When is a control system mechanical

In this work we present a geometric setting for studying mechanical control systems. We distinguish a special class: the class of geodesically accessible mechanical systems, for which the uniqueness of the mechanical structure is guaranteed (up to an extended point transformation). We characterise nonlinear control systems that are state equivalent to a system from this class and we describe the canonical mechanical structure attached to them. Several illustrative examples are given.


1.
Introduction. Mechanical control systems provide an important and challenging research area whose roots come from classical mechanics and modern nonlinear control. Both mechanics and control theory have independent and rich histories which overlap in a very interesting way: mechanics provides challenging control problems and nonlinear control brings new ideas and results which lead to the progress of mechanics.
An extensive overview of the history of mechanics can be found in [1]. A crucial period in the development of mechanics, which occurred together with the development of mathematics, dates back to Newton, Euler, Lagrange, Laplace, Poisson, Hamilton and Jacobi. After this an important impulse was carried out by Poincaré, who introduced new ideas and methods based on his geometric point of view of mechanics. Mathematical control theory is a more recent subject. Newer developments in nonlinear control theory can be found for example in the books [4,7,10,22,26,42,55]. Bibliographical notes in the overlapping history of mechanics and control theory are described in [7,10,28,45].
Two aspects make mechanical control systems a very attractive subject of research. On one hand, they are abundant in real life and, on the other hand, they offer very interesting mathematical problems. Examples of mechanical control systems can be found in robotics, automation, autonomous vehicles in marine, aerospace, flight control, problems in nuclear magnetic resonance, micro-electromechanical systems, and fluid mechanics (see [6,7,10,11,21,35,36,38,41,45,46,57,60]). Questions like controllability, accessibility, observability, motion planning, stabilization and tracking of these systems are of major interest for applications and the emerging related questions provide the starting point for very interesting mathematical theories.
Throughout this paper we shall rest on the Lagrangian point of view on mechanical control systems [7,10,40,45]. For an approach based on the Hamiltonian formalism we refer the reader to Chapter 12 of [42] (see also [1,7,16]). This paper is a study of mechanical control systems, which we define to be systems described by the second-order differential equations on a smooth manifold Q, called the configuration manifold. Here x = (x 1 , . . . , x n ) are local coordinates on Q and u = (u 1 , . . . , u m ) are inputs (controls) of the system. We use the summation convention throughout the paper, except for terms involving controls. The expression Γ i jk (x)ẋ jẋk corresponds to Coriolis and centrifugal terms, the terms d i j (x)ẋ j correspond to dissipative-type (or gyroscopic-type) forces acting on the system, g 0 represents an uncontrolled force (which can be potential or not) and g 1 , . . . , g m represent controlled forces.
Among systems of the above class are mechanical control systems which are neither subject to dissipative-type (or gyroscopic-type) forces nor uncontrolled ones. These systems are known in the literature as affine connection control systems (see for instance, [10,30,31,32,34]). This subclass of mechanical control systems is particularly interesting and has attracted a lot of attention in recent years. On one hand, many applications fit to this subclass, and, on the other hand, the equations of motion of affine connection control systems are simpler than for general mechanical control systems (1), but still they retain the crucial features of general mechanical control systems.
Given a nonlinear control system of the form on a smooth manifold M, we will be interested in answering the main question: (i) When is the control system Σ mechanical, that is, when is it equivalent via a diffeomorphism to a mechanical system (MS) of the form (1)? Related questions immediately arise: (ii) Can a control system admit more than one mechanical structure? (iii) Which types of mechanical control systems do we have? (iv) How to geometrically characterise different types of mechanical control systems? Our problem (i) is analogous to the classical inverse problem in the calculus of variations (in Lagrangian mechanics), which is to decide whether the solutions of a given system of second order ordinary differential equations are the solutions of Euler-Lagrange equations for some Lagrangian function L(x,ẋ), see, e.g., [5,14,17,37,39,53,54]. On one hand, our problem is more restrictive because we look for a mechanical Lagrangian only (L being the kinetic minus potential energy). On the other hand, our problem is more general. First, we do not assume any a priori tangent bundle structure on M (in other words, we do not assume that the original equations are of second order) and its construction constitutes a part of the problem. Secondly, we allow for dissipative-like terms. Thirdly, we require that also controlled vector fields take a form compatible with the Lagrangian (mechanical) structure. It is actually the presence of the controlled vector fields (and their interactions with the vector field defining the considered differential equation) that allow us to solve the problem.
Our solution of the problem (i) is the central result of the paper being a source of further investigations and results and for the convenience of the reader we will formulate it now. We will call a zero-velocity point for the mechanical control system (MS) of the form (1) any point of the form (x 0 ,ẋ 0 ) = (x 0 , 0), that is, any point of the zero section of the tangent bundle T Q. Let V denote the smallest vector space, over R, containing the vector fields G 1 , . . . , G m and satisfying Theorem. Let M be a smooth 2n-dimensional manifold. A system Σ is locally, at z 0 ∈ M, equivalent via a diffeomorphism to a geodesically accessible mechanical system (MS) of the form (1)  See Section 3 for the definition of geodesic accessibility and an equivalent computable definition of V and Section 4 for a detailed geometric and mechanical interpretation of the conditions (MS0)-(MS2). Here we just want to emphasize that they can be easily verified in terms of the control system Σ and that they encode all structure information about the mechanical system equivalent to Σ.
The paper is organized as follows. Section 2 provides some preliminary notions, notation, and, in particular, gives the background on mechanical control systems. Also, some illustrative examples of mechanical control systems are presented. In Section 3, we define geodesically accessible mechanical control systems (GAMS) and illustrate this notion with some examples. Then we characterise this special subclass of mechanical control systems establishing our main result, which is Theorem 3.2 in Subsection 3.2. A geometric interpretation of that characterisation is given in Section 4. The uniqueness of the mechanical structure is studied in Section 5. Also a characterisation of mechanical control systems (MS) that are not necessarily geodesically accessible is given. Section 6 is devoted to special forms of (GAMS) systems and Section 7 presents the proof of our main result. Section 8 contains two examples exhibiting a local nature of the results in this paper. Finally, we give conclusions and some future lines of research in Section 9.

2.
Preliminaries. In this section we present some mathematical tools and establish some notation that will be used throughout the paper.
2.1. Control-affine systems, vector fields and distributions. We shall consider control-affine systems, which we define as a pair Σ = (M, C), where (i) M is a finite dimensional smooth manifold, (ii) C = {F, G 1 , . . . , G m } is an (m + 1)-tuple of smooth vector fields on M.
The differential equations describing the evolution of a control-affine system are written asγ in which u : I → R m is the control or input vector; γ : I → M, I ⊂ R, is the corresponding trajectory of Σ in the state manifold M ; G 1 , . . . , G m are the control (or input) vector fields, and finally, F is the drift vector field, which describes the dynamics of the system in the absence of controls. Given a smooth manifold M, we shall denote by X(M ) the module of smooth vector fields on M and by C ∞ (M ) the ring of smooth real-valued functions on M. We will denote by z = (z 1 , . . . , z n ) local coordinates on M.
Let f ∈ X(M ) and λ ∈ C ∞ (M ). We denote by L f λ the Lie derivative of λ along f, which is defined as Given f, g ∈ X(M ), the Lie bracket of f and g is a new smooth vector field denoted by [f, g] and defined, in coordinates, as where Dg(z) and Df (z) denote the Jacobi matrix of g and f in z-coordinates, respectively. We will use often the notation ad 0 f g = g and, inductively, ad j+1 f g = [f, ad j f g], j ≥ 0. Let M and N be smooth manifolds of the same finite dimension and consider a diffeomorphism Φ : M → N. Given a vector field f on M, we define the transformed vector field Φ * f on N (the image of f under Φ) as A distribution D on M is a map that assigns to each point p in M a linear subspace D(p) of the tangent space at this point T p M . If D is a subbundle of the tangent bundle T M, that is, if the dimension dim D(p) is constant, and equal to k, we say that D is of constant rank k. In what follows we will assume that all distributions are of constant rank. Locally a smooth distribution D of constant rank k can be spanned by k linearly independent smooth vector fields, which we will denote as D = span{f 1 , . . . , f k }. Given a vector field f, we say that f belongs to the distribution D if f (p) ∈ D(p) for all p ∈ M. In this case we write f ∈ D.
2.2. Affine connections. Let Q be an n-dimensional smooth manifold.
An affine connection on Q is a map which satisfies the following conditions: for all X, Y, Z ∈ X(Q) and all α 1 , α 2 , α ∈ C ∞ (Q). The vector field ∇ X Y is called the covariant derivative of the vector field Y with respect to the vector field X.
In a system of local coordinates (x 1 , . . . , x n ) on Q, an affine connection is uniquely determined by its Christoffel symbols Γ i jk , 1 ≤ i, j, k ≤ n, which are defined by Here the summation convention is used with a sum on the repeated index i.
Using the properties of an affine connection, the definition of the Christoffel symbols and writing the vector fields X and Y as X = X j ∂ ∂x j and Y = Y k ∂ ∂x k , we obtain the covariant derivative ∇ X Y written in coordinates as Since the covariant derivative ∇ X Y at a point q ∈ Q depends only on the value of X at q, we define the covariant derivative of Y along a smooth curve γ : I ⊂ R → Q, as a vector field along γ given by A geodesic of an affine connection ∇ on Q is a smooth curve γ on Q satisfying ∇γ (t)γ (t) = 0. In local coordinates, γ(t) = x(t) = (x 1 (t), . . . , x n (t)), a geodesic is given as the solution of the system of second-order differential equations: This second-order system is equivalent to a system of first-order equations on the tangent bundle T Q, with the coordinates (x, y) = (x 1 , . . . , x n , y 1 , . . . , y n ) on T Q induced by the x-coordinates (x 1 , . . . , x n ) on Q : jk y j y k . These equations define a vector field S on T Q which is called the geodesic spray of ∇ and is given in local coordinates by The integral curves of S, being curves on T Q, are projected onto the geodesics on Q under the canonical projection π : T Q → Q, π(q, v q ) = q. Associated with an affine connection, there is a symmetric real-bilinear operation called the symmetric product: In coordinates, we have

SANDRA RICARDO AND WITOLD RESPONDEK
A smooth distribution D on Q is called geodesically invariant with respect to a smooth affine connection ∇ if every geodesic γ : I → Q, having the property that γ (t 0 ) ∈ D(γ(t 0 )) for some t 0 ∈ I, satisfies γ (t) ∈ D(γ(t)) for all t ∈ I. The following geometric interpretation of the symmetric product was given by Lewis in [30]. A distribution D on a manifold Q, equipped with an affine connection ∇, is geodesically invariant if and only if X : Y ∈ D, for every X, Y ∈ D.
Given an affine connection ∇, we define the maps which are called, respectively, the torsion tensor and the curvature tensor of the connection ∇.
We say that ∇ is symmetric if T (X, Y ) = 0 for all X, Y ∈ X(Q). In this case, for any local coordinates, the corresponding Christoffel symbols are symmetric, that is, Given a vector field k on Q, we define its vertical lift K = k vlift ∈ X(T Q), which is the vector field on T Q, given as In coordinates, if Hence K is a vector field which is annihilated when projected by the canonical projection π.
2.3. Mechanical control systems. We define a mechanical control system (MS) as a 4-tuple (Q, ∇, g 0 , d), in which (i) Q is an n-dimensional configuration manifold; (ii) ∇ is a symmetric affine connection on Q; (iii) g 0 = (g 0 , g 1 , . . . , g m ) is an (m + 1)-tuple of smooth vector fields on Q; (iv) d : T Q → T Q is a map sending the fiber T q Q into the fiber T q Q, for any q ∈ Q, linear on fibers. A curve γ : I → Q, I ⊂ R, is a trajectory of (MS) if it satisfies the equation Our definition of (MS) is motivated by the following considerations. We consider a kinetic energy Lagrangian L : T Q → R, L = K(x, y), with K the kinetic energy, that is, a positive definite symmetric bilinear form on T Q, defining a Riemannian metric. The tangent bundle T Q is called the velocity phase space. Assume that the system is subject to a field of external forces F : T Q → T * Q, with T * Q the cotangent bundle of Q, which is, by definition, a smooth map that sends the fiber T q Q into the fiber T * q Q, for any q ∈ Q. Then, the equations of motion (Euler-Lagrange equations with external forces) are given by We consider the force field F of the form where (i) η 0 is a differential 1-form on Q, representing a positional external force that cannot be controlled (which can be conservative or not); (ii) η 1 , . . . , η m are differential 1-forms on Q representing positional input forces (i.e., external controlled forces); (iii) d : T Q → T * Q is a map sending T q Q into T * q Q and linear on fibers. In (x, y)-coordinates on T Q the external force F is then expressed by The forces η 0 and η r , 1 ≤ r ≤ m, are positional, that is, for any q ∈ Q, Our force field F contains thus arbitrary forces that depend on configurations only, and arbitrary forces that are linear in velocities. We observe that, if L = K − V, with V : Q → R the potential energy, then the force η 0 will consist of an external uncontrolled force together with the potential force −dV.
The force d generalizes the notion of a dissipative or gyroscopic force. To see this, let D be a (0, 2)-tensor, that is, a bilinear form on each T x Q, i.e., x Q is linear on fibers, and in coordinates, with D ij being the same matrix as that defining D. We see that, indeed, d generalises the notion of dissipative forces (for which the matrix D ij is supposed to be symmetric negative definite) and that of gyroscopic forces (if D ij is antisymmetric). We do not put any assumption on the matrix D ij . The force d will be called a d-force.
The generalized Newton law gives where µ : T Q → T * Q, given by µ(v q )(·) = K(v q , ·) for all v q ∈ T Q, is a diffeomorphism associated to the kinetic energy K, which is called the Legendre transformation or mass operator. Then, µ −1 : T * Q → T Q and we denote This yields which justifies our definition of (MS), given by (7). We observe that g 0 : Q → T Q and g r : Q → T Q, 1 ≤ r ≤ m, are vector fields on Q, whereas d : T Q → T Q is a map sending T q Q into T q Q and linear on fibers, that is, in coordinates In local coordinates (x 1 , . . . , x n ) on Q, equation (7) is equivalent to the secondorder system of differential equations (1) or, to the first-order system of differential equations on T Q, equipped with coordinates (x 1 , . . . , x n , y 1 , . . . , y n ) : In a condensed form, the latter equation can also be written aṡ whereΓ(x, y) is an R n -valued function, homogeneous of degree two with respect to y, i.e., Γ(x, y) = (Γ 1 (x, y), . . . ,Γ n (x, y)) T , withΓ i (x, y) = y T Γ i (x)y, 1 ≤ i ≤ n, and each element Γ i is a symmetric n × n matrix with components Γ i jk (x). As we already explained d(x, y) = d i j (x)y j is homogeneous of degree one with respect to y.

2.4.
Some examples of mechanical control systems. We present now some simple examples to illustrate our class of mechanical control systems. These examples are simplified models constructed in order to have a small number of states and controls. We briefly introduce each example, outlining information about the configuration of the system, the kinetic and potential energy, the forces acting on the system and the equations of motion. Then, we exhibit the objects of the definition of a mechanical control system given in the last subsection.
Example 2.4.1 (The planar rigid body). The presentation of this example closely follows [10]. This system can be thought of as a model for a planar rigid body or, equivalently, for a simplified hovercraft, as depicted in Figure 1.
We consider the inertial reference frame R spatial = (O spatial , {s 1 , s 2 }), and the body reference frame R body = (O body , {b 1 , b 2 }), in which O body coincides with the center of mass of the body. Two movements are allowed: the planar rigid body can translate in the plane and rotate about its center of mass. Let q = (θ, x, y) be the configuration of the system, with θ describing the relative orientation of the body reference frame R body with respect to the inertial reference frame R spatial , and the vector (x, y) denoting the position of the center of mass measured with respect to the inertial reference frame R spatial . 1 Letq = (θ,ẋ,ẏ) be the spatial velocity, that is, the velocity of the system with respect to the inertial coordinate system. The state of the system is given by (q,q), that is, the state consists of the configuration along with the spatial velocity. We assume that the system moves in a plane perpendicular to the direction of the gravitational forces, so that the potential energy is zero. Concerning the kinetic energy, it is given by Here "T " stands for transpose, m is the mass of the body and J denotes its moment of inertia about the center of mass. We consider a force F applied to a point on the body that is at a distance h > 0 from the center of mass, along the body b 1 -axis, as shown in the figure. Let u 1 be the component of F in the body b 1 -direction and u 2 be the component in the b 2 -direction. The equations of motion for the planar rigid body system are thus given by [10]). The kinetic energy defines the Riemannian structure G = Jdθ⊗dθ+m(dx⊗dx+dy⊗dy). Thus, the Christoffel symbols of the corresponding Levi-Civita connection ∇ (see e.g., [10,44]) are Γ i jk = 0. Also, η 0 = 0 and d = 0 (implying that, respectively, g 0 = 0 and d = 0) and the control forces are η 1 = cos θdx + sin θdy and η 2 = −hdθ − sin θdx + cos θdy, 1 In agreement with the notation used in the Section 2.3, it would be logical to denote the configuration of the system by q = (x 1 , x 2 , x 3 ). However, physically it is more natural to use q = (θ, x, y). giving the input vector fields The mechanical control system is then Example 2.4.2 (The robotic Leg). The presentation of this example follows closely [10] and [33]. This system consists of a main body which rotates about a fixed point, and at that fixed point is attached an extensible leg with a point mass m at its tip (see Figure 2). The moment of inertia of the base rigid body about the pivot point is J and the direction of the gravitational field is assumed to be orthogonal to the plane of motion of the system (see [10]). We denote q = (r, θ, ψ) the configuration of the system, with r describing the extension of the leg, θ the angle of the leg with respect to an inertial reference frame and ψ the angle of the body. The configuration manifold is Q = R + × S 1 × S 1 . Two input forces are considered: the torque η 1 = dθ −dψ applied at the point of rotation that controls the angle between the body and the leg, and the force η 2 = dr that extends the leg. There exists neither uncontrolled forces nor dissipative-type forces (η 0 = 0 and d = 0). The equations of motion describing the system are [10,33]: The potential energy of this system is zero and the kinetic energy is given by which defines the Riemannian structure G = mdr ⊗ dr + mr 2 dθ ⊗ dθ + Jdψ ⊗ dψ. The corresponding affine connection ∇ is the Levi-Civita connection and the only nonzero Christoffel symbols for ∇ are computed to be (compare with [10]) The input vector fields are obtained to be the uncontrolled vector field is g 0 = 0 and also d = 0. The mechanical control system is then Example 2.4.3 (The PVTOL Aircraft). We consider now a simple model of a planar vertical take off and landing (PVTOL) aircraft as depicted in Figure 3. This mechanical control system was introduced in [21] and has attracted a lot of attention in recent years (see for example [36,43]). Similar to the rigid body example, the configuration of the system is q = (θ, x, y), with θ the angle the PVTOL makes with the horizontal line and (x, y) the position of the aircraft center of mass. Let u 1 be the control corresponding to the body vertical force and let u 2 be the control corresponding to forces on the tips of the wings.
The dynamics of this simplified version of the PVTOL, after normalisation of m and J, are given as (see [36,43]) where the constant a g is the gravity acceleration and = 0 is a fixed constant related to the geometry of the aircraft. We obtain for the control vector fields, g 0 = −a g ∂ ∂y being the uncontrolled vector field and d = 0.

Example 2.4.4 (Pendulum with Damping)
. All examples that we have presented so far are with no d-forces. Now we will consider the pendulum with dampinġ with θ the angle and ω the angular velocity. We consider (θ, ω) ∈ T Q = S 1 × R. By m, l, and a g we denote, respectively, the mass, the length, and the gravitational acceleration, and k is the damping-coefficient. The controlled force applied to the pendulum is the torque η 1 = udθ, the uncontrolled force is the gravitational force η 0 = −ma g l sin θdθ, and the map d : T θ Q → T * θ Q associates the covector −kω θ to the angular velocity ω θ at θ. Represented as a mechanical control system, the pendulum takes the formθ ∂θ is the controlled vector field and the fiber-linear map d : T θ Q → T θ Q associates the vector − k ml 2 ω θ to the velocity ω θ . 2.5. State equivalence to mechanical control systems. In this paper, we shall consider state equivalence transformations. We will study the problem of when a control system admits a mechanical structure, that is, when it is state equivalent to a mechanical system. Feedback equivalence of a nonlinear control system to a given Hamiltonian system has been considered in [12,29,58] and equivalence to a gradient system in [13].
Two control systems are said to be state equivalent if they are related by a diffeomorphism (and then also their trajectories, corresponding to the same controls, are related by that diffeomorphism) [22,42,48]. More precisely, consider two control systems Definition 2.1. We say that Σ andΣ are state equivalent, shortly S-equivalent, if there exists a diffeomorphism Φ : M →M such that Recall that Φ * stands for the tangent map of Φ (see Subsection 2.1). The systems Σ andΣ are called locally S-equivalent, at z 0 ∈ M andz 0 ∈M , respectively, if there exists neighborhoods U of z 0 andŨ ofz 0 , such that Σ restricted to U andΣ restricted toŨ are S-equivalent.
The crucial property of state equivalence is that the diffeomorphism Φ establishing the S-equivalence yields a one-to-one correspondence between trajectories of Σ andΣ corresponding to the same controls.
If one of the systems is mechanical, we obtain the following definition, fundamental for our considerations. Assume that the dimension of M is 2n. Definition 2.2. We say that Σ is S-equivalent to a mechanical control system if there exists a mechanical system of the form where (x, y) = (x 1 , . . . , x n , y 1 , . . . , y n ) are local coordinates on T Q, such that Σ and (MS) are S-equivalent. In this case, we may also say that Σ admits a mechanical structure.
In other words, Σ is S-equivalent to (MS) if the diffeomorphism Φ, establishing the S-equivalence, satisfies Analogously, we define local S-equivalence of Σ at z 0 to a mechanical system (MS) at (x 0 , y 0 ) ∈ T Q. In this case, we say that Σ locally admits a mechanical structure.

Definition and examples.
Consider the mechanical control system (MS) = (Q, ∇, g 0 , d). Let SYM(g 1 , . . . , g m ) be the smallest distribution on Q containing the input vector fields g 1 , . . . , g m and such that it is closed under the symmetric product defined by the connection ∇.
and geodesically accessible if the above equality holds for all x 0 ∈ Q. A geodesically accessible mechanical system will be denoted by (GAMS).
Example 3.1.1. Let (x 1 , x 2 ) be coordinates of R 2 in which a mechanical system (MS) reads as 2ẍ and is defined by the affine connection ∇ on R 2 , given by the Christoffel symbols, which in these coordinates are Γ 1 22 = 1 2 and Γ i jk = 0, otherwise. We have g = ∂ ∂x 2 . By (5), we obtain g : and we conclude that this system is a (GAMS).
Consider the linear mechanical system on R 2 , written in coordinates (x 1 , x 2 ),ẍ 1 = u, We consider the affine connection determined by the Christoffel symbols which in these coordinates vanish (that is, the affine connection is the Euclidean). For g := ∂ ∂x 1 , we have SYM(g) = span{g}. We conclude that this system is not a (GAMS). It is noteworthy that this linear system is controllable and so, in particular, accessible and strongly accessible (definitions and a detailed analysis of accessibility, strong accessibility and controllability can be found in [10,22,42,56]; see also [4,7,26]).
Example 3.1.3 (Planar rigid body cont'd). As in Example 2.4.1, we consider the configuration coordinates q = (θ, x, y) ∈ Q = S 1 × R 2 and the input vector fields We get the symmetric product of these vector fields as (recall that all Christoffel symbols vanish) .
from which we conclude that the system is a (GAMS).
Example 3.1.4 (Robotic Leg cont'd). As observed in Example 2.4.2, the configuration of the system is q = (r, θ, ψ) ∈ Q = R + × S 1 × S 1 and the input vector fields are given by We obtain (using the Christoffel symbols calculated in Example 2.4.2) ∂ ∂r and g 1 : g 2 = g 2 : g 2 = 0.
Clearly, g 1 : g 1 is colinear with g 2 (compare with [10]). Direct computations show that all symmetric products of higher order are either colinear with g 2 or vanish. Therefore, SYM(g 1 , g 2 ) = span{g 1 , g 2 } from which we conclude that the system is not a (GAMS), although it is accessible.

Statement of the main result.
For any vector field F and any collections G 1 and G 2 of vector fields, we will put Define the following sequence of families of vector fields on M : and, inductively, A point z 0 ∈ M is said to be an equilibrium point for the system Σ if F (z 0 ) = 0. We will call a zero-velocity point for the mechanical control system (MS) any point of the form (x 0 , y 0 ) = (x 0 , 0), that is any point of the zero section of T Q. An equilibrium point for (MS) is a zero-velocity point such that, additionally, we have g 0 (x 0 ) = 0. (Recall that g 0 is the uncontrolled vector field in the definition of the mechanical control system (MS)). The condition (MS0) implies that the diffeomorphism establishing the S-equivalence (if it exists) will map z 0 into a zero-velocity point. A mechanical system (more generally, a control system that is S-equivalent to a mechanical system (MS)) is geodesically accessible around a zero-velocity point if and only if it satisfies (MS0) and (MS1) (see Corollary 4.5 in Section 4). The condition (MS2) is always necessary for S-equivalence to a mechanical system (MS) (see Proposition 5.2 below) and sufficient provided that (MS1) and (MS2) hold. It says that the Lie algebra L = {F, G 1 , . . . , G m } LA contains an abelian subalgebra V which is the structural condition reflecting the existence of a mechanical structure of Σ.
Observe that the definition of V, given in Section 1, as the smallest vector space, over R, containing the vector fields G 1 , . . . , G m and satisfying coincides with the constructive one given by (12). Indeed, let us denote by U the smallest vector space, over R, containing the vector fields G 1 , . . . , G m and satisfying (12). By construction, V satisfies [V, ad F V] ⊂ V and, since U is the smallest vector space with that property, it follows that U ⊂ V. To prove V ⊂ U, notice that , in which, as usual, a sum is understood over the index i, with i = 1, 2. We have Thus dim V(z) = 2 and dim (V + [F, V]) (z) = 4, for any z ∈ R 4 , and [V, V] = 0. By Theorem 3.2, the system is S-equivalent to a (GAMS) around a zero-velocity point (x 0 , 0). Indeed, the diffeomorphism 3 2 (y 2 ) 2 , y 2 = y 2 transforms the above system into the system of Example 3.1.1.
Example 3.2.2 (Planar rigid body cont'd). We have seen in Example 3.1.3 that the planar rigid body corresponds to a (GAMS). We shall see now that, indeed, it satisfies the conditions of Theorem 3.2. Denote the coordinates of T Q by (θ, x, y,θ,ẋ,ẏ). Set Direct Lie brackets computations show that Also, Hence . Indeed, as we shall see below, under these three conditions, we are able to find the configuration manifold, to define suitable vector fields on it, to find a connection and a map (linear on fibers) defining the d-forces.
Objects defined for the system Σ due to the conditions (MS0)-(MS2) (i.e., in particular, due to the structural condition [V, V] = 0) will be denoted by an upper index Σ : Q Σ , g Σ r , ∇ Σ and d Σ . 4 These objects will be obtained locally around a point z 0 ∈ M.
The proof of the sufficiency part of Theorem 3.2 (see Section 7) shows that, under the conditions (MS0)-(MS2), there exist n independent and commuting vector fields V 1 , . . . , V n in the set V and we can find a local system of coordinates (x, y) such that in a neighborhood O z0 of z 0 ∈ M the vector fields V j are given by V j = ∂ ∂y j , 1 ≤ j ≤ n. We will continue our considerations in that open set O z0 , where the vector fields V j are rectified. In particular, span V = span ∂ ∂y 1 , . . . , ∂ ∂y n . We will define the configuration manifold as (see Figure 4) We can easily check that Q Σ is, indeed, an n-dimensional smooth submanifold of M. To see this, consider, in (x, y)-coordinates, the vector field F represented as Let us consider the smooth map defined on O z0 : x, y), . . . , f n (x, y)) T .
We observe that the differentials df 1 (x, y), . . . , df n (x, y) are independent on O z0 (since the vector fields ad F V 1 , . . . , ad F V n are independent mod span V). Thus, 4 Since the objects Q Σ , g Σ r , ∇ Σ , d Σ are defined by V, it would be appropriate to denote them by Q Σ,V , g Σ,V r , ∇ Σ,V , d Σ,V , but we will skip the symbol V for compactness of notation.
The tangent space of Q Σ at its arbitrary point q = (x, y) is the set and the distribution span V = span ∂ ∂y 1 , . . . , ∂ ∂y n defines the canonical foliation The configuration manifold Q Σ is transversal to the leaves of F. Indeed, the matrix is of full rank (which is a direct consequence of the independence of the vector fields ad F V 1 , . . . , ad F V n mod spanV). Therefore, Define the surjective submersion π : O z0 → Q Σ by attaching to any z ∈ O z0 ⊂ M the unique point q = π(z), such that q ∈ Q Σ ∩ L z , where L z is the leaf passing through z.
Any vector field V ∈ V gives rise to a vector field v on Q Σ . Indeed, by definition of V, the vector field ad F V satisfies the condition [ad F V, V] ⊂ V, which implies that ad F V projects to the vector field on Q Σ : v := −π * (ad F V ).
In particular, since G r ∈ V, 1 ≤ r ≤ m, the above formula defines the control vector fields on Q Σ by: g Σ r := −π * (ad F G r ). In this way, given n locally independent vector fields V 1 , . . . , V n ∈ V we obtain a local frame v 1 , . . . , v n on Q Σ . Alternatively, given a local frame v 1 , . . . , v n on Q Σ , there exists a unique collection of independent vector fields V 1 , . . . , V n ∈ spanV such that V j = (v j ) vlift , 1 ≤ j ≤ n (recall the identity (6) for the vertical lift of a vector field on Q Σ ).
Let V denote the module generated by all vector fields belonging to V, over the ring of smooth functions on O z0 that are constant on the leaves of the canonical foliation F. Clearly, any vector field V ∈ V projects to a vector field v on Q Σ given by the above formula v = −π * (ad F V ) but, conversely, for any vector field v ∈ Q Σ there exists a unique vector field V ∈ V such that v = −π * (ad F V ), which defines the vertical lift of v, namely v vlift = V.
Using that one-to-one correspondence of the modules V and X(Q Σ ), we will proceed now to define an affine connection on Q Σ for any system satisfying (MS1) and (MS2).
Consider the map ∇ Σ : with v, w arbitrary vector fields on Q Σ and V, W their corresponding vertical lifts. We will prove that ∇ Σ defined by the above equality is, indeed, an affine connection.
This observation, together with the fact that any vector field from V is annihilated by π, that is π * V = 0, for all V ∈ V, allows to obtain: Moreover, we have As we have just seen, the conditions (MS0)-(MS2) encode enough information that allows to define canonically the configuration manifold Q Σ , the control vector fields g Σ 1 , . . . , g Σ m and to define the affine connection ∇ Σ . To completely define a mechanical structure, we still have to identify the uncontrolled vector field g 0 and the fibers-linear map d : Recall that by V we denote the module of vector fields generated by V over the ring of smooth functionsᾱ on M such that L Vᾱ = 0, for all V ∈ V. Note that there exists a unique vector field G 0 ∈ V, such that G 0 and F coincide on Q Σ (clearly, F is not tangent to Q Σ ). In (x, y)-coordinates, G 0 is of the form We have [ad F G 0 , V] ⊂ V, which allows to consider the vector field g Σ 0 defined on Q Σ as g Σ 0 = −π * (ad F G 0 ). Take n vector fields V 1 , . . . , V n ∈ V, defined on O z0 , and the corresponding projected vector fields (14)). So, for any q ∈ Q Σ ⊂ M, we decompose where v 1 (q) ∈ T q Q Σ , v 2 (q) ∈ V(q), and we define the value of the map d Σ (q) on v(q) ∈ T q Q Σ by where v 2 (q) ∈ V(q) is interpreted as an element of T q Q due to the identification of V with X(Q).
In the remaining part of the section, we will express, in terms of the vector fields F and G 1 , . . . , G m , the symmetric product and then we will calculate the torsion and the curvature of the canonical connection ∇ Σ .
Recall the definition of the symmetric product (see Section 2.2) and let g = {g 1 , . . . , g m } be the collection of input vector fields for system (7). We consider the following sequence of families of vector fields on Q : Note that the smallest distribution on Q containing the vector fields g 1 , . . . , g m and closed under the symmetric product is the distribution spanned by Sym(g), that is, we have SYM(g) = span Sym(g). For an alternative but equivalent definition of Sym(g) see [10].
The following formula generalises the analogous one proved in [33] for the case of F being a geodesic spray (see also [10]).
We recall the notation k vlift for the vertical lift of a vector field k on Q (see (6)).
Lemma 4.2. Let X and Y be arbitrary vector fields on the configuration manifold Q and Proof. Given X and Y, vector fields on Q, written in local coordinates as , their respective vertical lifts X vlift and Y vlift are expressed in local coordinates (x, y) of T Q, by We calculate The expression at the right-hand side of the last equality correspondes to the coordinate representation of X : Y vlift (see (5) and (6)).
Proof. We proceed by induction on i. The result is trivially true for i = 1. Now we suppose that the lemma is true for 1 ≤ j ≤ i − 1 and i ≥ 2. By definition of Sym i (g), for any v ∈ Sym i (g) there exist vector fields v 1 ∈ Sym p (g) and v 2 ∈ Sym l (g) such that v = v 1 : v 2 and p + l = i, 1 ≤ p, l ≤ i − 1.
The induction assumption yields LetF = Φ * F be the drift of the system (MS). Then, by construction of V i . On the other hand, lemma 4.2 yields Proof. Consider a geodesically accessible mechanical system around a zero-velocity point. Then there exist n independent vector fields v 1 , . . . , v n in Sym(g 1 , . . . , g m ), and we have for some vector fields V i ∈ V, because of Lemma 4.3 and Proposition 4.4. It follows that ad F V i , 1 ≤ i ≤ n, are independent mod span V. Clearly, we have F (z 0 ) ∈ V(z 0 ), for z 0 = (x 0 , 0) ∈ T Q, that is (MS0) holds. Moreover, this condition implies that V 1 , . . . , V n are independent, which yields (MS1). Indeed, F (z 0 ) ∈ V(z 0 ), implies that we can represent Since the vector fields ad F V i , 1 ≤ i ≤ n, are independent mod span V, it follows b i = 0 and hence n i=1 a i V i (z 0 ) = 0, a i ∈ R.

WHEN IS A CONTROL SYSTEM MECHANICAL? 287
We have where the first bracket vanishes since F 1 (z 0 ) = 0 and n i=1 a i V i (z 0 ) = 0, while the second bracket vanishes because of condition [V, V] = 0 (which is always necessary for S-equivalence to a mechanical system (MS)). It follows that Conversely, if a mechanical system satisfies (MS1), then there exist n independent vector fields V 1 , . . . , V n ∈ V. Put v i = −π * (ad F V i ), 1 ≤ i ≤ n. By Lemma 4.3 and Proposition 4.4, v 1 , . . . , v n ∈ Sym(g 1 , . . . , g m ) and they are independent, thus proving geodesic accessibility, which holds around a zero-velocity point because of condition (MS0).
Recall the definitions of torsion and curvature tensors of a connection ∇ given in Subsection 2.2.
Proof. We have and so, the connection has zero torsion. We get the expression for the curvature by simple application of equality (15).

(MS)-structures: uniqueness and characterisation.
In Section 3 we have found necessary and sufficient conditions for the system Σ to admit a (GAMS)structure around a zero-velocity point; in Section 4 we have shown how to construct canonically a (GAMS)-structure for systems admitting one. Now, in this section, we are concerned with the uniqueness of a (MS)-structure in general and the uniqueness of a (GAMS)-structure in particular. We start with a definition. Let (MS) and ( MS) be two mechanical systems. We say that they are mechanically state-equivalent, shortly, MS-equivalent (respectively, locally mechanical state equivalent at points (x 0 , y 0 ) and (x 0 ,ỹ 0 )) if they are S-equivalent (respectively, locally S-equivalent at points (x 0 , y 0 ) and (x 0 ,ỹ 0 )) under an extended point transformation Φ = (φ 1 , φ 2 ) : where (x, y) and (x,ỹ) are local coordinates, respectively, of (MS) and ( MS).

5.1.
Uniqueness of the (GAMS) structure. If we are given a system Σ that is already a (GAMS) = (Q, ∇, g 0 , d), around a zero-velocity point, then it is clear that it satisfies conditions (MS0)-(MS2), since they are necessary for S-equivalence to a (GAMS). But this means, in view of the considerations of the last section, that we can determine canonically the 4-tuple defining the structure of the system, that is, to construct Q Σ , ∇ Σ , g Σ 0 and d Σ . A natural question is to ask what is the relation between the original structure (Q, ∇, g 0 , d) and the canonical structure (Q Σ , ∇ Σ , g Σ 0 , d Σ ). It turns out that both structures coincide. More precisely, we have the following result.
Theorem 5.1. (i) For any system Σ that is already a (GAMS) = (Q, ∇, g 0 , d), the original mechanical structure coincides with the canonical one, that is, Proof. (i) Consider a (GAMS) given by (8) (or, equivalently, by (9)). We have span V = span{ ∂ ∂y 1 , . . . , ∂ ∂y n } (by geodesic accessibility) and hence Q Σ = {y = 0} which coincides with Q. Now, we will show that ∇ = ∇ Σ . The drift F of the (GAMS) is where Γ i jk are the Christoffel symbols of the connection ∇, defining the (GAMS). Let V i and V j be arbitrary vector fields in V: It is straightforward to get We observe that the vector field [V j , ad F V i ] defined by (20) gives rise to the vector field on Q : We have We obtain from which we conclude: which coincides with the formula (3). This shows that the functions Γ k rs (x), with 1 ≤ k, r, s ≤ n, are also the Christoffel symbols for the affine connection ∇ Σ . It follows that ∇ = ∇ Σ .

5.2.
Non-uniqueness of (MS) structures. In the last subsection we proved that, if a control system Σ admits a (GAMS)-structure, then it is unique up to an extended point transformation. The geodesic accessibility assumption plays a crucial role in guaranteeing the uniqueness of the mechanical structure. Indeed, as the next example shows, if the system is not geodesically accessible, then it can admit multiple mechanical structures. = (x 1 ∂ ∂x 2 ) vlift = x 1 ∂ ∂y 2 and without d-forces. We have G = ∂ ∂y 1 and F = The system is accessible (even strongly accessible) since the Lie algebra L generated by F and G is and hence dim L(p) = 4, for any p ∈ R 4 . The system is not geodesically accessible since V = Vect R {G}, which by Lemma 4.3 implies that SYM(g) = span Sym(g) = span{g}, in which g := −π * (ad F G).
The local diffeomorphism Ψ defined as 5 This shows that (MS) 1 and (MS) 2 are S-equivalent (as control systems) around the origin (0, 0) ∈ R 4 . Now we will prove that they are not MS-equivalent. To this end, assume that an extended point diffeomorphism (x,ỹ) = Φ(x, y) of the form We have Φ * G = Φ * ( ∂ ∂y 1 ) = ∂ ∂ỹ 1 implying that the components (φ 1 , φ 2 ) of φ are φ 1 = x 1 + a(x 2 ) and φ 2 = b(x 2 ), for some smooth functions a and b. Calculating we conclude that The third equation implies a = 0 thus givingỹ 1 = y 1 . Now pluggingỹ 1 = y 1 and y 2 = b y 2 into the fourth equation yields a contradiction. Therefore the above systems are, indeed, not MS-equivalent and we conclude that the system (MS) 1 is bi-mechanical (and so is (MS) 2 ). Indeed, it admits two nonequivalent mechanical structures: the original structure (MS) 1 and the structure (MS) 2 . Notice that for both mechanical structures, the configuration manifolds are the same: Q 1 = {y 1 = y 2 = 0} = {ỹ 1 =ỹ 2 = 0} = Q 2 but they have two different bundle structures. In the coordinates (x 1 , x 2 , y 1 , y 2 ) of (MS) 1 , the fibers T x Q 1 of the first structure are defined by {x 1 = c 1 , x 2 = c 2 , c 1 , c 2 ∈ R}, while the fibers of the second structure are given by {x 1 = c 1 , x 2 + 1 2 (y 2 ) 2 = c 2 , c 1 , c 2 ∈ R} and are mapped, via Ψ, into The non-MS-equivalence of the systems (MS) 1 and (MS) 2 is also seen if we compute the curvature tensors of the connections defining the systems. In fact, the connection defining (MS) 1 is flat (it is defined by the Euclidean metric) thus having zero curvature tensor, whereas the curvature tensor of the connection for (MS) 2 (defined by Γ 2 12 = Γ 2 21 = − 1 2(1+x 1 ) , remaining Γ i jk being zero) is non-zero. Indeed, using the definition of curvature tensor in Subsection 2.2, we calculate (for simplicity, we omit the "tildas"): The proof of the above proposition is actually included in the proof of Theorem 3.2 (see Section 7) and will be omitted. Item (i) implies that [V, V] = 0 is a structural condition and it is always necessary for the existence of a mechanical structure. If in the abelian Lie algebra V there are independent vector fields V 1 , . . . , V n that, together with ad F V i , are linearly independent everywhere, then (accordingly to Theorem 3.2), the system Σ admits a mechanical structure which is actually a (GAMS) and, as such, unique. In general, however, we may not be able to find n independent vector fields in V (see (ii)). In this case, the following result, which describes (not necessarily geodesically accessible) mechanical control systems (MS), still holds. Proof. The proof of the sufficiency part follows the same line as that of Theorem 3.2 (see Section 7) and will be omitted. To prove necessity, we start by observing that any V ∈ V is of the form with, as usual, a sum is understood over the index i, 1 ≤ i ≤ n, and, by Proposition 5.2, we have dim V(z 0 ) = k ≤ n. Let us choose k vector fields V 1 , . . . , V k in V which are independent at z 0 and denote W i = V i , for 1 ≤ i ≤ k. We can complete them by W k+1 = ∂ ∂y k+1 , . . . , W n = ∂ ∂y n (after permuting, if necessary, the coordinates x 1 , . . . , x n , y 1 , . . . , y n , accordingly) such that in a neighborhood of z 0 we have  1 , but also to W 2 = Vect R { ∂ ∂y 1 , 1 1+x 1 (−y 2 ∂ ∂x 2 + ∂ ∂y 2 )} which defines (when passing to (x,ỹ)-coordinates) the structure (MS) 2 .
6. When is a (GAMS) of special form? Theorem 3.2 describes all (GAMS)'s, that is, all control systems that admit a geodesically accessible mechanical structure.
In this section, we analyse systems that exhibit particular forms of the mechanical structure.
We shall use the notations and objects introduced in Section 4. In particular, recall that Q Σ = {z ∈ M | F (z) ∈ V(z)} stands for the configuration manifold of the control-affine system Σ.
for any z in a neighborhood of z 0 .
(ii) (GAMS) without uncontrolled forces, i.e. with g 0 = 0, around an equilibrium point, if and only if, for any z in a neighborhood of z 0 . Remark 6.2. As mentioned in Section 1, mechanical control systems for which d = 0 and g 0 = 0 are known in the literature as affine connection control systems and have been the subject of intensive research in recent years, see for instance, [10,30,31,32,34]. The case (iii) of Proposition 6.1 gives a characterisation of those affine connection control systems that are geodesically accessible. They are studied in detail in [49].
Proof of Proposition 6.1.
We recall that an arbitrary elementṼ = Φ * V ofṼ = Φ * V, has the form It follows, Since spanṼ = span ∂ ∂y 1 , . . . , ∂ ∂y n  x 1 = y 1 , x 2 = y 2 ,ẏ 1 = −y 2 ( 1 2 y 2 + x 1 + u), y 2 = x 1 + u, satisfies the conditions (MS1) and (MS2) for any z ∈ R 4 , thus being S-equivalent to a (GAMS) around any z 0 satisfying (MS0), that is, z 0 = (x 1 0 , x 2 0 , 0, 0). Now we compute We have thus showing that the conditions (MSND) and (MSNU) of Proposition 6.1 are not satisfied by this system. This is in accordance with the fact, mentioned in Example 3.2.1, that the above system is S-equivalent around any zero-velocity point to the system of Example 3.1.1, which is a (GAMS) with d-forces and an uncontrolled vector field.

Proof of the main result.
Proof. (Proof of Theorem 3.2) To prove necessity, let Φ : O z0 → T Q, O z0 ⊂ M, be a local diffeomorphism transforming the system Σ, restricted to a neighborhood O z0 of z 0 , into a (GAMS), with Φ(z 0 ) =z 0 = (x 0 , 0). We observe that in coordinates Φ(z) = (x, y) = (x 1 , . . . , x n , y 1 , . . . , y n ) ∈ T Q, we have for the mechanical system F = Φ * F andG r = Φ * G r as given by equalities (10) and (11), respectively. We constructṼ accordingly to (12), with G r and F being replaced byG r and F , respectively. Clearly,Ṽ = Φ * V. Direct Lie-bracket computations show that any elementṼ ofṼ is of the formṼ with, as usual, a sum understood over the index i, 1 ≤ i ≤ n. Therefore, Ṽ ,Ṽ = 0 and, clearly, Also, by (22), dimṼ(x, y) = dim V(z) ≤ n, for any z ∈ O z0 . Since dim M = dim T Q = 2n and the system (MS) is geodesically accessible around x 0 ∈ Q, we have (by definition) with O x0 an open neighborhood of x 0 . Therefore, there exist n independent vector fieldsṽ 1 , . . . ,ṽ n in Sym(g). Then, the vector fields on T Q given bỹ are independent and, by Lemma 4.3, of the formṼ j = Φ * V j , where each V j ∈ V i , for some i. This proves that dim V(z) ≥ n. But, again by Lemma 4.3, all elements of V are of that form, thus proving that, indeed, dim V(z) = n, for all z ∈ O z0 .
SinceF is of the form (10), and we have also dim Ṽ + [F ,Ṽ] (z) = dim (V + [F, V]) (z) = 2n, for any z, in a neighborhood of z 0 , thus completing the proof of (MS1). Moreover, the form of (10) implies that Φ * F (x 0 , 0) = g i 0 (x 0 ) ∂ ∂y i proving (MS0). To prove sufficiency, we start, due to condition (MS1), by choosing n vector fields in V, that are linearly independent at any z in a neighborhood O z0 of z 0 . Denote them by V 1 , . . . , V n . We claim that V 1 , . . . , V n , [F, V 1 ], . . . , [F, V n ] are linearly independent at any point of O z0 . Indeed, because of dim V(z) = n, any V ∈ V is of the form Then showing that V + ad F V is spanned, at any point z ∈ O z0 , by the vector fields V 1 , . . . , V n , [F, V 1 ], . . . , [F, V n ], thus proving our claim.
In the system of local coordinates (x, y), we express the drift as where f i (x 0 , 0) = 0, because of (MS0). To simplify notation, we will denote φ * F, φ * G r and φ * V j by F, G r and V j respectively. We have implying that all third order derivatives ∂ 3 F ∂y l ∂y k ∂y j vanish. The drift becomes with c i (x, y) andc i (x, y) being homogeneous functions of degree two with respect to y and everywhere we take sums over repeated indices i and j.
Since [V k , ad F V j ] ∈ span V = span ∂ ∂y 1 , . . . , ∂ ∂y n , for any 1 ≤ j, k ≤ n, then we get c i (x, y) = 0. It follows, where e i (x 0 ) = 0. We have wherec i (x,ỹ) is homogeneous of degree two with respect to y, and the control vector fields intoG Omitting the tilda notation and denotingc i (x,ỹ) = −Γ i jk (x)y j y k , we obtain the system in the form (8), which is geodesically accessible (by assumption (MS1) and Lemma 4.3). 8. The local nature of our results.
The following examples show that the presented results are of local nature. We would like to thank an anonymous reviewer for raising that issue.
By a direct calculation we get ad F G = − ∂ ∂x and hence [G, ad F G] = 0. It follows that V = Vect R {G} = Vect R {−2y ∂ ∂x + ∂ ∂y }. Now it is immediate to see that everywhere on M we have [V, V] = 0, implying the condition (MS2), and that dim V(z) = 1 and dim (V + [F, V]) (z) = 2, for any z ∈ M, implying (MS1). By Theorem 3.2, the system Σ is locally S-equivalent to a geodesically accessible mechanical system around any z 0 satisfying (MS0), that is, F (z 0 ) ∈ V(z 0 ). Notice that the latter defines the configuration manifold Q Σ = {z ∈ M | F (z) ∈ V(z)}, whose open subsets are configuration manifolds of local mechanical structures of Σ, as discussed in Section 4. Now assume that there exists a tangent bundle T Q, a geodesically accessible mechanical system (GAMS) and a global diffeomorphism Φ, from M onto T Q, transforming Σ into (GAMS). Because of the uniqueness of the geodesically accessible mechanical structure, it is clear that the restriction of Φ to Q Σ would establish a diffeomorphism between Q Σ and Q. Moreover, Φ would transform the integral leaves of the line-distribution (spanned by V) passing through any z 0 ∈ Q Σ into the fibers T q Q, where Φ(z 0 ) = (q, 0), with (q, 0) in the zero section of T Q evaluated at q. In particular, the union of integral leaves passing through all points of Q Σ should be mapped, via Φ, onto T Q. Now observe that the integral leaves are of the form {x + y 2 = c}, where c ∈ R, and the leaves corresponding to c < 0 cross Q Σ = {(x, y) | x < 0, y = 0} while those corresponding to c ≥ 0 do not. It follows that Φ(M − ) = T Q, where M − = {(x, y) | x + y 2 < 0}, thus contradicting the injectivity of Φ defined globally on M . Notice that the system Σ restricted to M − can be globally transformed by the diffeomorphism x = x + y 2 , y = y, into the linear geodesically accessible mechanical systeṁ x =ỹ, y = u, on T Q, where Q = {(x,ỹ) |x < 0,ỹ = 0}. In other words, the manifold M is too "large": we are not able to map the leaves {x + y 2 = c | c ≥ 0} into the fibers of T Q, but if we eliminate them, by restricting the manifold M to M − , the restriction of Σ to M − becomes globally equivalent to a mechanical system.
In the previous example, the topology of leaves of V is trivial (but there are leaves not intersecting Q Σ ). Now we will give an example of a locally mechanical system whose topology of leaves of V is nontrivial. We have ad F G = e y cos x ∂ ∂x − e y sin x ∂ ∂y and [G, ad F G] = 0. It follows that the system Σ satisfies everywhere on M = R 2 the conditions (MS1) and (MS2) of Theorem 3.2. Therefore, around any point z 0 satisfying (MS0), that is, any point z 0 of Q Σ = {z ∈ M | F (z) ∈ V(z)}={x = π 2 + kπ, k ∈ Z}, the system is locally equivalent to a geodesically accessible mechanical system. Notice that Q Σ , the candidate for the global configuration manifold, is not connected. Now assume that there exists a tangent bundle T Q, a geodesically accessible mechanical system (GAMS), and a global diffeomorphism Φ mapping M onto T Q and transforming the system Σ into (GAMS). By the uniqueness of the geodesically accessible mechanical structure, the diffeomorphism Φ should map the leaves of the line-distribution spanned by V into the fibers T q Q. As a consequence, the quotient space M/ ∼ , where z ∼z if they belong to the same leaf, should be diffeomorphic to the manifold Q, but M/ ∼ is not a smooth manifold. So Φ cannot be a global diffeomorphism. 9. Conclusions and further research. In this paper we have studied the geometry of general mechanical control systems (MS), which we define as a 4-tuple (MS) = (Q, ∇, g 0 , d), where Q is an n-dimensional configuration manifold; ∇ is a symmetric affine connection on Q; g 0 is an (m + 1)-tuple of vector fields which includes an uncontrolled vector field g 0 and input vector fields g 1 , . . . , g m ; and d : T Q → T Q is a smooth bundle map over Q (i.e., covering the identity) linear on fibers, that physically represents dissipative-type or gyroscopic-type forces (and their generalisations) acting on the system.
The subclass of geodesically accessible mechanical control systems (GAMS) has been of main interest for our considerations. We define a (GAMS) as a mechanical control system for which the smallest distribution on Q, containing the input vector fields and closed under the symmetric product coincides with T Q, at each point q ∈ Q.
We gave necessary and sufficient conditions for a control-affine system Σ to be state equivalent to a (GAMS) around a point (x 0 , 0) ∈ T Q. These conditions encode all the necessary information about the mechanical structure of the (GAMS). Studying the class of geodesically accessible mechanical systems reveals to be very natural, since, as we have shown in Section 5, if Σ admits a (GAMS)-structure then it is unique (up to an extended point transformation) and if the system is not geodesically accessible it can admit multiple mechanical structures.
Finally, we gave a characterisation of particular classes of (GAMS) systems. Namely, we have characterised (GAMS) that are subject neither to dissipativetype nor gyroscopic-type forces (that is, d = 0) and (GAMS) without uncontrolled forces (that is, g 0 = 0). A combination of these two cases leads to a characterisation of geodesically accessible affine connection control systems, a class of mechanical control systems that has recently attracted a lot of attention, see, e.g., [10,30,31,32,34]).
Also a characterisation of mechanical control systems (MS) that are not necessarily (GAMS) is provided.
In our opinion the presented study has opened many potential future directions of research, that form a natural continuation of this work. In an upcoming paper [49], the authors have studied the class of geodesically accessible affine connection control systems, constructing two families of structure functions which are equivariants for the state and mechanical state equivalence. Also, the results in the present paper have been used to characterise systems in a special form, namely a subclass of mechanical control systems subject to second-order nonholonomic constraints that includes, in particular, the second-order chained form [52].
A challenging problem is to use the methods and techniques developed in the paper in order to get a characterisation of Hamiltonian control systems and to relate that characterisation with the work obtained by Crouch and van der Schaft [15,16,59]. Another important class to study are mechanical control systems for which the uncontrolled vector field is a gradient vector field. Exploring relations between our approach with a recent characterisation of that class in [13] is a very natural and interesting problem. Refining the results in this paper to systems on Lie groups and their homogeneous spaces and obtaining global results, instead of local ones, are also very challenging future lines of research.
The equivalence studied in this paper is state equivalence and mechanical state equivalence. A very challenging and open problem is that of feedback equivalence [22,42,48] of a control-affine system to a mechanical system, in particular, a search of feedback invariants of mechanical systems. For a particular problem of feedback equivalence of a nonlinear control system to a given Hamiltonian system see [12,29,58]. Four approaches have been developed in order to study feedback equivalence and feedback invariants. First is based on studying equivalence of the distributions and vector fields defining the control systems and their singularities [23,47,51,61]. The second approach, proposed by Gardner, uses Cartan's method of equivalence [18,19,20]. The third method, inspired by the Hamiltonian formalism of optimal control, has been developed by Agrachev, Bonnard, and Jakubczyk [2,3,8,24,25]. Finally, a fourth approach was proposed by Kang and Krener [27] (see the survey [50] and references therein) and is based on formal equivalence. Perhaps using not just one of those approaches but combining them will allow to describe and understand nonlinear control systems that are feedback equivalent to a mechanical system.