed it in his fundamental work Die Prinzipien der Mechanik in neuem Zusammenhaange dargestellt [17]. Hertz's ideas were developed by Poincaré in [39]. At the same time, various aspects of nonholonomic systems need to be studied such as (a) The problem of the realization of nonholonomic constraints (see for instance [25,27]). (b) The stability of nonholonomic systems (see for instance [35,43]).
(c) The role of the so called transpositional relations (see [20,34,35,42]) where d dt denotes the differentiation with respect to the time, δ is the virtual variation, and x = (x 1 , . . . , x N ) is the vector of the generalized coordinates. Indeed the most general formulation of the Hamiltonian principle is the Hamilton-Suslov principle suitable for constrained and unconstrained Lagrangian systems, whereL is the Lagrangian of the mechanical system. Clearly, the equations of motion obtained from the Hamilton-Suslov principle depend on the point of view on the transpositional relations. This fact shows the importance of these relations. (d) The relation between nonholonomic mechanical systems and vakonomic mechanical systems.
There was some confusion in the literature between nonholonomic mechanical systems and variational nonholonomic mechanical systems also called vakonomic mechanical systems. Both kinds of systems have the same mathematical "ingredients": a Lagrangian function and a set of constraints. But the way in which the equations of motion are derived differs. As we observe the equations of motion in nonholonomic mechanic are deduced using the d'Alembert-Lagrange's principle. In the case of vakonomic mechanics, the equations of motion are obtained through the application of a constrained variational principle. The term vakonomic ("variational axiomatic kind") is due to Kozlov (see [22][23][24]), who proposed this mechanics as an alternative set of equations of motion for a constrained Lagrangian system.
The distinction between the classical differential equations of motion and the equations of motion of variational nonholonomic mechanical systems has a long history going back to Korteweg's survey (1899) [21], and discussed in a more modern context in [10,19,29,49]. In these papers, the authors have analyzed the domain of the vakonomic and nonholonomic mechanics. In the paper Critics of some mathematical model to describe the behavior of mechanical systems with differential constraints [19], Kharlamov studied the Kozlov model and in a concrete example showed that the subset of solutions of the studied nonholonomic systems is not included in the set of vakonomic model and proved that the principle of determinacy is not valid in the Kozlov model. In [28], the authors put in evidence the main differences between the d'Alembertian and the vakonomic approaches. From the results obtained in several papers, it follows that in general the vakonomic model is not applicable to the nonholonomic constrained Lagrangian systems.
The equations of motion for the constrained mechanical systems deduced by Kozlov (see for instance [2]) from the Hamiltonian principle with the Lagrangian L : where L j = 0 for j = 1, . . . , M < N are the given constraints, and L 0 is the classical Lagrangian. These equations are for k = 1, . . . , N , see for more details [2]. Clearly, Eq.
(3) differ from the classical equations by the presence of the terms λ j E k L j . If the constraints are integrable, i.e., L j = d dt g j (t, x), then the vakonomic mechanics reduces to the holonomic one.
In this paper, we give a modification of the vakonomic mechanics. This modification is valid for the holonomic and nonholonomic constrained Lagrangian systems. We apply the generalized constrained Hamiltonian principle with non-zero transpositional relations. By applying this constrained variational principle, we deduce the equations of motion for the nonholonomic systems with constraints which in general are nonlinear in the velocity. These equations coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced from the d'Alembert-Lagrange principle.

Statement of the main results
In this paper, we state and solve the following inverse problem of the constrained Lagrangian systems (see [31]) We consider the constrained Lagrangian systems with configuration space Q and phase space T Q.
Let L : R × T Q × R M −→ R be a smooth function such that where = (λ 1 , . . . , λ M ) are the additional coordinates (Lagrange multipliers), L j : in all the points of R × T Q, except perhaps in a zero Lebesgue measure set, L j and λ 0 j are arbitrary functions and constants, respectively, for j = M + 1, . . . , N .
We must determine the smooth functions L j , constants λ 0 j for j = M + 1, . . . , N and the matrix A in such a way that the differential equations describing the behavior of the constrained Lagrangian systems are obtained from the Hamiltonian principle with transpositional relation given by where We give the solutions of this problem in two steps. First we obtain the differential equations along the solutions satisfying (6). Second we shall contrast the obtained equations and classical differential equations which described the behavior of the constrained mechanical systems. The solution of this inverse problem is presented in Sect. 4.
Note that the function L is singular, due to the absence ofλ.
We observe that the arbitrariness of the functions L j , of the constants λ 0 j for j = M + 1, . . . , N , and of the matrix A will play a fundamental role in the construction of the mathematical model which we propose in this paper.
In particular if the constraints are given by the formulȧ then systems (20) become which are the classical Voronets differential equations. Consequently, Eq. (20) are an extension of the Voronets differential equations for the case when the constraints are nonlinear in the velocities. (20) describe the motion of the constrained Lagrangian systems with the constraints L α =ẋ α − α (y,ẏ) = 0 and Lagrangian L * = L * (y,ẏ). Under these assumptions Eq. (20) take the form

Proposition 6 Differential equations
In particular if the constraints are given by the formulȧ then systems (23) become From (5) and in view of the Implicit Function Theorem, we can locally express the constraints (reordering coordinates if is necessary) aṡ for α = 1, . . . , M. We note that Propositions 5 and 6 are also valid for every constrained mechanical systems with constraints locally given by (26), this follows from  The proof of this corollary is given in Sect. 9.
We have the following conjecture.

Hamiltonian principle
We introduce the following results, notations, and definitions which we will use later on (see [2]).
A Lagrangian system is a pair (Q,L) consisting of a smooth manifold Q, and a smooth functionL : The point x = (x 1 , . . . , x N ) ∈ Q denotes the position (usually its components are called generalized coordinates) of the system and we call each tangent vectoṙ x = (ẋ 1 , . . . ,ẋ N ) ∈ T x Q the velocity (usually called generalized velocity) of the system at the point x. A pair (x,ẋ) is called a state of the system. In Lagrangian mechanics it is usual to call Q, the configuration space, the tangent bundle T Q is called the phase space,L is the Lagrange function or Lagrangian and the dimension N of Q is the number of degrees of freedom.
Let a 0 and a 1 be two points of Q. The map such that γ (t 0 ) = a 0 , γ (t 1 ) = a 1 is called a path from a 0 to a 1 . We denote the set of all these path by (Q, a 0 , a 1 , t 0 , t 1 ) := .
We shall derive one of the most simplest and general variational principles the Hamiltonian principle (see [40]).
The functional F : −→ R defined by is called the action. We consider the path Let the variation of the path γ (t) be defined as a smooth mapping

By definition we have
This function is called the virtual displacement or virtual variation corresponding to the variation of γ (t) and it is a function of time, all its components are functions of t of class C 2 (t 0 , t 1 ) and vanish at t 0 and t 1 i.e., δx(t 0 ) = δx(t 1 ) = 0. A varied path is a path which can be obtained as a variation path.
The first variation of the functional F at γ (t) is and it is called the differential of the functional F (see [2]). The path γ (t) ∈ is called the critical point of is known as the Lagrangian derivative.
It is easy to show the following property of the Lagrangian derivative for arbitrary smooth function f = f (t, x). We observe that in view of (27) we obtain that the Lagrangian derivative is unchanged if we replace the function g by g + d f dt , for any function f = f (t, x). This reflects the gauge invariance. We shall say that the functions , and we shall write g ĝ.

Proposition 9
The differential of the action can be calculated as follows: Proof We have that Hence, by considering that the virtual variation vanishes at the points t = t 0 and t = t 1 we obtain the proof of the proposition.

Corollary 10
The differential of the action for a Lagrangian system Q,L can be calculated as follows: Proof Indeed, for the Lagrangian system the transpositional relation is equal to zero (see for instance [32] page 29), i.e., Thus, from Proposition 9, it follows the proof of the corollary.
The path γ (t) ∈ is called a motion of the Lagrangian systems Q,L if γ (t) is a critical point of the action F, i.e., This definition is known as the Hamiltonian variational principle or Hamiltonian variational principle of least action or simple Hamiltonian principle. Now we need the Lagrange lemma or fundamental lemma of calculus of variations (see for instance [1]) Lemma 11 Let f be a continuous function of the interval [t 0 , t 1 ] satisfying the equation

Corollary 12
The Hamiltonian principle for Lagrangian systems is equivalent to the Lagrangian equations Proof Clearly, if (30) holds, by Corollary 2, δ F = 0. The reciprocal result follows from Lemma 11.
From the formal point of view, the Hamiltonian principle is equivalent to the problem of variational calculus [13,40]. However, despite the superficial similarity, they differ essentially. Namely, in mechanics the symbol δ stands for the its virtual variation, i.e., it is not an arbitrary variation but a displacement compatible with the constraints imposed on the systems. Thus, only in the case of the holonomic systems, for which the number of degrees of freedom is equal to the number of generalized coordinates, the virtual variations are arbitrary and the Hamiltonian principle is completely equivalent to the corresponding problem of the variational calculus. An important difference arises for the systems with nonholonomic constraints, when the variations of the generalized coordinates are connected by the additional relations usually called Chetaev conditions which we give later on. Let M * be the submanifold of R × T Q defined by the equations (??), i.e.,

D'Alembert-Lagrange principle
The constraint is called integrable if it can be written Otherwise the constraint is called nonintegrable. According to Hertz [17] the nonintegrable constraints are also called nonholonomic.
The Lagrangian systems with nonintegrable constraints are usually called (also following to Hertz) the nonholonomic mechanical systems or nonholonomic constrained mechanical systems, and with integrable constraints are called the holonomic constrained mechanical systems or holonomic constrained Lagrangian systems. The systems free of constraints are called Lagrangian systems or holonomic systems.
Sometimes it is also useful to distinguish between constraints that are dependent on or independent of time. Those that are independent of time are called scleronomic, and those that depend on time are called rheonomic. This therminology can also be applied to the mechanical systems themselves. Thus, we say that the constrained Lagrangian systems are scleronomic (reonomic) if the constraints and Lagrangian are time independent (dependent).
The constraints where a k j = a k j (t, x), a k = a k (t, x), are called linear constraints with respect to the velocity. For simplicity we shall call linear constraints. We observe that (31) admits an equivalent representation as a Pfaffian equations (for more details see [38]) We shall consider only two classes of systems of equations, the equations of constraints linear with respect to the velocity (ẋ 1 , . . . ,ẋ N ), or linear with respect to the differential (dx 1 , . . . , dx N , dt). In order to study the integrability or nonintegrability problem of the constraints the last representation of a Pfaffian system is the more useful. This is related with the fact that for the given 1-forms we have the Frobenius theorem which provides the necessary and sufficient conditions under which the 1-forms are closed and consequently the given set of constraints is integrable.
The constrains L j (t, x,ẋ) = 0 are called perfect constraints or ideal if they satisfy the Chetaev conditions (see [8]) In what follows, we shall consider only perfect constraints.
If the constraints admit the representation (26) then the Chetaev conditions take the form The virtual variations of the variables x α for α = 1, . . . , M are called dependent variations and of the variable The admissible path is called the motion of the constrained Lagrangian systems (Q,L, The following result is well known (see for instance [2,5,15,35]).

Proposition 13 The d'Alembert-Lagrange principle for constrained Lagrangian systems is equivalent to the Lagrangian differential equations with multipliers
where μ α for α = 1, . . . , M are the Lagrangian multipliers.

The varied path
The varied path produced in Hamiltonian's principle is not in general an admissible path if the perfect constraints are nonholonomic, i.e., the mechanical systems cannot travel along the varied path without violating the constraints. We prove the following result, which shall play an important role in the all assertions below.

Proposition 14
If the varied path is an admissible path then the following relations hold Proof Indeed, the original path γ (t) = x(t) by definition satisfies the Chetaev conditions and constraints, i.e., L j (t, x(t),ẋ(t)) = 0. If we suppose that the variation path γ * (t) = x(t) + εδx(t) also satisfies the constraints i.e., Thus, restricting only to the terms of first order with respect to ε and by the Chetaev conditions we have (for simplicity we omitted the argument) Subtracting these relations from (35) we obtain (34). Consequently, if the varied path is an admissible path, then relations (34) must hold.
From (34) and (7) it follows that the elements of the matrix A satisfy This property will be used below.
Clearly, the equalities (37) are satisfied if (29) holds. We observe that in general for holonomic constrained Lagrangian systems relation (29) cannot hold (see example 2).

Transpositional relations
As we observe in the previous subsection for nonholonomic constrained Lagrangian systems the curves, obtained doing a virtual variation in the motion of the systems, in general are not kinematical possible trajectories when (29) is not fulfilled. This leads to the conclusion that the Hamiltonian principle cannot be applied to nonholonomic systems, as it is usually employed for holonomic systems. The essence of the problem of the applicability of this principle for nonholonomic systems remains unclarified (see [35]). In order to clarify this situation, it is sufficient to note that the question of the applicability of the principle of stationary action to nonholonomic systems is intimately related to the question of transpositional relation.
The key point is that the Hamiltonian principle assumes that the operations of differentiation with respect to the time d dt and the virtual variation δ commute in all the generalized coordinate systems.
For the holonomic constrained Lagrangian systems relations (29) cannot hold (see Corollary 3). For a nonholonomic systems the form of the Hamiltonian principle will depend on the point of view adopted with respect to the transpositional relations.
What are then the correct transpositional relations? Until now, there does not exist a common point of view concerning to the commutativity of the operation of differentiation with respect to the time and the virtual variation when there are nonintegrable constraints. Two points of view have been maintained. According to one (supported, for example, by Volterra, Hamel, Hölder, Lurie, Pars,…), the operations d dt and δ commute for all the generalized coordinates, independently if the systems are holonomic or nonholonomic, i.e., According to the other point of view (supported by Suslov, Voronets, Levi-Civita, Amaldi,…) the operations d dt and δ commute always for holonomic systems, and for nonholonomic systems with the constraintṡ the transpositional relations are equal to zero only for the generalized coordinates x M+1 , . . . , x N , ( for which their virtual variations are independent). For the remaining coordinates x 1 , . . . , x M , (for which their virtual variations are dependent), the transpositional relations must be derived on the basis of the equations of the nonholonomic constraints, and cannot be identically zero, i.e., The second point of view acquired general acceptance and the first point of view was considered erroneous (for more details see [35]). The meaning of the transpositional relations (1) can be found in [20,32,34,35].
In the results given in the following section the equalities play a key role (34). From these equalities and from the examples it will be possible to observe that the second point of view is correct only for the so called Voronets-Chaplygin systems, and in general for locally nonholonomic systems. There exist many examples for which the independent virtual variations generated non-zero transpositional relations. Thus, we propose a third point of view on the transpositional relations: the virtual variations can generate the transpositional relations given by the formula (7) where the elements of the matrix A satisfy the conditions (see formula (36)) we observe that here the L α = 0 are constraints which in general are nonlinear in the velocity.

Hamiltonian-Suslov principle
After the introduction of the nonholonomic mechanics by Hertz, it appeared the question of extending to the nonholonomic mechanics the results of the holonomic mechanics. Hertz [17] was the first in studying the problem of applying the Hamiltonian principle to systems with nonintegrable constraints. In [17] Hertz wrote: "Application of Hamilton's principle to any material systems does not exclude that between selected coordinates of the systems rigid constraints exist, but it still requires that these relations could be expressed by integrable constraints. The appearance of nonintegrable constraints is unacceptable. In this case the Hamilton's principle is not valid." Appell [3] in correspondence with Hertz's ideas affirmed that it is not possible to apply the Hamiltonian principle for systems with nonintegrable constraints Suslov [48] claimed that "Hamilton's principle is not applied to systems with nonintegrable constraints, as derived based on this equation are different from the corresponding equations of Newtonian mechanics".
The applications of the most general differential principle, i.e., the d'Alembert-Lagrange and their equivalent Gauss and Appel principle, is complicated due to the presence of the terms containing the secondorder derivative. On the other hand the most general variational integral principle of Hamilton is not valid for nonholonomic constrained Lagrangian systems. The generalization of the Hamiltonian principle for nonholonomic mechanical systems was deduced by Voronets and Suslov (see for instance [48,53]). As we can observe later on from this principle it follows the importance of the transpositional relations to determine the correct equations of motion for nonholonomic constrained Lagrangian systems.

Proposition 16 The d'Alembert-Lagrangian principle for the contrained Lagrangian systems
Proof From the d'Alembert-Lagrangian principle we obtain the identity where δL is a variation of the LagrangianL. After the integration and assuming that δx k (t 0 ) = 0, δx k (t 1 ) = 0 we easily obtain (2), which represent the most general formulation of the Hamiltonian principle (Hamilton-Suslov principle) suitable for constrained and unconstrained Lagrangian systems.
Suslov determines the transpositional relations only for the case when the constraints are of Voronets type, i.e., given by the formula (22). Assume that Voronets and Suslov deduced that Thus, we obtain This is the Hamiltonian principle for nonholonomic systems in the Suslov form (see for instance [48]). We observe that the same result was deduced by Voronets in [53].
It is important to observe that Suslov and Voronets require a priori that the independent virtual variations produce the zero transpositional relations. Sometimes these authors consider only linear constraints with respect to the velocity of the type (22).

Modification of the vakonomic mechanics (MVM)
As we observe in the introduction, the main objective of this paper was to construct the variational equations of motion describing the behavior of the constrained Lagrangian systems in which the equalities (34) take place in the most general possible way. We shall show that the d'Alembert-Lagrange principle is not the only way to deduce the equations of motion for the constrained Lagrangian systems. Instead of it we can apply the generalization of the Hamiltonian principle, whereby the motions of such systems are extremals of the variational Lagrange problem (see for instance [13]), i.e., the problem of determining the critical points of the action in the class of curves with fixed endpoints and satisfying the constraints. The solution of this problem as we shall see will give the differential equations of second order which coincide with the well-known classical equations of the mechanics except perhaps in a zero Lebesgue measure set. From the previous section we deduce that in order to generalize the Hamiltonian principle to nonholonomic systems we must take into account the following relations pgwhere L α = 0 for α = 1, . . . , M are the constraints.
A lot of authors consider that (C) is always fulfilled (see for instance [32,38]), together with the conditions (A) and (B). However, these conditions are incompatible in the case of the nonintegrable constrains. We observe that these authors deduced that the Hamiltonian principle is not applicable to the nonholonomic systems.
To obtain a generalization of the Hamiltonian principle for the nonholonomic mechanical systems, some of these three conditions must be excluded.
In particular for the Hölder principle conditions (A) is excluded and keep (B) and (C) (see [18]). For the Hamiltonian-Suslov principle condition (A) and (B) hold, and (C) only holds for the independent variations.
In this paper we extend the Hamiltonian principle by supposing that conditions (A) and (B) hold and (C) does not hold . Instead of (C) we consider that (7) holds where elements of matrix A satisfy the relations (38).

Solution of the inverse problem of the constrained Lagrangian systems
We shall determine the equations of motion of the constrained Lagrangian systems using the Hamiltonian principle with non-zero transpositional relations, whereby the motions of the systems are extremals of the variational Lagrange's problem (see for instance [13]), i.e., are the critical points of the action functional in the class of path with fixed endpoints satisfying the independent constraints L j (t, x,ẋ) = 0, for j = 1, . . . , M.
In the classical solution of the Lagrange problem usually we apply the Lagrange multipliers method which consists in the following. We introduce the additional coordinates = (λ 1 , . . . , λ M ) , and Lagrangian L : Under this choice we reduce the Lagrange problem to a variational problem without constraints, i.e., we must determine the extremal of the action functional  L (t, x,ẋ, )) dt = 0 under the additional condition that the transpositional relations are given by the formula (7). The solution of the inverse problem stated in Sect. 2 is the following. Differential equations obtained from (6) are given by the formula (8) (see Theorem 1). We choose the arbitrary functions L j in such a way that the matrices W 1 and W 2 given in Theorems 2 and 3 are nonsingular, except perhaps in a zero Lebesgue measure set. The constants λ 0 j for j = M + 1, . . . , N are arbitrary in Theorem 2, and λ 0 j for j = 1, . . . , N − 1 are arbitrary and λ 0 N = 0 in Theorem 3. The matrix A is determined from the equalities (11) and (15)  Using the arbitrariness of the constants λ 0 j we can always take that , x). Consequently, we can always suppose that L L. Thus, the only difference between the classical and the modified Lagrangian multipliers method consists only on the transpositional relations: for the classical method the virtual variations produce zero transpositional relations (i.e., the matrix A is the zero matrix) and for the modified method in general it is determined by the formulae (7) and (36). A very important subcase is obtained when the constraints are given in the form (Voronets-Chapliguin constraints type)ẋ α − α (t, x,ẋ M+1 , . . . ,ẋ N ) = 0, for α = 1, . . . , M. As we shall show under these assumptions the arbitrary functions are determined as follows: L j =ẋ j for j = M + 1, . . . , N . Consequently, the action of the modified Lagrangian multipliers method and the action of the classical Lagrangian multipliers method are equivalently. In view of (26) this equivalence always locally holds for any constrained Lagrangian systems.

Proof of Theorems 1, 2 and 3
Proof of Theorem 1 In view of the equalities Consequently, Here we use the equalities δx(t 0 ) = δx(t 1 ) = 0. Hence if (8) holds then (6) is satisfied. The reciprocal result is proved by choosing where ζ(t) is a positive function in the interval (t * 0 , t * 1 ), and it is equal to zero in the intervals [t 0 , t * 0 ] and [t * 1 , t 1 ], and applying Corollary .
From the definition (8) we have that where a is a constant. Now we shall write (8) in a more convenient way From these relations and since the constraints L j = 0 for j = 1, . . . , M, we easily obtain Eq. (9) or equivalently Thus, the theorem is proved. Now we show that the differential Eq. (39) for convenient functions L j constants λ 0 j for j = M + 1, . . . , N and for convenient matrix A describe the motion of the constrained Lagrangian systems.

Proof of Theorem 2
The matrix Eq. (11) can be rewritten in components as follows: for α, k = 1, . . . , N . Consequently, the differential Eq. (39) become which coincide with the first systems (12). In view of the condition |W 1 | = 0 we can solve Eq. (11) with respect to A and obtain A = W −1 1 1 . Hence, by considering (40) we obtain the second systems from (12) and the transpositional relation (13).

Proof of Theorem 3 The matrix Eq. (15) is equivalent to the systems
for k = 1, . . . , N , and α = 1, . . . , N − 1. Thus, by considering that λ 0 N = 0 we deduce that systems (39) take the form Hence we obtain systems (16). On the other hand from (15) we have that A = W −1 2 2 . Hence we deduce that the transpositional relation (7) can be rewritten in the form (17).
The mechanics based on the Hamiltonian principle with non-zero transpositional relations given by formula (7), Lagrangian (4), and equations of motion (8) are called here the modification of the vakonomic mechanics and we shortly write MVM.
From the proofs of Theorems 2 and 3 it follows that the relations (36) hold identically in MVM.

Corollary 18 Differential equations (12) are invariant under the change
where the a j s are constants for j = 1, . . . , N .

Remark 19
The following interesting facts follow from Theorems 2 and 3.
(1) The equations of motion obtained from Theorem 2 are more general than the equations obtained from Theorem 3. Indeed in (12) where W 1 and W 2 are the matrixes defined in Theorems 2 and 3.
If the constraints are nonlinear in the velocity and |W 2 | = 0 everywhere in M * then we have the equivalence The equivalence with respect to the equations D ν L 0 = M j=1 dλ j dt ∂ L j ∂ẋ ν in general is not valid in this case because

Application of Theorems 2 and 3 to the Appell-Hamel mechanical systems
As a general rule the constraints studied in classical mechanics are linear with respect to the velocities, i.e., L j can be written as (31). However, Appell and Hamel (see [3,16]), in 1911, considered an artificial example of nonlinear nonholonomic constrains. A big number of investigations have been devoted to the derivation of the equations of motion of mechanical systems with nonlinear nonholonomic constraints see for instance [9,16,35,36]. The works of these authors do not contain examples of systems with nonlinear nonholonomic constraints differing essentially from the example given by Appell and Hamel.

Corollary 20
The equivalence (42) also holds for the Appell -Hamel system i.e., for the constrained Lagrangian systems where a and g are positive constants.
Proof The classical equations (33) for the Appell-Hamel system arë where μ is the Lagrangian multiplier. Now we apply Theorem 3. Hence, in order to obtain that |W 2 | = 0 everywhere we choose the functions L j for j = 1, 2, 3 as follows: In this case the matrices W 2 , 2 , and A are By considering that |W 2 | L 1 =0 = 1 + a 2 , we obtain that the Eq. (16) in this case describe the global behavior of the Appell-Hamel systems and take the form Clearly, this system coincides with classical differential equations (44) withλ = μ. After the derivation of the constraintż −a ẋ 2 +ẏ 2 = 0 along the solutions of (45), we obtain 0 =z −aẍ ẋ 2 +ẏ 2 + aÿ ẋ 2 +ẏ 2 = −g + (1 + a 2 )λ.
Therefore,λ = g 1 + a 2 . Hence the equations of motion (45) becomë In this case the Lagrangian (14) writes where C and λ 0 2 are arbitrary constants.
Under the condition L 1 = 0 we obtain that the transpositional relations are From this example we obtain that the independent virtual variations δx and δy produce non-zero transpositional relations. This result is not in accordance with with the Suslov point on view on the transpositional relations. Now we apply Theorem 2. The functions L 0 , L 1 , L 2 , and L 3 are determined as follows: Thus, the matrices W 1 and 1 are where q = a(ẍẏ −ẍẏ) ẋ 2 +ẏ 2 3 . Therefore, |W 1 | = −1. Hence, after some computations from (11) we have that The equations of motion (12) becomë By solving these equations with respect toẍ,ÿ, andz we obtain the equations We observe in this case that |W 1 | = −1, consequently these equations, obtained from Theorem 2, give a global behavior of the Appell-Hamel systems, i.e., coincide with the classical equations (44) witḣ The transpositional relations (13) can be written as From this corollary we observe that the independent virtual variations δx and δy produce non-zero transpositional relations (47) and zero transpositional relations (49).
The Lagrangian (10) in this case takes the form
In the next sections we show the importance of the equations of motion (12) and (16) contrasting them with the classical differential equations of nonholonomic mechanics.

Modificated vakonomic mechanics versus vakonomic mechanics
Now we show that the equations of the vakonomic mechanics (3) can be obtained from Eq. (9). More precisely, if in (7) we require that all the virtual variations of the coordinates produce the zero transpositional relations, i.e., the matrix A is the zero matrix and we require that λ 0 j = 0 for j = M + 1, . . . , N , then from (9) by considering that D k L = E k L , we obtain the vakonomic equations (3), i.e., In the following example in order to contrast Theorem 2 with the vakonomic model we study the skate or knife edge on an inclined plane.
Example 1 To set up the problem, consider a plane with cartesian coordinates x and y, slanted at an angle α. We assume that the y-axis is horizontal, while the x-axis is directed downward from the horizontal and let (x, y) be the coordinates of the point of contact of the skate with the plane. The angle ϕ represents the orientation of the skate measured from the x-axis. The skate is moving under the influence of the gravity. Here the acceleration due to gravity is denoted by g. It also has mass m, and the moment inertia of the skate about a vertical axis through its contact point is denoted by J, (see page 108 of [35] for a picture). The equation of nonintegrable constraint is With these notations the Lagrangian function of the skate iŝ Thus, we have the constrained mechanical systems For appropriate choice of mass, length, and time units, we reduce the LagrangianL to here for simplicity we leave the same notations for the all variables. The question is, what is the motion of the point of contact? To answer this question we shall use the vakonomic equations (3) and the Eq. (12) proposed in Theorem 2.
We study the motion of the skate in the following three cases: For the first case (ω = 0), after the change of variables X = cos ϕ 0 x − sin ϕ 0 y, Y = cos ϕ 0 x + sin ϕ 0 y, the differential equations (9) and the constraint becomë respectively. Consequently, thus the trajectories are straight lines. For the second case (ω = 0), we take ϕ 0 =ẏ 0 = x 0 = x 0 = y 0 = 0 in order to simplify the computations. In view of the equalityφ =φ| t=0 = ω and denoting by the derivation with respect ϕ we get that (54) becomes x + y = g sin α ω 2 cos 2 ϕ, which are easy to integrate and we obtain x = − g sin α 4ω 2 cos (2ϕ), y = − g sin α 4ω 2 sin (2ϕ) + g 2ω 2 ϕ, ϕ = ωt, which correspond to the equation of the cycloid. Hence the point of contact of the skate follows a cycloid along the plane, but does not slide down the plane.

The study of the skate applying vakonomic model
Now we consider instead of Theorem 2 the vakomic model for studying the motion of the skate.
The equations of motion (3) for the skate are We shall study only the case when α = 0. After integration we obtain the differential systemṡ ϕ (a cos ϕ + b sin ϕ) , where a =ẋ 0 −λ 0 sin ϕ 0 , b =ẏ 0 +λ 0 cos ϕ 0 and λ 0 = λ| t=0 is an arbitrary parameter. After the integration of the third equation we obtain that where h is an arbitrary constant which we choose in such a way that From (57) we get sin ϕ = sn t h + a 2 + b 2 2 , , where sn and cn are the Jacobi elliptic functions . Hence, if we takeẋ 0 = 1,ẏ 0 = ϕ 0 = 0, then the solutions of the differential equations (56) are It is interesting to compare this amazing motions with the motions that we obtained above. For the same initial conditions the skate moves sideways along the circles. By considering that the solutions (58) depend on the arbitrary parameter λ 0 we obtain that for the given initial conditions, there do not exist a unique solution of the differential equations in the vakonomic model. Consequently, the principle of determinacy is not valid for vakonomic mechanics with nonintegrable constraints (see the Corollary of page 36 in [2]).

MVM versus Lagrangian mechanics
The Lagrangian equations which describe the motion of the Lagrangian systems can be obtained from Theorem 2 by supposing that M = 0, i.e., there is no constraints We choose the arbitrary functions L α for α = 1, . . . , N as follows: Hence the Lagrangian (10) takes the form In this case we have that |W 1 | = 1. By considering the property of the Lagrangian derivative (see (27)) we obtain that 1 is a zero matrix . Hence the matrix A 1 is the zero matrix. As a consequence the Eq. (12) become The transpositional relations (13) in this case are δ dx dt − dδx dt = 0, which are the wellknown relations in the Lagrangian mechanics (see formula (29)).

MVM versus constrained Lagrangian systems
From the equivalences (42) we have that in the case when the constraints are linear in the velocity the equations of motions of the MVM coincide with the Lagrangian equations with multipliers (33) except perhaps in a zero Lebesgue measure set |W 2 | = 0 or |W 1 | = 0. When the constraints are nonlinear in the velocity, we have the equivalence (43). Consequently, equations of motions of the MVM coincide with the Lagrangian equations with multipliers (33) except perhaps in a zero Lebesgue measure set |W 2 | = 0. We illustrate this result in the following example.

Example 2 Let
be the constrained Lagrangian systems. In order to apply Theorem 2 we choose the arbitrary function L 1 and L 2 as follows: Thus, the matrices W 1 and 1 are Consequently, Eq. (12) describe the motion everywhere for the constrained Lagrangian systems. Equations (12) becomë Transpositional relations take the form
From this example we obtain that for the holonomic constrained Lagrangian systems the transpositional relations can be non-zero (see (59)), or can be zero (see (60)). We observe that from condition (34) it follows the relation This equality holds identically if (60) and (59) take place.
The equations of motions (33) in this case arë Example 3 To contrast the MVM with the classical model we apply Theorems 2 to the Gantmacher's systems (see for more details [12,45]).
Two material points m 1 and m 2 with equal masses are linked by a metal rod with fixed length l and small mass. The systems can move only in the vertical plane and so the speed of the midpoint of the rod is directed along the rod. It is necessary to determine the trajectories of the material points m 1 and m 2 .
Let (q 1 , r 1 ) and (q 2 , r 2 ) be the coordinates of the points m 1 and m 2 , respectively. Clearly, (q 1 − q 2 ) 2 + (r 1 − r 2 ) 2 = l 2 . Thus, we have a constrained Lagrangian system in the configuration space R 4 with the Lagrangian function L = 1 2 q 2 1 +q 2 2 +ṙ 2 1 +ṙ 2 2 − g/2(r 1 + r 2 ), and with the linear constraints Introducing the following change of coordinates: we obtain . Hence we have the constrained Lagrangian The equations of motion (33) obtained from the d'Alembert-Lagrange principle arë where μ 1 , μ 2 are the Lagrangian multipliers such that For applying Theorem 2 we have the constraints and we choose the arbitrary functions L 3 and L 4 as follows: For the given functions we obtain that Consequently, differential equations (12) take the form Derivating the constraints we obtain that the multipliersλ 1 andλ 2 arė Inserting these values into (63) we deducë These equations coincide with equations (61) every- , where l is the length of the rod. The transpositional relations in this case are From this example we again get that the virtual variations produce the non-zero transpositional relations.

Remark 21
From the previous example we observe that the virtual variations produce zero or non-zero transpositional relations, depending on the arbitrary functions which appear in the construction of the proposed mathematical model. Thus, the following question arises: Can be chosen the arbitrary functions L j for j = M + 1, . . . , N in such a way that for the nonholonomic systems only the independent virtual variations would generate zero transpositional relations?
The positive answer to this question is obtained locally for any constrained Lagrangian systems and globally for the Chaplygin-Voronets mechanical systems and for the generalization of these systems studied in the next section.

MVM and nonholonomic generalized Voronets-Chaplygin systems. Proofs of Theorem 4 and Proposition 5 and 6.
It was pointed out by Chaplygin [7] that in many conservative nonholonomic systems the generalized coordinates (x, y) := x 1 , . . . , x s 1 , y 1 , . . . , y s 2 , s 1 + s 2 = N , can be chosen in such a way that the Lagrangian function and the constraints take the simplest form.
In particular Voronets in [53] studied the constrained Lagrangian systems with LagrangianL =L (x, y,ẋ,ẏ) and constraints (22). This systems is called the Voronets mechanical systems. We shall apply Eq. (12) to study the generalization of the Voronets systems, which we define now.
An example of generalized Voronets systems is Appell-Hamel systems analyzed in the previous subsection.

Corollary 22 Every Nonholonomic constrained Lagrangian mechanical systems locally is a generalized Voronets mechanical systems.
Proof Indeed, the independent constraints can be locally represented in the form (26). Thus, by introducing the coordinates then we have that any constrained Lagrangian mechanical systems is locally a generalized Voronets mechanical systems.
Proof of Theorem 4 For simplicity we shall study only scleronomic generalized Voronets systems.
To determine Eq. (12) we suppose that It is evident from the form of the constraint equations that the virtual variations, δy, are independent by definition. The remaining variations, δx, can be expressed in terms of them by the relations (Chetaev's conditions) We shall apply Theorem 2. To construct the matrix W 1 . We first determine L s 1 +1 , . . . , L s 1 +s 2 = L N as follows: Hence, the Lagrangian (4) becomes y,ẏ)) .
respectively, where a α j = ∂ L α ∂ẏ j , and the matrices 1 respectively. Consequently, the differential equations (12) take the form (18). The transpositional relations (13) in view of (67) take the form (21). As we can observe from (21) the independent virtual variations δy for the systems with the constraints (66) produce the zero transpositional relations. The fact that the transpositional relations are zero follows automatically and it is not necessary to assume it a priori, and it is valid in general for the constraints which are nonlinear in the velocity variables.
We observe that the relations (34) in this case take the form for α = 1, . . . , s 1 . Clearly, from (21) these relations hold identically.
From differential equations (18), eliminating the Lagrangian multipliers we obtain Eq. (19). After some computations we obtain d dt for k = 1, . . . , s 2 . By introducing the function = L 0 | L 1 =...=L s 1 =0 , Eq. (71) can be written as beingL the Lagrangian of (65). Now we establish the relations between Eq. (18) and the classical Voronets differential equations with the Lagrangian function L * =L . The functionsL and L * are determined in such a way that Eq. (19) take place in view of the equalities for k = 1, . . . , s 2 , which in view of equalities d dt ∂ L * ∂ẋ ν = 0 for ν = 1, . . . , s 1 , take the form (20). (20) describe the motion of the constrained generalized Voronets systems with Lagrangian L * and constraints (66). The classical Voronets equations for scleronomic systems are easy to obtain from (20) with α = s 2 k=1 a α k (x, y)ẏ k .

Proof of Proposition 5 Equations
Finally by considering Corollary 4 we get that differential equations (20) describe locally the motions of any constrained Lagragian systems.

Proof of Proposition 6
To determine the differential equations which describe the behavior of the generalized Chaplygin systems we apply Theorem 2, with for α = 1, . . . , s 1 and β = s 1 + 1, . . . , s 2 and consequently the matrix W 1 is given by the formula (69) and Therefore, the differential equations (12) take the form The transpositional relations are By excluding the Lagrangian multipliers from (75) we obtain the equations In this case Eq. (73) take the form Analogously to the Voronets case we study the subcase when = 0. We choose L 0 =L (y,ẋ,ẏ) − L (y, ,ẏ) :=L − L * . We assume that the functions L and L * are such that For the generalized Chaplygin systems the Lagrangian L takes the form for j = 1, . . . , s 1 where the constants C j for j = 1, . . . , s 1 are arbitrary. Indeed, from (75) it follows that By inserting in (4) L 0 =L − L * and λ j for j = 1, . . . , s 1 we obtain function L of (79). We note that Vorones and Chaplygin equations with nonlinear constraints in the velocity were also obtained by Rumiansev and Sumbatov (see [44,47]).

Example 4
We shall illustrate the above results in the following example.
In the Appel's and Hamel's investigations the following mechanical system was analyzed. A weight of mass m hangs on a thread which passes around the pulleys and is wound round the drum of radius a. The drum is fixed to a wheel of radius b which rolls without sliding on a horizontal plane, touching it at the point B with the coordinates (x B , y B ). The legs of the frame support the pulleys and keep the plane of the wheel vertical slide on the horizontal plane without friction. Let θ be the angle between the plane of the wheel and the Ox axis; ϕ the angle of the rotation of the wheel in its own plane; and (x, y, z) the coordinates of the mass m. Clearly, The coordinates of the point B and the coordinates of the mass are related as follows (see page 223 of [35] for a picture): x = x B + ρ cos θ, y = y B + ρ sin θ.
The condition of rolling without sliding leads to the equations of nonholonomic constraints: x B = a cos θφ,ẏ B = a sin θφ b > 0.
We observe that the constraintsż = bφ admit the representatioṅ Denoting by m 1 , A and C the mass and the moments of inertia of the wheel and neglecting the mass of the frame, we obtain the following expression for the Lagrangian functioñ L = m + m 1 2 ẋ 2 +ẏ 2 + m 2ż 2 +m 1 ρθ (sin θẋ − cos θẏ) + A + m 1 ρ 2 2θ 2 + C 2φ 2 − mgz.
Consequently, if ρ = 0 then Hamel in [16] neglects the mass of the wheel (m 1 = J = C = 0). Under these conditions the previous equations become ρ 2ÿ 1 + aρ bẏ 1ẏ2 = 0, (a 2 + b 2 )ÿ 2 − abρẏ 2 1 = −gb Appell and Hamel obtained the example of nonholonomic system with nonlinear constraints by means of the passage to the limit ρ → 0. However, as a result of this limiting process, the order of the system of differential equations is reduced, i.e., they become degenerate. In [35] the authors study the motion of the nondegenerate system for ρ > 0 and ρ < 0. From these studies it follows that the motion of the nondegenerate system (ρ = 0) and degenerate system (ρ → 0) differ essentially. Thus, the Appell-Hamel example with nonlinear constraints is incorrect. Clearly, these relations are independent of , A, C, and m 1 .

Consequences of Theorems 2 and 3 and the proof of Corollary 1.
We observe the following important aspects from Theorems 2 and 3.
(I) Conjecture 8 is supported by the following facts. (a) As a general rule the constraints studied in classical mechanics are linear in the velocities. However Appell and Hamel, in 1911, considered an artificial example with a constraint nonlinear in the velocity . As it follows from [35] (see example 4) this constraint does not exist in the Newtonian mechanics.
(b) The idea developed for some authors (see for instance [4]) to construct a theory in Newtonian mechanics, by allowing that the field of force depends on the acceleration, i.e., function ofẍ as well as of the position x, velocityẋ, and the time t, is inconsistent with one of the fundamental postulates of the Newtonian mechanics: when two forces act simultaneously on a particle the effect is the same as that of a single force equal to the resultant of both forces (for more details see [38] pages [11][12]. Consequently, the forces depending on the acceleration are not admissible in Newtonian dynamics. This does not preclude their appearance in electrodynamics, where this postulate does not hold.
(c) Let T be the kinetic energy of the constrained Lagrangian systems. We consider the generalization of the Newton law: the acceleration (see [37,46] ∂x is equal to the force F. Then in the differential equations (12) with L 0 = T we obtain that the field of force F generated by the constraints is If the constraints are nonlinear in the velocity, then F 1 depends onẍ. Consequently, in Newtonian mechanics does not exist a such field of force. Therefore, the existence of nonlinear constraints in the velocity and the meaning of force F 1 must be sought outside of the Newtonian model.
For example, for the Appel-Hamel constrained Lagrangian systems studied in the previous subsection we have that a 2ẋ x 2 +ẏ 2 (ẋÿ −ẏẍ), a 2ẏ x 2 +ẏ 2 (ẋÿ −ẏẍ), 0 . For the generalized Voronets systems and locally for any nonholonomic constrained Lagrangian systems from the Eq. (18) we obtain that the field of force F 1 has the following components: for k = 1 . . . , N and s 1 = M. Consequently, such field of force does not exist in Newtonian mechanics if the constraints are nonlinear in the velocity.
(II) Equations (12) can be rewritten in the form with the Lagrangian mechanics we obtain that for the unconstrained Lagrangian systems the transpositional relations are always zero. Thus, we have the proof of the corollary.