Symmetries and Properties of the Energy-Casimir Mapping in the Ball-Plate Problem

In this paper a system derived by an optimal control problem for the ball-plate dynamics is considered. Symplectic and Lagrangian realizations are given and some symmetries are studied. The image of the energy-Casimir mapping is described and some connections with the dynamics of the considered system are presented.


Introduction
In 1983 the problem of optimal rolling of a sphere over a horizontal plane without slipping was posed by Hammersley [1].Using quaternion calculus of variations and optimal control theory, Arthurs and Walsh [2] obtained a system of differential equations that can be integrated analytically in terms of elliptic integrals.
A similar problem, named the ball-plate problem, was considered by Brockett and Dai [3].They have considered the problem of optimal rolling of a ball without slipping between two horizontal plates, the lower plate fixed, and the upper plate mobile.More precisely, by moving of the upper plate, the ball is rolling from the prescribed initial state into some terminal state along a path which minimizes  ∫  0 ‖V()‖ 2  among all possible path which satisfy these boundary conditions, where V() is the velocity of the moving plate,  is the time of transfer, and  is a positive constant.The control is the velocity of the center of the ball.The state of such a system is described by the position of the center of the ball and by the orientation of the ball in the threedimensional space.The nonslipping assumptions mean that the instantaneous velocity of the point of contact of the ball with the lower plate is equal to zero, and the velocity of the point of the contact with the moving plate is equal to the velocity of the moving plate.Such kind of motion is of great interest in mechanics, robotics, and control theory [4][5][6][7], making it a widely investigated problem.
The ball-plate problem has been formulated by Jurdjevic [8] as an optimal control problem on the Lie group R 2 × SO(3).Among the results obtained by Jurdjevic in his study we state here the equations of motion, their integrals of motion, and the integrability properties.Hamilton-Poisson formulations, a study of stability of the equilibrium points and numerical integration of the system obtained by Jurdjevic were presented in [9,10].We mention other topics regarding the problem of optimal rolling sphere such as exponential map and the corresponding group of symmetries, Maxwell points [11], parametrization of extremal trajectories by using elliptic functions, and asymptotics of extremal trajectories [12,13].
According to the Jurdjevic's paper [8], in the following, the derivation of the equations of motion in the ball-plate problem is recalled.
Let us consider two right-handed orthonormal frames in the space R 3 .One is a fixed frame e 1 , e 2 , e 3 centered at a point  in the lower plane, with e 3 perpendicular to this plane and pointing upward, and the other one is a moving frame a 1 , a 2 , a 3 fixed to the ball at its center.Let q = ( 1 ,  2 ,  3 ) be the coordinates of a point  on the surface of the ball relative to the fixed frame and let Q = ( 1 ,  2 ,  3 ) be the coordinates of the same point relative to the moving frame.

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Thus q = Q, where  ∈ SO(3) denotes the rotation matrix which transforms the coordinates relative to the moving frame onto the coordinates relative to the stationary frame.Also, if the center  of the ball has the coordinates q 0 = ( 1 ,  2 , 1) (the radius of the ball is 1) and r is the coordinates of   → , both relative to the fixed frame, then q = q 0 + r.It gives that the motion of the point  is described by a curve q() = q 0 () + r() relative to the fixed frame, or by a path (( 1 (),  2 ()), ()) in R 2 × SO (3), where ( 1 (),  2 ()) describes the motion of the center of the ball relative to e 1 and e 2 , and r() = Q() for some path () in SO (3).
The control problem is to transfer the ball from a given initial position and orientation to a prescribed final position and orientation along a path which minimizes the functional Using the Maximum Principle, Jurdjevic obtained that the controls which minimize  are given by where  and  are the solutions of the system and ,  ∈ R.
If  =  = 0, then  =  0 and it is easy to see that the trajectory of the center of the ball is a circle if  0 ̸ = 0, respectively, and a line if  0 = 0.In the sequel we assume that  2 +  2 > 0.
The paper is organized as follows.At the beginning of the second section we recall Hamilton-Poisson structures of system (8).We give two symplectic realizations of system (8) and their corresponding Lagrangian realizations.In the end of this section we determine some symmetries of the considered system.More precisely, using the Euler-Lagrange's equations that give a Lagrangian realization, we obtain a one-parameter Lie group of transformations that leaves invariant these equations.The third section is dedicated to the energy-Casimir mapping (EC).We characterize the equilibrium points as critical points of EC.Moreover, we give a description of the image of the energy-Casimir mapping and we remark that the boundary of this set is the union of the image through EC of the stable equilibrium points.Also, we decompose this set into a disjoint union of the connected semialgebraic sets which are related to the image through EC of the stable and unstable equilibrium points, obtaining semialgebraic stratifications.From the topological point of view, we prove that the fibers corresponding to the strata of these semialgebraic stratifications are, respectively, stable equilibrium points, homoclinic orbits, and periodic orbits.Moreover, we give parametric representations of these fibers.Also, employing energetic methods based on Lyapunov functions, we prove some known stability results and new ones.In the last section of this paper, using the parametric representations of the aforementioned fibers, we mention parametric equations of the trajectory of the center of the ball.
In this section, we give symplectic and Lagrangian realizations of system (8).Also, some symmetries of the considered system are pointed out.
We recall that the following constants of motion of system (8) were given in [8].
In [9], two Poisson structures were considered.Moreover, system (8) has the following Hamilton-Poisson realizations (R In the sequel we present two symplectic realizations of system (8).
We remark that ( 2  2 is a first integral for system (13).

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As in the proof of Theorem 1, one obtains that the application is a surjective submersion, ( 17) are mapped onto (8), and the structure {⋅, ⋅}  induced by  is mapped onto the Poisson structure Π 1,−1 .Also,  1 ∘  = H1 , which finishes the proof.
) is a first integral for system (17).
Remark 4. Considering the first integrals H2 and  1 ∘ , respectively, H1 and  1,−1 ∘ , it is easy to see that system (13) and system (17) respectively are completely integrable in the sense of Arnold-Liouville.
We mention that, in the particular case  = 1,  = 0, a symplectic realization and its complete integrability were given in [14].
Analogously one obtains the following.
Proposition 6.The Hamiltonian mechanical system (R 4 , , H1 ) considered in Theorem 2 has a Lagrangian realization on the tangent bundle R 2 given by the Euler-Lagrange's equations generated by the Lagrangian In the last part of this section we study some symmetries of system (8).For this purpose, we will use the Lagrangian realization given in Proposition 6.
A symmetry group of a system of differential equations is a group of transformations which maps any solution to another solution of the system.Following [15], let us search for a one-parameter () Lie group of transformations which leaves ( 21) invariant.The corresponding infinitesimal generator is given by a vector field k: with the property that the action of its second prolongation on (21) vanishes.The second prolongation of the vector field k is given by pr (2) A method to find k is given, for example, in [15,16].More precisely, taking into account the chain rule for the computation of ξ , ξ , η 1 , η 1 , η 2 , η 2 and replacing q 1 , q 2 by using ( 21), the relations obtained by applying the second prolongation of k on (21) become two equations in ,  1 , q 1 ,  2 , q 2 , which are all independent.These equations must be satisfied identically in ,  1 , q 1 ,  2 , q 2 , which leads to the finding of k.Detailed utilization of the aforementioned method can be found, for example, in [17,18].
Proof.It is easy to see that the action of pr 2 (k) on ( 21) vanishes, where k is given by (26).Now, some variational symmetries of system ( 21) can be presented.
Remark 8. (i) For  =  = 0 and  ̸ = 0, we have k 1 = (/ 2 ) that represents a translation in the cyclic  2 direction which is related to the conservation of ( (ii) For  =  = 0 and  ̸ = 0, we have k 2 = (/) that represents the time translation symmetry which generates the conservation of energy H1 given by (16).
In [12], Mashtakov and Sachkov gave a partition of the cylinder C into subsets corresponding to motions of the pendulum (28) of the same type, and using this partition, they introduced elliptic coordinates rectifying the phase flow of the pendulum on a subset of full measure of the cylinder C. Furthermore, parametrization of the solutions of ( 28) is obtained.
The qualitative information about the Hamiltonian mechanical system ( * R 2 , , ),  =  1 ∧  1 +  2 ∧  2 , can be given by the so-called energy momentum mapping (see, e.g., [15]).A corresponding of the energy momentum mapping in the case of the Hamilton-Poisson system (R 3 , {⋅, ⋅}, ) is the energy-Casimir mapping (see, e.g., [20]).In [20], Tudoran et al. formulated the following open questions."Is there any connection between the dynamical properties of a given dynamical system and the geometry of the image of the energy-Casimir mapping, and if yes, how can one detect as many dynamical elements as possible (e.g., equilibria, periodic orbits, and homoclinic and heteroclinic connections) and dynamical behavior (e.g., stability, bifurcation phenomena for equilibria, periodic orbits, and homoclinic and heteroclinic connections) by just looking at the image of this mapping?"Particularly, for some dynamical systems, the answers to these questions were given (see, e.g., [21][22][23]).
The goal of this section is to give an answer to the above questions in the case of system (8).Consequently, in this section we study some connections between the dynamics of the considered system and the corresponding energy-Casimir mapping.Our approach regards stability problem, homoclinic and periodic orbits, and their connections with the image of the energy-Casimir mapping and the fibers of the energy-Casimir mapping, respectively.
In the following, we consider system (8) with the Hamilton-Poisson realization (R 3 , Π 1 , ), where Π 1 is given by (10) and the Hamiltonian  =  1 and a Casimir  =  2 , given by (9).Consequently, the energy-Casimir mapping corresponding to the considered system is defined by EC : Using the critical points of mapping (29), we present a characterization of the equilibrium points of system (8).
On the other hand, a point  0 fl ( 0 ,  0 ,  0 ) ∈ R 3 is a critical point of EC if and only if the derivative of EC at  0 is not surjective; that is, EC( 0 ) has rank less than two.Taking into account that the conclusion follows.
In order to present the connections between the equilibrium points and the image of the energy-Casimir mapping, we give a description of the image of the energy-Casimir mapping.Also, we present a study of the stability of equilibrium points.
By the image of the energy-Casimir mapping EC is understood by the set Now, we describe the image of the energy-Casimir mapping.
The graphical description of Im(EC) is given in Figure 1.
Remark 12.The curve of the equation One gets the image of the energy-Casimir mapping (32) is the disjoint union of the following connected semialgebraic sets (Figure 2): The image of the energy-Casimir mapping as a union of connected semialgebraic sets.
Following, for example, [24], we obtain the following semialgebraic stratifications: Here, Σ  1 and Σ  2 are principal strata and, as we will see in the following, the superscripts  and  stand, respectively, for stable and unstable.By Proposition 7, the union E 1 ∪E 2 gives the equilibrium points of system (8), where In [10] the stability problem was treated by using the energy-Casimir method.However, the stability of the equilibrium points (0, 0, 0) and (−, , 0) was not established.In the following we complete the study of the stability.Also, using Lyapunov functions, we give another proof of the known stability results.
we have for some neighborhood  of (−, , ).Moreover, using (8), it follows for all (, , ) ∈ .By [25,26], we deduce that all the equilibrium states from the family E 1 are nonlinearly stable.Proposition 14 (see, also, [10]).The equilibrium points from E 2 are nonlinearly stable for  ∈ (−∞,0] and unstable otherwise. Proof.The characteristic polynomial associated with the linear part of system (8) at the equilibrium   fl (−−, + , 0) ∈ E 2 is given by If  > 0 we notice that a root of    is strictly positive, whence   is an unstable equilibrium state.Therefore, all the equilibrium states from the family E 2 are unstable in the case  ∈ (0, ∞).
If  = −1, it is obvious that the above system has a unique solution, namely, (0, 0, 0).Thus this equilibrium state is nonlinearly stable.
Let now  ∈ (−∞, 0) \ {−1}.From geometric point of view, the solution of system (43) is the curve of intersection between a circular cylinder and a sphere.This curve is reducing at a point if and only if the sphere and the cylinder are tangent.In this geometrical configuration, it is enough to prove that the circles obtained by the intersection of the aforementioned surfaces with the plane  are tangent that is the following system has a unique solution: By straightforward computation, one obtains  = − − ,  =  + , which finishes the proof.
The diagram of equilibrium points is given in Figure 3.
In the next result we give some connections between the image through the energy-Casimir mapping of the families of equilibrium states and the set Im(EC).

Proposition 15. Let E 𝑠
1 and E  2 and E  2 be the families of nonlinearly stable equilibrium points and unstable equilibrium points, respectively, and let EC be the energy-Casimir mapping (29) of system (8).Then, , where  denotes the boundary of the set ;  Proof.The conclusions result by simple computations.
In the sequel we study the topology of the fibers of the energy-Casimir mapping (29).A fiber of the energy-Casimir mapping EC is the preimage of an element (ℎ, ) ∈ Im(EC) through EC, that is, By fixing an element (ℎ, ) from Im(EC), it belongs to one stratum (36).Therefore the classification of the fibers will be done taking into account the stratification of Im(EC).
Since the dynamics has place to the intersection between the surfaces (, , ) = ℎ and (, , ) = , one obtains that these fibers are, respectively, a stable equilibrium point, a periodic orbit, a pair of homoclinic orbits, a pair of periodic orbits, and two stable equilibrium points (Figures 4, 5, 6, and  7).Now we prove the observed results.The first proposition refers to the fibers related to the stable equilibrium states.
Just looking to the image of the energy-Casimir mapping (Figure 2) and taking into account the papers [20,22,23], we expect that there exist homoclinic orbits in dynamics (8).Moreover, Figure 6 suggests that the fibers corresponding to the stratum Σ  2 are homoclinic orbits.More precisely, we have the following.

The Trajectory of the Center of the Ball
Using a connection with the elastica problem, Jurdjevic [8] gave a completely geometric analysis of the motion of the center of the ball; he described various qualitative types of rolling for a sphere along elastics of various forms: inflectional, noninflectional, a circle, and a straight line.
In [12], using the coordinates obtained by integration of the pendulum motion on the partition of a cylinder, a parametrization of extremal trajectories is obtained.
In this section, we classify the types of the trajectory of the center of the ball taking into account the fibers of the energy-Casimir mapping.Using the parametric representations of the studied fibers, we obtain the parametric representations of the trajectory of the center of the ball.These representations are similarly with the ones discussed in [8], respectively [12].
As we have seen in the Introduction, namely, ( 3) and ( 7), the controlled motion of the center of the ball is described by the following system: where  and  are given by system (8) and  0 1 ,  0 2 ∈ R. In the following, considering that (, , ) belongs to a specific fiber F (ℎ,) , we will integrate system (58): .Then the ball rolls along the line Proof.By Proposition 17 (ii), it immediately follows.
In the following results we are using the Jacobi's epsilon function [28]: where  is the modulus.
In Figure 9 we present some trajectories of the center of the ball when ℎ is fixed and  increases such that (ℎ, ) ∈ Σ Proof.Replacing (56) with  0 = 0 in (59) and taking into account dn 2 V +  2 sn 2 V = 1, making use of (62) and the property (dn V)  = − 2 sn V cn V, we obtain (64).
In Figure 10 some trajectories of the center of the ball are presented when ℎ is fixed and  increases such that (ℎ, ) ∈ Σ  1 , first near the stratum Σ  2 .
Remark 26.We notice that the relations (63) and (64) can be written by using elliptic integrals.Indeed, we have E(, ) = (am , ) [28], where am  =  is the Jacobi amplitude and (, ) = ∫  0 √ 1 −  2 sin 2   is the elliptic integral of the second kind with the modulus .Using Wolfram Mathematica software, one can draw a variety of possible trajectories of the center of the ball by customizing  and  and taking (ℎ, ) belonging to different strata.

Conclusions
In our work some properties of the dynamic of the ballplate problem are studied.We particularly refer here to the connections between the dynamical properties of the considered system of differential equations and the geometrical properties of the corresponding energy-Casimir mapping EC.Thus, the equilibrium points are the critical points of EC, and the boundary of the image of the energy-Casimir mapping is the image through EC of the stable equilibrium states.Also, using the image through EC of the equilibrium states, semialgebraic partitions of the image of the energy-Casimir mapping are obtained.In addition, the fibers corresponding to the strata of the presented semialgebraic stratifications are, respectively, stable equilibrium points, homoclinic orbits, and periodic orbits.More precisely, the fibers corresponding to the unstable stratum Σ  2 are a pair of homoclinic orbits and the fibers corresponding to the principal strata are periodic orbits.It is an open problem to establish a class of three-dimensional system of differential equations with the same properties.In the end of the paper, using the parametric representations of the aforementioned fibers, parametric equations of the trajectory of the center of the ball are mentioned.

Figure 1 :Proposition 11 .
Figure 1: The image of the energy-Casimir mapping.

Figure 3 :
Figure 3: The diagram of equilibrium points.