The Hamilton-Jacobi theory, from the variational point of view, was originally developed by Jacobi in 1866, and it states that the integral of the Lagrangian of a mechanical system along the solution of its Euler-Lagrange equation satisfies the Hamilton-Jacobi equation.The classical description of this problem from the generating function and the geometrical point of view was given by Abraham and Marsden in[1]as follows: letting Q be a smooth manifold and TQ the tangent bundle, T?Q is the cotangent bundle with a canonical symplectic form ω, and the projection πQ:T?Q →Q induces the map TπQ:TT?Q →TQ.

Theorem 1.1 Assume that the triple (T?Q,ω,H) is a Hamiltonian system with the Hamiltonian vector field XH, and W : Q →R is a given generating function.Then the following two assertions are equivalent:

(i) For every curve σ :R →Q satisfying that ˙σ(t)=TπQ(XH(dW(σ(t)))), ?t ∈R, dW·σ is an integral curve of the Hamiltonian vector field XH.

There are three reasons for us to pursue this line of research.First, from the proof of the theorem given in Abraham and Marsden [1], we know that the assertion (i), equivalent to Hamilton-Jacobi equation(ii)by the generating function,gives a geometric constraint condition of the canonical symplectic form on the cotangent bundle T?Q for the Hamiltonian vector field of the system.Thus, the Hamilton-Jacobi equation reveals the deeply internal relationships of the generating function, the canonical symplectic form, and the dynamical vector field of a Hamiltonian system.

However, from Marsden et al.[8] we also know that the set of Hamiltonian systems with symmetries on a cotangent bundle is not complete under the Marsden-Weinstein reduction,and the symplectic reduced system of a Hamiltonian system with symmetry defined on the cotangent bundle T?Q may not be a Hamiltonian system on a cotangent bundle, so we cannot give the Hamilton-Jacobi equation for the Marsden-Weinstein reduced Hamiltonian system as in Theorem 1.1; we have to look for a new way.transforms the dynamical vector field of a time-dependent Hamiltonian system to equilibrium

Third, we know that a regular controlled Hamiltonian (RCH) system is a Hamiltonian system with external force and control, which was defined in Marsden et al.[8].In general,under the actions of external force and control, an RCH system is not Hamiltonian, however,it is a dynamical system closely related to a Hamiltonian system, and it can be explored and studied by extending the methods for external force and control used in the study of Hamiltonian systems.Thus,we can emphasize explicitly the impact of external force and control in the study of the RCH systems.In particular, in Marsden et al.[8], the authors gave the regular point reduction and the regular orbit reduction for an RCH system with symmetry and a momentum map by analyzing carefully the geometrical and topological structures of the phase space and the reduced phase space of the corresponding Hamiltonian system.This work not only gave a variety of reduction methods for RCH systems,but also showed a variety of relationships of the controlled Hamiltonian equivalence of these systems.However,since an RCH system defined on the cotangent bundle T?Q, in general, may not be a Hamiltonian system, and as it has yet no generating function, we cannot give the Hamilton-Jacobi equations for the RCH system and its regular reduced systems as in Theorem 1.1.Thus,a natural problem is describing precisely the geometric constraint conditions of a canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of an RCH system and its regular reduced systems, and describing explicitly the relationship between the RCH-equivalence and the solutions of the corresponding Hamilton-Jacobi equations.Addressing this problem is our goal in this paper.

The research of this paper is organized as follows: in the second section,we first review some relevant definitions and basic facts about the RCH systems and RCH-equivalence,then prove a key lemma that is an important tool for the proofs of the two types of Hamilton-Jacobi theorems for the RCH system and its regular reduced systems.Then we derive precisely the geometric constraint conditions of the canonical symplectic form for the dynamical vector fields of an RCH system on the cotangent bundle of a configuration manifold;that is,the two types of Hamilton-Jacobi equations for an RCH system.In the third and fourth sections we begin to discuss the regular reducible RCH system with symmetry and a momentum map,by combining it with the Hamilton-Jacobi theory and a regular symplectic reduction theory for the RCH system, and derive the two types of Hamilton-Jacobi equations for the Rp-reduced and Ro-reduced RCH systems by using the reduced symplectic forms and the regular reducible dynamical vector fields.These results are similar to the development of the two types of Hamilton-Jacobi equations for a Hamiltonian system and its reduced systems given in Wang[11].Moreover,we prove that the RCH-equivalence for the RCH system, and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries, leave the solutions of the corresponding Hamilton-Jacobi equations invariant.Finally,as an application of the theoretical results,in fifth section we consider the regular point reducible RCH system on the generalization of a Lie group,and derive the regular point reduction and the two types of Hamilton-Jacobi equations for the reduced system.In particular, we show the Type I and Type II Hamilton-Jacobi equations for the Rp-reduced controlled rigid body-rotor system and the Rp-reduced controlled heavy top-rotor system on the generalization of the rotation group SO(3), and on the generalization of the Euclidean group SE(3).This research reveals the deeply internal relationships among the geometrical structures of phase spaces, the dynamical vector fields and the controls of the RCH systems, and develops the theory and applications of the regular symplectic reduction and the Hamilton-Jacobi theory for RCH systems with symmetries; affirmatively, it gives us a much deeper understanding of the structure of Hamiltonian systems and RCH systems.

2 Hamilton-Jacobi Equations for an RCH System

In this paper, our goal is to study the Hamilton-Jacobi theory for an RCH system with symplectic structure and symmetry,to prove the two types of Hamilton-Jacobi theorems for an RCH system and its regular reduced RCH systems, and to describe the relationship between the RCH-equivalence for the RCH systems and the solutions of the corresponding Hamilton-Jacobi equations.In order to do this, we first review some relevant definitions and basic facts about RCH systems and RCH-equivalence.Then we give the geometric constraint conditions of a canonical symplectic form for the dynamical vector field of the RCH system; that is, the two types of Hamilton-Jacobi equations, and state that the solution of the Hamilton-Jacobi equation for the RCH system remains invariant under the conditions of RCH-equivalence.We shall follow some of the notations and conventions introduced in Abraham and Marsden [1],Marsden and Ratiu [4], Marsden et al.[8],Wang [11], Libermann and Marle [12], Marsden[13]and Ortega and Ratiu [14].In this paper, we assume that all manifolds are real, smooth and finite dimensional, and that all actions are smooth left actions.

From Marsden et al.[8], we know that the symplectic reduced space of a Hamiltonian system defined on the cotangent bundle of a configuration manifold may not be a cotangent bundle, and hence, the set of Hamiltonian systems with symmetries on the cotangent bundle is not complete under the Marsden-Weinstein reduction.This is a serious problem.If we define directly a controlled Hamiltonian system with symmetry on a cotangent bundle, then it is possible that the Marsden-Weinstein reduced RCH system may not have a definition.In order to describe, uniformly, RCH systems defined on a cotangent bundle and on the regular reduced spaces, in this section we first define an RCH system on a symplectic fiber bundle;see Marsden et al.[8].Then we can obtain the RCH system on the cotangent bundle of a configuration manifold as a special case, and discuss RCH-equivalence.As a consequence, we can regard the associated Hamiltonian system on the cotangent bundle as a special case of the RCH system without external force and control; as such we can study the RCH systems with symmetries by combining them with the regular symplectic reduction theory of Hamiltonian systems.For convenience, we assume that all of the controls appearing in this paper are the admissible controls.

Let (E,M,π) be a fiber bundle, and for each point x ∈M, assume that the fiber Ex=π?1(x) is a smooth submanifold of E with a symplectic form ωE(x); that is, (E,ωE) is a symplectic fiber bundle.If, for any function H : E →R, we have a Hamiltonian vector field XHwhich satisfies the Hamilton’s equation; that is,iXHωE=dH, then (E,ωE,H) is a Hamiltonian system.Moreover,considering the external force and control,we can define a kind of regular controlled Hamiltonian (RCH) system on the symplectic fiber bundle E as follows:

Definition 2.1(RCH system) An RCH system on E is a 5-tuple(E,ωE,H,F,W),where(E,ωE,H) is a Hamiltonian system, and the function H : E →R is called the Hamiltonian, a fiber-preserving map F :E →E is called the (external) force map, and a fiber submanifold W of E is called the control subset.

Sometimes, W is also denoted as the set of fiber-preserving maps from E to W.When a feedback control law u:E →W is chosen,the 5-tuple(E,ωE,H,F,u)is a closed-loop dynamic system.In particular, when Q is a smooth manifold, and T?Q is its cotangent bundle with a symplectic form ω (not necessarily canonical symplectic form), then (T?Q,ω) is a symplectic vector bundle.If we take that E = T?Q, from the above definition we can obtain an RCH system on the cotangent bundle T?Q; that is, the 5-tuple (T?Q,ω,H,F,W).Here the fiberpreserving map F : T?Q →T?Q is the (external) force map, which is the reason that the fiber-preserving map F :E →E is called an (external) force map in above definition.

In order to describe the dynamics of the RCH system (E,ωE,H,F,W) with a control law u, we need to give a good expression of the dynamical vector field of the RCH system.First,we introduce notations for vertical lift maps of a vector along a fiber.For a smooth manifold E, its tangent bundle TE is a vector bundle, and for the fiber bundle π :E →M, we consider the tangent mapping Tπ : TE →TM and its kernel ker(T π) = {ρ ∈T E|T π(ρ) = 0}, which is a vector subbundle of TE.Denote that V E :=ker(T π), which is called a vertical bundle of E.Assume that there is a metric on E, take a Levi-Civita connection A on TE, and denote that HE := ker(A), which is called a horizontal bundle of E such that TE = HE ⊕V E.For any x ∈M, ax,bx∈Ex, any tangent vector ρ(bx) ∈TbxE can be split into horizontal and vertical parts; that is, ρ(bx) = ρh(bx)⊕ρv(bx), where ρh(bx) ∈HbxE and ρv(bx) ∈VbxE.Let γ be a geodesic in Exconnecting axand bx, and denote by ρvγ(ax) a tangent vector at ax,which is a parallel displacement of the vertical vector ρv(bx) along the geodesic γ from bxto ax.Since the angle between two vectors is invariant under a parallel displacement along a geodesic, Tπ(ρvγ(ax)) = 0, and hence ρvγ(ax) ∈VaxE.Now, for ax,bx∈Exand tangent vector ρ(bx)∈TbxE, we can define the vertical lift map of a vector along a fiber given by

It is easy to check from the basic facts of differential geometry that this map does not depend on the choice of γ.If F : E →E is a fiber-preserving map, for any x ∈M, we have that Fx: Ex→Exand TFx: T Ex→TEx, so, for any ax∈Exand ρ ∈T Ex, the vertical lift of ρ under the action of F along a fiber is defined by

where γ is a geodesic in Exconnecting Fx(ax) and ax.

where αx∈T?xQ, x ∈Q, and γ is a straight line in T?xQ connecting Fx(αx) and αx.In the same way, when a feedback control law u : T?Q →W is chosen, the change of XHunder the action of u is such that

As a consequence, we can give an expression of the dynamical vector field of the RCH system as follows:

Theorem 2.2The dynamical vector field of an RCH system (T?Q,ω,H,F,W) with a control law u is the synthesis of a Hamiltonian vector field XHand its changes under the actions of the external force F and the control u; that is,

In what follows,we shall derive the geometric constraint conditions of a canonical symplectic form for the dynamical vector field of the RCH system;that is,the Type I and Type II Hamilton-Jacobi equations for the RCH system.We also state that the solution of the corresponding Hamilton-Jacobi equation remains invariant under the conditions of RCH-equivalence.In order to do this, we first give an important notion and prove a key lemma.

Denote by ?i(Q) the set of all i-forms on Q, i = 1,2.For any γ ∈?1(Q), q ∈Q, so γ(q)∈T?qQ,and we can define a map γ :Q →T?Q, q →(q,γ(q)).Hence we say often that the map γ : Q →T?Q is a one-form on Q.If the one-form γ is closed, thendγ(x,y)=0, ?x,y ∈TQ.Note that, for any v,w ∈TT?Q, we have thatdγ(TπQ(v),TπQ(w)) = π?(dγ)(v,w) is a two-form on the cotangent bundle T?Q, where π?:T?Q →T?T?Q.Thus, in the following, we can give a weaker notion:

Definition 2.3The one-form γ is said to be closed with respect to TπQ: T T?Q →T Q if, for any v,w ∈TT?Q, we have thatdγ(TπQ(v),T πQ(w))=0.

From the above definition we know that,if γ is a closed one-form,then it must be closed with respect to TπQ: TT?Q →TQ.Conversely, if γ is closed with respect to TπQ: T T?Q →T Q,then it may not be closed.We can prove a general result as follows,which states that the notion that γ is closed with respect to TπQ: TT?Q →T Q is not equivalent to the notion that γ is closed (see Wang [11]):

Proposition 2.4Assume that γ :Q →T?Q is a one-form on Q and that is not closed.We define the set N, which is a subset of TQ,such that the one-form γ on N satisfies the condition that, for any x,y ∈N,dγ(x,y)/= 0.Denote that Ker(T πQ)= {u ∈TT?Q|T πQ(u)= 0}, and that Tγ :TQ →TT?Q.If Tγ(N)?Ker(TπQ),then γ is closed with respect to TπQ:T T?Q →

TQ.

Now we prove Lemma 2.5.It is worth noting that this lemma is obtained by a careful modification for the corresponding result of Abraham and Marsden in [1]; also see Wang [11].This lemma is very important for our research, and we also give its proof here.

Lemma 2.5Assume that γ :Q →T?Q is a one-form on Q,and that λ=γ·πQ:T?Q →T?Q.Then we have that the following two assertions hold:

(i) For any x,y ∈TQ, γ?ω(x,y) = ?dγ(x,y), and for any v,w ∈T T?Q, λ?ω(v,w) =?dγ(TπQ(v), TπQ(w)), since ω is the canonical symplectic form on T?Q.

(ii) For any v,w ∈TT?Q, ω(Tλ·v,w)=ω(v,w ?T λ·w)?dγ(T πQ(v), T πQ(w)).

ProofWe first prove the assertion(i).Since ω is the canonical symplectic form on T?Q,we know that there is a unique canonical one-form θ such that ω =?dθ.From the Proposition 3.2.11 in Abraham and Marsden [1], we have that, for the one-form γ : Q →T?Q, γ?θ = γ.Then we can obtain that

However, the second term on the right-hand side is given by

Thus, the assertion (ii) holds.?

From the expression of the dynamical vector field of an RCH system, we know that under the actions of the external force F and the control u, in general, the dynamical vector field is not Hamiltonian, and the RCH system is not yet a Hamiltonian system, and hence, we cannot describe the Hamilton-Jacobi equation for an RCH system from the viewpoint of a generating function the same as in Theorem 1.1 given by Abraham and Marsden in [1].However, for a given RCH system (T?Q,ω,H,F,W) in which ω is the canonical symplectic form on T?Q,by using Lemma 2.5 and the dynamical vector field X(T?Q,ω,H,F,u), we can derive two types of geometric constraint conditions of the canonical symplectic forms for the dynamical vector field of the RCH system; that is, the two types of geometric Hamilton-Jacobi equations for the RCH system.First, by using the fact that the one-form γ : Q →T?Q is closed with respect to TπQ: TT?Q →TQ, we can prove the Type I geometric Hamilton-Jacobi theorem for the RCH system.For convenience, the maps involved in the theorem and its proof are shown in Diagram-1.

Theorem 2.6(Type I Hamilton-Jacobi Theorem for an RCH system) For the RCH system (T?Q,ω,H,F,W) with the canonical symplectic form ω on T?Q, assume that γ :Q →T?Q is a one-form on Q,and that ?Xγ=TπQ·?X·γ,where ?X =X(T?Q,ω,H,F,u)is the dynamical vector field of the RCH system (T?Q,ω,H,F,W) with a control law u.If the one-form γ :Q →T?Q is closed with respect to TπQ: TT?Q →TQ, then γ is a solution of the equation Tγ·?Xγ=XH·γ, where XHis the Hamiltonian vector field of the corresponding Hamiltonian system (T?Q,ω,H), and the equation is called the Type I Hamilton-Jacobi equation for the RCH system (T?Q,ω,H,F,W) with a control law u.

ProofSince ?X = X(T?Q,ω,H,F,u)= XH+vlift(F)+vlift(u) and TπQ·vlift(F) = T πQ·vlift(u)=0, we have that TπQ·?X·γ =TπQ·XH·γ.If we take that v =XH·γ ∈T T?Q, and that for any w ∈TT?Q, TπQ(w)/=0, from Lemma 2.5 (ii) we have that

since the one-form γ :Q →T?Q is closed with respect to TπQ:T T?Q →TQ.However,because the symplectic form ω is non-degenerate,the left side of (2.3) equals zero only when γ satisfies the equation Tγ·?Xγ= XH·γ.Thus, if the one-form γ : Q →T?Q is closed with respect to TπQ: TT?Q →TQ, then γ must be a solution of the Type I Hamilton-Jacobi equation Tγ·?Xγ=XH·γ.?

Next, for any symplectic map ε : T?Q →T?Q, we can prove the Type II geometric Hamilton-Jacobi theorem for the RCH system (T?Q,ω,H,F,W).For convenience, the maps involved in the theorem and its proof are shown in Diagram-2.

Theorem 2.7(Type II Hamilton-Jacobi Theorem for an RCH system) For the RCH system (T?Q,ω,H,F,W) with the canonical symplectic form ω on T?Q, assume that γ :Q →T?Q is a one-form on Q, and that λ = γ·πQ: T?Q →T?Q.For any symplectic map ε : T?Q →T?Q, denote that ?Xε= TπQ·?X·ε, where ?X = X(T?Q,ω,H,F,u)is the dynamical vector field of the RCH system(T?Q,ω,H,F,W)with a control law u.Then ε is a solution of the equation Tε·XH·ε=Tλ·?X·ε if and only if it is a solution of the equation T γ·?Xε=XH·ε,where XHand XH·ε∈TT?Q are the Hamiltonian vector fields of the functions H and H·ε:T?Q →R,respectively.The equation Tγ·?Xε=XH·ε is called the Type II Hamilton-Jacobi equation for the RCH system (T?Q,ω,H,F,W) with a control law u.

ProofSince ?X = X(T?Q,ω,H,F,u)= XH+vlift(F)+vlift(u) and TπQ·vlift(F) = T πQ·vlift(u)= 0, we have that TπQ·?X·ε =TπQ·XH·ε.If we take that v =XH·ε ∈T T?Q, for any w ∈TT?Q, Tλ(w)/=0, from Lemma 2.5 we have that

Because the symplectic form ω is non-degenerate,it follows that Tγ·?Xε=XH·ε is equivalent to Tε·XH·ε= Tλ·?X·ε.Thus, ε is a solution of the equation T ε·XH·ε= T λ·?X·ε if and only if it is a solution of the Type II Hamilton-Jacobi equation Tγ·?Xε=XH·ε.?

Remark 2.8It is worth noting that the Type I Hamilton-Jacobi equation Tγ·?Xγ=XH·γ is the equation of the differential one-form γ, and that the Type II Hamilton-Jacobi equation Tγ·?Xε=XH·ε is the equation of the symplectic diffeomorphism map ε.If both the external force and the control of an RCH system(T?Q,ω,H,F,W) are zero;that is, F =0 and W =?,then the RCH system is just a Hamiltonian system (T?Q,ω,H), and from the proofs of the Theorems 2.6 and 2.7 above, we can obtain two types of Hamilton-Jacobi equations for the associated Hamiltonian system (given in Wang [11]).Thus, Theorems 2.6 and 2.7 can be regarded as an extension of the two types of Hamilton-Jacobi equations for a Hamiltonian system to that for the system with external force and control.

Next, we note that when an RCH system is given, the force map F is determined, but the feedback control law u:T?Q →W can be chosen.In order to describe the feedback control law to modify the structure of the RCH system, the controlled Hamiltonian matching conditions and RCH-equivalence are induced as follows:

Definition 2.9(RCH-equivalence) Suppose that we have two RCH systems(T?Qi,ωi,Hi,Fi,Wi), i=1,2.We say that they are RCH-equivalent, or simply, that (T?Q1,ω1,H1,F1,W1)RCH～ (T?Q2,ω2,H2,F2,W2), if there exists a diffeomorphism ? : Q1→Q2such that the following controlled Hamiltonian matching conditions hold:

RCH-1The control subsets Wi, i = 1,2 satisfy the condition W1= ??(W2), where the map ??=T??:T?Q2→T?Q1is a cotangent lifted map of ?;

RCH-2For each control law u1:T?Q1→W1, there exists the control law u2:T?Q2→W2such that the two closed-loop dynamical systems produce the same equations of motion;that is,X(T?Q1,ω1,H1,F1,u1)·??=T(??)X(T?Q2,ω2,H2,F2,u2),where the map T(??):TT?Q2→T T?Q1is the tangent map of ??.

Moreover,considering the RCH-equivalence of the RCH systems,we can prove the following theorem, which states that the solutions of two types of Hamilton-Jacobi equations for the RCH systems remain invariant under the conditions of RCH-equivalence if the corresponding Hamiltonian systems are equivalent:

Theorem 2.10Suppose that two RCH systems,(T?Qi,ωi,Hi,Fi,Wi),i=1,2 are RCHequivalent with an equivalent map ? : Q1→Q2, and that the corresponding Hamiltonian systems, (T?Qi,ωi,Hi), i = 1,2 are also equivalent.Under the hypotheses and notations of Theorems 2.6 and 2.7, we have that the following two assertions hold:

(i) if the one-form γ2: Q2→T?Q2is closed with respect to T πQ2: TT?Q2→T Q2, then γ1=??·γ2·?:Q1→T?Q1is also closed with respect to TπQ1:TT?Q1→TQ1,and hence,γ1is a solution of the Type I Hamilton-Jacobi equation for the RCH system(T?Q1,ω1,H1,F1,W1);

(ii) if the symplectic map ε2:T?Q2→T?Q2is a solution of the Type II Hamilton-Jacobi equation for the RCH system (T?Q2,ω2,H2,F2,W2), then ε1=??·ε2·??: T?Q1→T?Q1is a symplectic map, and hence, ε1is a solution of the Type II Hamilton-Jacobi equation for the RCH system (T?Q1,ω1,H1,F1,W1).

ProofWe first prove the assertion (i).If two RCH systems, (T?Qi,ωi,Hi,Fi,Wi), i =1,2, are RCH-equivalent with an equivalent map ? : Q1→Q2, from the definition of RCHequivalence, we know that, for each control law u1: T?Q1→W1, there exists the control law u2:T?Q2→W2such that the two closed-loop dynamical systems produce the same equations of motion; that is, ?X1·??= T(??)·?X2, where ?Xi= X(T?Qi,ωi,Hi,Fi,ui), i = 1,2.From the commutative Diagram-3; that is,

we have that γ1= ??·γ2·?,dγ1= ??·dγ2·? and T ?·TπQ1·T??= T πQ2.For x ∈Q1and v, w ∈TT?xQ1, so ?(x) ∈Q2and T??(v), T ??(w) ∈T T??(x)Q2.Since the one-form γ2:Q2→T?Q2is closed with respect to TπQ2:T T?Q2→TQ2,

where we have used that T(??)·XH2= XH1·??, because the corresponding Hamiltonian systems (T?Qi,ωi,Hi), i=1,2 are equivalent.Thus, the one-form γ1=??·γ2·? is a solution of the Type I Hamilton-Jacobi equation for the RCH system (T?Q1,ω1,H1,F1,W1).Note that the map ? : Q1→Q2is a diffeomorphism, and that ??: T?Q2→T?Q1is a symplectic isomorphism, and vice versa.It follows that the assertion (i) of Theorem 2.10 holds.

Next, we prove the assertion (ii).From the commutative Diagram-4; that is,

we have that ε1=??·ε2·??:T?Q1→T?Q1.Since ε2:T?Q2→T?Q2is symplectic with respect to ω2, then for x ∈Q1, v, w ∈TT?xQ1, and ?(x)∈Q2, T ??(v), T??(w) ∈T T??(x)Q2, we have that ε?2·ω2(T??(v), T??(w))(?(x)) =ω2(T??(v), T??(w))(?(x)).Note that the corresponding Hamiltonian systems(T?Qi,ωi,Hi), i=1,2 are equivalent, and hence that ??:T?Q2→T?Q1is symplectic, and that (??)?ω1(v,w)(x)=ω2(T ??(v), T??(w))(?(x)), so we have that

that is, the symplectic map ε1= ??·ε2·??is a solution of the Type II Hamilton-Jacobi equation for the RCH system(T?Q1,ω1,H1,F1,W1).In the same way,the map ?:Q1→Q2is a diffeomorphism, and ??:T?Q2→T?Q1is a symplectic isomorphism, and vice versa.Hence we have proven the assertion (ii) of Theorem 2.10.?

In Sections 3 and 4, we shall generalize the above results to regular point and regular orbit reducible RCH systems with symmetries and momentum maps, and derive a variety of Hamilton-Jacobi equations for the regular reduced RCH systems.

3 Hamilton-Jacobi Equations of the Rp-reduced RCH System

The reduction theory for the mechanical system with symmetry is an important subject and has been widely studied in mathematics and mechanics.The main goal of reduction theory in mechanics is to use conservation laws and the associated symmetries to reduce the number of dimensions of a mechanical system need to be described.Thus,reduction theory is regarded as a useful tool for simplifying and studying concrete mechanical systems.Over forty years ago,the regular symplectic reduction for the Hamiltonian system with symmetry and a coadjoint equivariant momentum map was set up by professors Jerrold E.Marsden and Alan Weinstein.This work is called Marsden-Weinstein reduction, and great developments have been obtained around it; see Abraham et al.[1, 2], Arnold [3], Marsden and Ratiu [4], Libermann and Marle[12], Marsden [13], Ortega and Ratiu [14], Le′on and Rodrigues [15], Marsden et al.[16, 17],Marsden and Perlmutter [18], Marsden and Weinstein [19], Meyer [20] and Nijmeijer and Van der Schaft [21], etc..

It is worth noting that the authors in Marsden et al.[8]set up the regular reduction theory for the RCH systems with symplectic structures and symmetries on a symplectic fiber bundle as an extension of the Marsden-Weinstein reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions, and did so from the viewpoint of the completeness of the regular symplectic reduction and by analyzing carefully the geometrical and topological structures of the phase space and the reduced phase space of the corresponding Hamiltonian system.Some developments around the work are given in Wang and Zhang[22], Ratiu and Wang [23], Wang [24] and Wang [25].In this section, we first give the regular point reducible RCH system with symmetry and a momentum map, then we give the geometric constraint conditions of the Rp-reduced symplectic form for the dynamical vector field of the regular point reducible RCH system, and prove the Type I and Type II Hamilton-Jacobi theorems for the Rp-reduced RCH system.Moreover,we state the relationship between the solutions of Type II Hamilton-Jacobi equations and regular point reduction.Finally, we consider the RpCH-equivalence, and state that the solutions of the two types of Hamilton-Jacobi equations for the RCH systems with symmetries remain invariant under the conditions of RpCH-equivalence if the corresponding Hamiltonian systems are equivalent.We shall follow the notations and conventions introduced in Marsden et al.[8],Wang[11],Wang[24]and Wang[25].

First, we consider the regular point reducible RCH system,which was given by Marsden et al.[8].Let Q be a smooth manifold and T?Q its cotangent bundle with the symplectic form ω.Let Φ : G×Q →Q be a smooth left action of the Lie group G on Q, which is free and proper.Assume that the cotangent lifted left action ΦT?: G×T?Q →T?Q is symplectic,free and proper, and that it admits an Ad?-equivariant momentum map J : T?Q →g?, where g is a Lie algebra of G and g?is the dual of g.Let μ ∈g?be a regular value of J and denote by Gμthe isotropy subgroup of the coadjoint G-action at the point μ ∈g?, which is defined by Gμ= {g ∈G|Ad?gμ = μ}.Since Gμ(?G) acts freely and properly on Q and on T?Q, then Qμ= Q/Gμis a smooth manifold and the canonical projection ρμ: Q →Qμis a surjective submersion.It follows that Gμacts also freely and properly on J?1(μ), so that the space (T?Q)μ=J?1(μ)/Gμis a symplectic manifold with the Rp-reduced symplectic form ωμuniquely characterized by the relation

Definition 3.1 (Regular Point Reducible RCH System) A 6-tuple (T?Q,G,ω,H,F,W)with the canonical symplectic form ω on T?Q, where the Hamiltonian H : T?Q →R, the fiber-preserving map F :T?Q →T?Q and the fiber submanifold W of T?Q are all G-invariant,is called a regular point reducible RCH system if there exists a point μ∈g?which is a regular value of the momentum map J such that the regular point reduced system; that is, the 5-tuple ((T?Q)μ,ωμ,hμ,fμ,Wμ), where (T?Q)μ= J?1(μ)/Gμ, π?μωμ= i?μω, hμ·πμ= H·iμ,F(J?1(μ))?J?1(μ), fμ·πμ=πμ·F·iμ, W ∩J?1(μ)/=?, Wμ=πμ(W ∩J?1(μ)), is an RCH system, which is simply written as the Rp-reduced RCH system.Where ((T?Q)μ,ωμ) is the Rp-reduced space, the function hμ: (T?Q)μ→R is called the Rp-reduced Hamiltonian, the fiber-preserving map fμ:(T?Q)μ→(T?Q)μis called the Rp-reduced(external)force map,and Wμis a fiber submanifold of (T?Q)μand is called the Rp-reduced control subset.

Denote by X(T?Q,G,ω,H,F,u)the dynamical vector field of the regular point reducible RCH system (T?Q,G,ω,H,F,W) with a control law u.Assume that it can be expressed by

From Marsden et al.[8] and Wang [24], we know that the set of Hamiltonian systems with symmetries on the cotangent bundle is not complete under the Marsden-Weinstein reduction,so the regular point reduced system of an RCH system with symmetry defined on the cotangent bundle T?Q may not be an RCH system on a cotangent bundle.On the other hand, from the expression of the dynamical vector field of an Rp-reduced RCH system, we know that under the actions of the Rp-reduced external force fμand the Rp-reduced control uμ, in general,the dynamical vector field is not Hamiltonian, and that the Rp-reduced RCH system is not yet a Hamiltonian system.Thus, we cannot describe the Hamilton-Jacobi equation for the Rp-reduced RCH system from the viewpoint of a generating function the same as in Theorem 1.1 given by Abraham and Marsden in [1].However, for a given regular point reducible RCH system(T?Q,G,ω,H,F,W)with an Rp-reduced RCH system((T?Q)μ,ωμ,hμ,fμ,uμ),by using Lemma 2.5,we can give the geometric constraint conditions of the Rp-reduced symplectic form for the dynamical vector field of the regular point reducible RCH system;that is,the Type I and Type II Hamilton-Jacobi equations for the Rp-reduced RCH system ((T?Q)μ,ωμ,hμ,fμ,uμ).First,by using the fact that the one-form γ :Q →T?Q is closed with respect to T πQ:T T?Q →TQ,that Im(γ)?J?1(μ),and that γ is Gμ-invariant,we can prove the Type I Hamilton-Jacobi theorem for the Rp-reduced RCH system ((T?Q)μ,ωμ,hμ,fμ,uμ).For convenience, the maps involved in the theorem and its proof are shown in Diagram-5.

Theorem 3.2 (Type I Hamilton-Jacobi Theorem for an Rp-reduced RCH system) For the regular point reducible RCH system(T?Q,G,ω,H,F,W)with an Rp-reduced RCH system((T?Q)μ,ωμ,hμ,fμ,uμ), assume that γ : Q →T?Q is a one-form on Q, and that ?Xγ=TπQ·?X·γ, where ?X = X(T?Q,G,ω,H,F,u)is the dynamical vector field of the regular point reducible RCH system(T?Q,G,ω,H,F,W)with a control law u.Moreover,assume thatμ∈g?is a regular value of the momentum map J,and that Im(γ)?J?1(μ),and that γ is Gμ-invariant,and that ˉγ = πμ(γ) : Q →(T?Q)μ.If the one-form γ : Q →T?Q is closed with respect to TπQ: TT?Q →TQ, then ˉγ is a solution of the equation Tˉγ·?Xγ= Xhμ·ˉγ, where Xhμis the Hamiltonian vector field of the Rp-reduced Hamiltonian function hμ: (T?Q)μ→R, and the equation is called the Type I Hamilton-Jacobi equation for the Rp-reduced RCH System((T?Q)μ,ωμ,hμ,fμ,uμ).

Proof First, from Theorem 2.6, we know that γ is a solution of the Type I Hamilton-Jacobi equation Tγ·?Xγ=XH·γ.Next,we note that Im(γ)?J?1(μ),and that γ is Gμ-invariant,in this case, π?μωμ= i?μω = ω, along Im(γ).Since ?X = X(T?Q,G,ω,H,F,u)= XH+vlift(F)+vlift(u), and TπQ·vlift(F) = TπQ·vlift(u) = 0, we have that TπQ·?X·γ = TπQ·XH·γ.By using the reduced symplectic form ωμ, if we take that v = XH·γ ∈T T?Q, and for any w ∈TT?Q, TπQ(w)/=0 and Tπμ(w)/=0, from Lemma 2.5 (ii) we have that

Because the reduced symplectic form ωμis non-degenerate,the left side of(3.5)equals zero only when ˉγ satisfies the equation Tˉγ·?Xγ= Xhμ·ˉγ.Thus, if the one-form γ : Q →T?Q is closed with respect to TπQ:TT?Q →TQ, then ˉγ must be a solution of the Type I Hamilton-Jacobi equation Tˉγ·?Xγ=Xhμ·ˉγ.?

Next, for any Gμ-invariant symplectic map ε : T?Q →T?Q, we can prove the Type II Hamilton-Jacobi theorem for the Rp-reduced RCH system ((T?Q)μ,ωμ,hμ,fμ,uμ).For convenience, the maps involved in the theorem and its proof are shown in Diagram-6.

Theorem 3.3 (Type II Hamilton-Jacobi Theorem for an Rp-reduced RCH system) For the regular point reducible RCH system(T?Q,G,ω,H,F,W)with an Rp-reduced RCH system((T?Q)μ,ωμ,hμ,fμ,uμ), assume that γ : Q →T?Q is a one-form on Q, and that λ = γ·πQ:T?Q →T?Q.For any symplectic map ε : T?Q →T?Q, we denote that ?Xε= T πQ·?X·ε,where ?X = X(T?Q,G,ω,H,F,u)is the dynamical vector field of the regular point reducible RCH system (T?Q,G,ω,H,F,W) with a control law u.Moreover, assume that μ ∈g?is a regular value of the momentum map J, that Im(γ) ?J?1(μ), and that γ and ε are Gμ-invariant,and that ε(J?1(μ)) ?J?1(μ).Denote that ˉγ = πμ(γ) : Q →(T?Q)μ, ˉλ = πμ(λ) : J?1(μ)(?T?Q)→(T?Q)μ,and ˉε=πμ(ε):J?1(μ)(?T?Q)→(T?Q)μ.Then ε and ˉε satisfy the equation Tˉε·(Xhμ·ˉε)=Tˉλ·?X·ε if and only if they satisfy the equation Tˉγ·?Xε=Xhμ·ˉε, where Xhμand Xhμ·ˉε∈TT?Q are the Hamiltonian vector fields of the Rp-reduced Hamiltonian functions hμand hμ·ˉε : T?Q →R, respectively.The equation Tˉγ·?Xε= Xhμ·ˉε is called the Type II Hamilton-Jacobi equation for the Rp-reduced RCH system ((T?Q)μ,ωμ,hμ,fμ,uμ).

Proof We note that Im(γ) ?J?1(μ), and that γ is Gμ-invariant.In this case, π?μωμ=i?μω = ω along Im(γ).Since ?X = X(T?Q,G,ω,H,F,u)= XH+ vlift(F) + vlift(u), and T πQ·vlift(F)=TπQ·vlift(u)=0, we have that TπQ·?X·ε=T πQ·XH·ε.By using the Rp-reduced symplectic form ωμ, if we take that v = XH·ε ∈T T?Q, and for any w ∈T T?Q, Tˉλ(w) /= 0,and Tπμ(w)/=0, from Lemma 2.5 (ii) we have that

where we have used that Tπμ(XH) = Xhμ.Note that ε : T?Q →T?Q is symplectic, and that π?μωμ= i?μω = ω along Im(γ), and hence, ˉε = πμ(ε) : J?1(μ)(?T?Q) →(T?Q)μis also symplectic along Im(γ), and Xhμ·ˉε = Tˉε·Xhμ·ˉεalong Im(γ)∩Im(ε).From the above arguments, we can obtain that

Because the Rp-reduced symplectic form ωμis non-degenerate,it follows that Tˉγ·?Xε=Xhμ·ˉε is equivalent to Tˉλ·?X·ε = Tˉε·Xhμ·ˉε.Thus, we know that the ε and ˉε satisfy the equation Tˉε·(Xhμ·ˉε) = Tˉλ·?X·ε if and only if they satisfy the Type II Hamilton-Jacobi equation Tˉγ·?Xε=Xhμ·ˉε.?

Moreover, for a given RCH system (T?Q,G,ω,H,F,W) with an Rp-reduced RCH system((T?Q)μ,ωμ,hμ,fμ,uμ),we know that the Hamiltonian vector fields XHand Xhμfor the corresponding Hamiltonian system (T?Q,G,ω,H) and its Rp-reduced system ((T?Q)μ,ωμ,hμ) are πμ-related; that is, that Xhμ·πμ=Tπμ·XH·iμ.Then we can prove the Theorem 3.4, which states the relationship between the solutions of Type II Hamilton-Jacobi equations and the regular point reduction.

Theorem 3.4 For a given RCH system (T?Q,G,ω,H,F,W) with an Rp-reduced RCH system ((T?Q)μ,ωμ,hμ,fμ,uμ), assume that γ : Q →T?Q is a one-form on Q, and that ε : T?Q →T?Q is a Gμ-invariant symplectic map, and ˉε = πμ(ε) : J?1(μ)(?T?Q) →(T?Q)μ.Under the hypotheses and notations of Theorem 3.3, we have that ε is a solution of the Type II Hamilton-Jacobi equation T γ·?Xε= XH·ε for the regular point reducible RCH system(T?Q,G,ω,H,F,W)if and only if ε and ˉε satisfy the Type II Hamilton-Jacobi equation Tˉγ·?Xε=Xhμ·ˉε for the Rp-reduced RCH system ((T?Q)μ,ωμ,hμ,fμ,uμ).Because both the symplectic form ω and the Rp-reduced symplectic form ωμare non-degenerate,it follows that the equation Tˉγ·?Xε=Xhμ·ˉε is equivalent to the equation T γ·?Xε=XH·ε.Thus,ε is a solution of the Type II Hamilton-Jacobi equation Tγ·?Xε=XH·ε for the regular point reducible RCH system(T?Q,G,ω,H,F,W)if and only if ε and ˉε satisfy the Type II Hamilton-Jacobi equation Tˉγ·?Xε=Xhμ·ˉε for the Rp-reduced RCH system ((T?Q)μ,ωμ,hμ,fμ,uμ).?

Remark 3.5It is worth noting that the Type I Hamilton-Jacobi equation Tˉγ·?Xγ=Xhμ·ˉγ is the equation of the Rp-reduced differential one-formˉγ,and that the Type II Hamilton-Jacobi equation Tˉγ·?Xε=Xhμ·ˉε is the equation of the symplectic diffeomorphism map ε and the Rp-reduced symplectic diffeomorphism map ˉε.If both the external force and the control of a regular point reducible RCH system (T?Q,G,ω,H,F,W) are zero; that is, F = 0 and W = ?, in this case, the RCH system is just a regular point reducible Hamiltonian system(T?Q,G,ω,H).From the proofs of the Theorems 3.2, 3.3 and 3.4 above, we can also get two types of Hamilton-Jacobi equations for the associated Marsden-Weinstein reduced Hamiltonian system (given in Wang [11]).Thus, Theorems 3.2, 3.3 and 3.4 can be regarded as an extension of the two types of Hamilton-Jacobi equations for the Rp-reduced Hamiltonian system to that for the Rp-reduced RCH system.

Moreover, for the regular point reducible RCH system we can also introduce the regular point reducible controlled Hamiltonian equivalence (RpCH-equivalence) as follows:

Definition 3.6(RpCH-equivalence) Suppose that we have two regular point reducible RCH systems, (T?Qi,Gi,ωi,Hi,Fi,Wi), i = 1,2.We say that they are RpCH-equivalent, or simply, that (T?Q1,G1,ω1,H1,F1,W1)RpCH～ (T?Q2,G2,ω2,H2,F2,W2), if there exists a diffeomorphism ?:Q1→Q2such that the following controlled Hamiltonian matching conditions hold:

RpCH-2For each control law u1:T?Q1→W1,there exists the control law u2:T?Q2→W2such that X(T?Q1,G1,ω1,H1,F1,u1)·??=T(??)X(T?Q2,G2,ω2,H2,F2,u2).

It is worth noting that,for the regular point reducible RCH system,the induced equivalent map ??μalso keeps the equivariance of G-action at the regular point.Moreover,we can obtain a regular point reduction theorem for an RCH system, which explains the relationship between the RpCH-equivalence for the regular point reducible RCH systems with symmetries and the RCH-equivalence for the associated Rp-reduced RCH systems; the proof is given in Marsden et al.[8] and Wang [24].This theorem can be regarded as an extension of Marsden-Weinstein reduction theorem of Hamiltonian system under regular controlled Hamiltonian equivalence conditions.

Theorem 3.7Two regular point reducible RCH systems, (T?Qi,Gi,ωi,Hi,Fi,Wi), i =1,2 are RpCH-equivalent if and only if the associated Rp-reduced RCH systems((T?Qi)μi,ωiμi,hiμi,fiμi,Wiμi), i=1,2 are RCH-equivalent.

Moreover, considering the RpCH-equivalence of the regular point reducible RCH systems and using Theorems 3.7, 3.4 and 2.10, we can obtain Theorem 3.8, which states that the solutions of the two types of Hamilton-Jacobi equations for the regular point reducible RCH systems remain invariant under the conditions of RpCH-equivalence if the corresponding Hamiltonian systems are equivalent.

Theorem 3.8Suppose that two regular point reducible RCH systems, (T?Qi,Gi,ωi,Hi,Fi,Wi), i = 1,2 are RpCH-equivalent with an equivalent map ? : Q1→Q2, and that the associated Rp-reduced RCH systems are ((T?Qi)μi,ωiμi,hiμi,fiμi,uiμi), i= 1,2.Assume that the corresponding Hamiltonian systems, (T?Qi,Gi,ωi,Hi), i = 1,2 are also equivalent.Then,under the hypotheses and notations of Theorems 3.2, 3.3 and 3.4, we have the following two assertions hold:

(i) If the one-form γ2: Q2→T?Q2is closed with respect to T πQ2: TT?Q2→T Q2, andˉγ2= π2μ2(γ2) : Q2→(T?Q2)μ2is a solution of the Type I Hamilton-Jacobi equation for the Rp-reduced RCH system((T?Q2)μ2,ω2μ2,h2μ2,f2μ2,u2μ2),then γ1=??·γ2·?:Q1→T?Q1is a solution of the Type I Hamilton-Jacobi equation for the RCH system(T?Q1,G1,ω1,H1,F1,W1),and ˉγ1= π1μ1(γ1) : Q1→(T?Q1)μ1is a solution of the Type I Hamilton-Jacobi equation for the Rp-reduced RCH system ((T?Q1)μ1,ω1μ1,h1μ1,f1μ1,u1μ1).

(ii) If the G2μ2-invariant symplectic map ε2: T?Q2→T?Q2and ˉε2=π2μ2(ε2):J?12(μ2)(?T?Q2)→(T?Q2)μ2satisfy the Type II Hamilton-Jacobi equation for the Rp-reduced RCH system ((T?Q2)μ2,ω2μ2,h2μ2,f2μ2,u2μ2), then ε1= ??·ε2·??: T?Q1→T?Q1and ˉε1=π1μ1(ε1):J?11(μ1)(?T?Q1)→(T?Q1)μ1satisfy the Type II Hamilton-Jacobi equation for the Rp-reduced RCH system ((T?Q1)μ1,ω1μ1,h1μ1,f1μ1,u1μ1).

Next, we prove the assertion (ii).If the G2μ2-invariant symplectic map ε2: T?Q2→T?Q2and ˉε2= π2μ2(ε2) : J2?1(μ2)(?T?Q2) →(T?Q2)μ2satisfy the Type II Hamilton-Jacobi equation for the Rp-reduced RCH system ((T?Q2)μ2,ω2μ2,h2μ2,f2μ2,u2μ2), from Theorem 3.4 we know that ε2is a solution of the Type II Hamilton-Jacobi equation for the RCH system (T?Q2,G2,ω2,H2,F2,W2).Since the two regular point reducible RCH systems,(T?Qi,Gi,ωi,Hi,Fi,Wi), i = 1,2 are RpCH-equivalent, and hence are also RCH-equivalent,from Theorem 2.10 we know that ε1= ??·ε2·??: T?Q1→T?Q1is a solution of the Type II Hamilton-Jacobi equation for the RCH system (T?Q1,G1,ω1,H1,F1,W1).Moreover, from Theorem 3.4 we know that ε1and ˉε1=π1μ1(ε1) satisfy the Type II Hamilton-Jacobi equation for the Rp-reduced RCH system ((T?Q1)μ1,ω1μ1,h1μ1,f1μ1,u1μ1).In the same way, because the map ? : Q1→Q2is a diffeomorphism, ??: T?Q2→T?Q1is a symplectic isomorphism,and vice versa.We have proved the assertion (ii) of Theorem 3.8.?equivalence, and prove the two types of Hamilton-Jacobi theorem for the Rp-reduced RCH system ((T?Q)μ,ωμ,hμ,fμ,uμ) in a way similar to that above, and state that the solutions of two types of Hamilton-Jacobi equations for the regular point reducible RCH systems with symmetries remain invariant under the conditions of RpCH-equivalence if the corresponding Hamiltonian systems are equivalent, where the Rp-reduced space ((T?Q)μ,ωμ) is determined by the affine action and regular point reduction.

4 Hamilton-Jacobi Equations of the Ro-reduced RCH System

We know that the orbit reduction for a Hamiltonian system with symmetry is an alternative approach to symplectic reduction given by Marle [26], and Kazhdan, Kostant and Sternberg[27], and that this is different from the Marsden-Weinstein reduction.Thus, the regular orbit reduction for an RCH system with symmetry is different from the regular point reduction.In this section, we first give the regular orbit reducible RCH system with symmetry and a momentum map.Then we give the geometric constraint conditions of the Ro-reduced symplectic form for the dynamical vector field of the regular orbit reducible RCH system, and prove the Type I and Type II Hamilton-Jacobi theorems for the Ro-reduced RCH system.Moreover,we state the relationship between the solutions of the Type II Hamilton-Jacobi equations and the regular orbit reduction.Finally, we consider the RoCH-equivalence, and state that the solutions of the two types of Hamilton-Jacobi equations for RCH systems with symmetries remain invariant under the conditions of RoCH-equivalence if the corresponding Hamiltonian systems are equivalent.We shall follow the notations and conventions introduced in Marsden et al.[8], Wang [11], Wang [24] and Wang [25].

First,we consider the regular orbit reducible RCH system,which is given by Marsden et al.[8].Assume that the cotangent lifted left action ΦT?:G×T?Q →T?Q is symplectic, free and proper,and admits an Ad?-equivariant momentum map J:T?Q →g?.Letμ∈g?be a regular value of the momentum map J and let Oμ=G·μ?g?be the G-orbit of the coadjoint G-action through the point μ.Since G acts freely, properly and symplectically on T?Q, the quotient space (T?Q)Oμ= J?1(Oμ)/G is a regular quotient symplectic manifold with the Ro-reduced symplectic form ωOμuniquely characterized by the relation

The maps iOμ:J?1(Oμ)→T?Q and πOμ:J?1(Oμ)→(T?Q)Oμare the natural injection and projection, respectively.The pair ((T?Q)Oμ,ωOμ) is called the Ro-reduced symplectic space of(T?Q,ω) at μ.In the general case, we thought that perhaps the structure of the Ro-reduced symplectic space ((T?Q)Oμ,ωOμ) are more complex than that of the Rp-reduced symplectic space ((T?Q)μ,ωμ), but, from the regular reduction diagram (see Ortega and Ratiu [14]), we know that the Ro-reduced space((T?Q)Oμ,ωOμ)is symplectic diffeomorphic to the Rp-reduced space ((T?Q)μ,ωμ), and hence is also symplectic diffeomorphic to a symplectic fiber bundle.

Denote by X(T?Q,G,ω,H,F,u)the dynamical vector field of the regular orbit reducible RCH system (T?Q,G,ω,H,F,W) with a control law u.Assume that it can be expressed by

If an Ro-reduced feedback control law uOμ:(T?Q)Oμ→WOμis chosen, the Ro-reduced RCH system((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ)is a closed-loop regular dynamical system with a control law uOμ.Assume that its dynamical vector field X((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ)can be expressed by

where XhOμis the Hamiltonian vector field of the Ro-reduced Hamiltonian hOμ,and vlift(fOμ)=vlift(fOμ)XhOμ, vlift(uOμ)=vlift(uOμ)XhOμare the changes of XhOμunder the actions of the Ro-reduced external force fOμand the Ro-reduced control law uOμ.The dynamical vector fields of the regular orbit reducible RCH system (T?Q,G,ω,H,F,u) and the Ro-reduced RCH system ((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ) satisfy the condition that

Since the set of Hamiltonian systems with symmetries on the cotangent bundle is not complete under the Marsden-Weinstein reduction, it is not yet complete under the regular orbit reduction, and the Ro-reduced system of an RCH system with symmetry defined on the cotangent bundle T?Q may not be an RCH system on a cotangent bundle.On the other hand, from the expression of the dynamical vector field of an Ro-reduced RCH system, we know that under the actions of the Ro-reduced external force fOμand the Ro-reduced control uOμ, in general, the dynamical vector field is not Hamiltonian, and the Ro-reduced RCH system is not yet a Hamiltonian system.Thus, we cannot describe the Hamilton-Jacobi equation for the Ro-reduced RCH system from the viewpoint of a generating function the same as in Theorem 1.1 given by Abraham and Marsden in [1].However, for a given regular orbit reducible RCH system (T?Q,G,ω,H,F,W) with an Ro-reduced RCH system((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ), by using Lemma 2.5,we can give the geometric constraint conditions of the Ro-reduced symplectic form for the dynamical vector field of the regular orbit reducible RCH system; that is, the Type I and Type II Hamilton-Jacobi equations for the Ro-reduced RCH system ((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ).By using Lemma 2.5 and the Roreduced symplectic form ωOμ, and the fact that the one-form γ : Q →T?Q is closed with respect to TπQ: TT?Q →TQ, and that γ is G-invariant, and a stronger assumption condition Im(γ)?J?1(μ), we can also prove the following Type I Hamilton-Jacobi theorem for the Ro-reduced RCH system ((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ):

Theorem 4.2(Type I Hamilton-Jacobi Theorem for an Ro-reduced RCH system) For a given regular orbit reducible RCH system (T?Q,G,ω,H,F,W) with an Ro-reduced RCH system((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ), assume that γ :Q →T?Q is a one-form on Q,and that ?Xγ=TπQ·?X·γ, where ?X =X(T?Q,G,ω,H,F,u)is the dynamical vector field of the regular orbit reducible RCH system (T?Q,G,ω,H,F,W) with a control law u.Moreover, assuming thatμ ∈g?is a regular value of the momentum mapJ, that Oμ, (μ ∈g?) is the regular reducible orbit of the corresponding Hamiltonian system(T?Q,G,ω,H),that Im(γ)?J?1(μ),and that γ is G-invariant, ˉγ =πOμ(γ):Q →(T?Q)Oμ.If the one-form γ :Q →T?Q is closed with respect to TπQ:TT?Q →TQ, then ˉγ is a solution of the equation Tˉγ·?Xγ=XhOμ·ˉγ, where XhOμis the Hamiltonian vector field of the Ro-reduced Hamiltonian function hOμ:(T?Q)Oμ→R, and the equation is called the Type I Hamilton-Jacobi equation for the Ro-reduced RCH system((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ).Here the maps involved in the theorem are shown in Diagram-8.

Because the Ro-reduced symplectic form ωOμis non-degenerate, the left side of (4.6) equals zero only when ˉγ satisfies the equation Tˉγ·?Xγ=XhOμ·ˉγ.Thus, if the one-form γ :Q →T?Q is closed with respect to TπQ:TT?Q →TQ,then ˉγ must be a solution of the Type I Hamilton-Jacobi equation Tˉγ·?Xγ=XhOμ·ˉγ.?

Next, for any G-invariant symplectic map ε : T?Q →T?Q, we can also prove the Type II Hamilton-Jacobi theorem for the Ro-reduced RCH system ((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ) as follows:

Theorem 4.3(Type II Hamilton-Jacobi Theorem for an Ro-reduced RCH system) For a given regular orbit reducible RCH system (T?Q,G,ω,H,F,W) with an Ro-reduced RCH system((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ), assume that γ :Q →T?Q is a one-form on Q,and that λ = γ·πQ: T?Q →T?Q.For any G-invariant symplectic map ε : T?Q →T?Q, denote that ?Xε=TπQ·?X·ε, where ?X =X(T?Q,G,ω,H,F,u)is the dynamical vector field of the regular orbit reducible RCH system(T?Q,G,ω,H,F,W)with a control law u.Moreover,assume thatμ∈g?is a regular value of the momentum mapJ,and that Oμ, (μ∈g?)is the regular reducible orbit of the corresponding Hamiltonian system (T?Q,G,ω,H), that Im(γ) ?J?1(μ), and that γ is G-invariant, and that ε(J?1(Oμ)) ?J?1(Oμ).Denote that ˉγ = πOμ(γ) : Q →(T?Q)Oμ,ˉλ = πOμ(λ) :J?1(Oμ)(?T?Q) →(T?Q)Oμ, and ˉε = πOμ(ε) :J?1(Oμ)(?T?Q) →(T?Q)Oμ.Then ε and ˉε satisfy the equation Tˉε·XhOμ·ˉε=Tˉλ·?X·ε if and only if they satisfy the equation Tˉγ·?Xε= XhOμ·ˉε, where XhOμand XhOμ·ˉε∈T T?Q are the Hamiltonian vector fields of the Ro-reduced Hamiltonian functions hOμand hOμ·ˉε : T?Q →R, respectively.The equation Tˉγ·?Xε= XhOμ·ˉε is called the Type II Hamilton-Jacobi equation for the Ro-reduced RCH system ((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ).Here the maps involved in the theorem are shown in Diagram-9.

Because the Ro-reduced symplectic form ωOμis non-degenerate,it follows that Tˉγ·?Xε=XhOμ·ˉε is equivalent to Tˉε·XhOμ·ˉε= Tˉλ·?X·ε.Thus, we know that ε and ˉε satisfy the equation Tˉε·XhOμ·ˉε= Tˉλ·?X·ε if and only if they satisfy the Type II Hamilton-Jacobi equation Tˉγ·?Xε=XhOμ·ˉε.?

Moreover, for the regular orbit reducible RCH system (T?Q,G,ω,H,F,W) with an Roreduced RCH system ((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ), we know that the Hamiltonian vector fields XHand XhOμfor the corresponding Hamiltonian system (T?Q,G,ω,H) and its Roreduced system ((T?Q)Oμ,ωOμ,hOμ) are πOμ-related; that is, XhOμ·πOμ= TπOμ·XH·iOμ.Then we can also prove Theorem 4.4, which states the relationship between the solutions of Type II Hamilton-Jacobi equations and the regular orbit reduction:

Theorem 4.4For the regular orbit reducible RCH system (T?Q,G,ω,H,F,W) with an Ro-reduced RCH system ((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ), assume that γ : Q →T?Q is a oneform on Q, and that ε:T?Q →T?Q is a G-invariant symplectic map, ˉε=πOμ(ε):J?1(Oμ)(?T?Q)→(T?Q)Oμ.Under the hypotheses and notations of Theorem 4.3,then we have that ε is a solution of the Type II Hamilton-Jacobi equation Tγ·?Xε=XH·ε for the regular orbit reducible RCH system (T?Q,G,ω,H,F,W) if and only if ε and ˉε satisfy the Type II Hamilton-Jacobi equation Tˉγ·?Xε=XhOμ·ˉεfor the Ro-reduced RCH system ((T?Q)Oμ,ωOμ,hOμ,fOμ,uOμ).

Remark 4.5It is worth noting that the Type I Hamilton-Jacobi equation Tˉγ·?Xγ=XhOμ·ˉγ is the equation of the Ro-reduced differential one-form ˉγ, and the Type II Hamilton-Jacobi equation Tˉγ·?Xε= XhOμ·ˉε is the equation of the symplectic diffeomorphism map ε and the Ro-reduced symplectic diffeomorphism map ˉε.If both the external force and control of a regular orbit reducible RCH system (T?Q,G,ω,H,F,W) are zero; that is, F = 0 and W = ?, the RCH system is just a regular orbit reducible Hamiltonian system (T?Q,G,ω,H).From the proofs of Theorems 4.2,4.3 and 4.4,we can also get the two types of Hamilton-Jacobi equations for the associated Ro-reduced Hamiltonian system(which given in Wang[11]).Thus,Theorems 4.2,4.3 and 4.4 can be regarded as an extension of the two types of Hamilton-Jacobi equations for the Ro-reduced Hamiltonian system to that for the Ro-reduced RCH system.

Moreover, for the regular orbit reducible RCH system we can also introduce the regular orbit reducible controlled Hamiltonian equivalence (RoCH-equivalence) as follows:

Definition 4.6 (RoCH-equivalence) Suppose that we have two regular orbit reducible RCH systems (T?Qi,Gi,ωi,Hi,Fi,Wi), i = 1,2.We say that they are RoCH-equivalent, or simply, that (T?Q1,G1,ω1,H1,F1,W1)RoCH～ (T?Q2,G2,ω2,H2,F2,W2), if there exists a diffeomorphism ?:Q1→Q2such that the following controlled Hamiltonian matching conditions hold:

It is worth noting that for the regular orbit reducible RCH system, the induced equivalent map ??Oμalso keeps the equivariance of G-action on the regular reducible orbit.Moreover, we can obtain a regular orbit reduction theorem for an RCH system that explains the relationship between the RoCH-equivalence for the regular orbit reducible RCH systems with symmetries and the RCH-equivalence for the associated Ro-reduced RCH systems (the proof is given in Wang [24] and Marsden et al.[8]).This theorem can be regarded as an extension of the regular orbit reduction theorem of Hamiltonian systems under the regular controlled Hamiltonian equivalence conditions.

Theorem 4.7 Two regular orbit reducible RCH systems, (T?Qi,Gi,ωi,Hi,Fi,Wi), i =1,2, are RoCH-equivalent if and only if the associated Ro-reduced RCH systems ((T?Q)Oμi,ωiOμi,hiOμi,fiOμi,WiOμi), i=1,2 are RCH-equivalent.

Moreover, considering the RoCH-equivalence of the regular orbit reducible RCH systems,and using Theorems 4.7, 4.4 and 2.10 above, we can obtain Theorem 4.8, which states that the solutions of the two types of Hamilton-Jacobi equations for the regular orbit reducible RCH systems remain invariant under the conditions of RoCH-equivalence if the corresponding Hamiltonian systems are equivalent.

Theorem 4.8 Suppose that two regular orbit reducible RCH systems, (T?Qi,Gi,ωi,Hi,Fi,Wi),i=1,2,are RoCH-equivalent with an equivalent map ?:Q1→Q2,with the associated Ro-reduced RCH systems are ((T?Q)Oμi,ωiOμi,hiOμi,fiOμi,WiOμi), i=1,2.Assume that the corresponding Hamiltonian systems, (T?Qi,Gi,ωi,Hi), i=1,2 are also equivalent.Under the hypotheses and notations of Theorems 4.2,4.3 and 4.4,we have that the following two assertions hold:

(i) If the one-form γ2: Q2→T?Q2is closed with respect to TπQ2: T T?Q2→T Q2,and ˉγ2= π2Oμ2(γ2) : Q2→(T?Q2)Oμ2is a solution of the Type I Hamilton-Jacobi equation for the Ro-reduced RCH system ((T?Q2)Oμ2,ω2Oμ2,h2Oμ2,f2Oμ2,u2Oμ2), then γ1= ??·γ2·? : Q1→T?Q1is a solution of the Type I Hamilton-Jacobi equation for the RCH system(T?Q1,G1,ω1,H1,F1,W1), and ˉγ1= π1Oμ1(γ1) : Q1→(T?Q1)Oμ1is a solution of the Type I Hamilton-Jacobi equation for the Ro-reduced RCH system ((T?Q1)Oμ1,ω1Oμ1,h1Oμ1,f1Oμ1,u1Oμ1).

(ii)If the G2-invariant symplectic map ε2:T?Q2→T?Q2and ˉε2=π2Oμ2(ε2):J2?1(Oμ2)(?T?Q2) →(T?Q2)Oμ2satisfy the Type II Hamilton-Jacobi equation for the Ro-reduced RCH system ((T?Q2)Oμ2,ω2Oμ2,h2Oμ2,f2Oμ2,u2Oμ2), then ε1= ??·ε2·??: T?Q1→T?Q1and ˉε1= π1Oμ1(ε1) : J1?1(Oμ1)(?T?Q1) →(T?Q1)Oμ1satisfy the Type II Hamilton-Jacobi equation for the Ro-reduced RCH system ((T?Q1)Oμ1,ω1Oμ1,h1Oμ1,f1Oμ1,u1Oμ1).

5 Applications

Now it is natural to ask if there is a practical RCH system, and if this may show the effect on controls in regular point reduction and the Hamilton-Jacobi theory of the system.In this section, as an application of the above theoretical results, we consider first the regular point reducible RCH system on the generalization of a Lie group, and give the geometric constraint conditions of the Rp-reduced symplectic form for the dynamical vector field of the regular point reducible RCH system; that is,the two types of Hamilton-Jacobi equations for the Rp- reduced RCH system.Then we show the Type I and Type II Hamilton-Jacobi equations for a rigid body and a heavy top with internal rotors on the generalization of the rotation group SO(3),and on the generalization of the Euclidean group SE(3), respectively.Note that these given equations are more complex than those of Hamiltonian systems without control,which describe explicitly the effect on controls in the regular point reduction and the Hamilton-Jacobi theory for the RCH systems.We shall follow the notations and conventions introduced in Marsden et al.[17], Marsden and Ratiu [4], Marsden et al.[8], and Wang [11].

5.1 RCH System on the Generalization of a Lie Group

In order to describe the two types of Hamilton-Jacobi equations of a rigid body and a heavy top with internal rotors, we need to first consider the regular point reducible RCH system (T?Q,G,ωQ,H,F,W) on the generalization of a Lie group Q = G×V, where G is a Lie group with Lie algebra g and V is a k-dimensional vector space.Define the left G-action Φ:G×Q →Q, Φ(g,(h,θ)):=(gh,θ) for any g,h ∈G, θ ∈V; that is, the G-action on Q is by the left translation on the first factor G and the trivial action on the second factor V.Because locally,T?Q ～=T?G×T?V and T?V ～=V×V?,and,by using the local left trivialization of T?G;that is,T?G ～=G×g?,where g?is the dual of g,we have that,locally,T?Q ～=G×g?×V ×V?.If the left G-action Φ : G×Q →Q is free and proper, then the cotangent lift of the action to its cotangent bundle T?Q, given by ΦT?: G×T?Q →T?Q, ΦT?(g,(h,μ,θ,λ)) :=(gh,μ,θ,λ)for any g,h ∈G, μ ∈g?, θ ∈V, λ ∈V?, is also a free and proper action, and the orbit space(T?Q)/G is a smooth manifold and π :T?Q →(T?Q)/G is a smooth submersion.Since G acts trivially on g?, V and V?, it follows that (T?Q)/G is diffeomorphic locally to g?×V ×V?.

We know that g?is a Poisson manifold with respect to the (±)-Lie-Poisson bracket {·,·}±defined by

If the Hamiltonian H(g,p,θ,l) : T?Q ～= G×g?×V ×V?→R is left cotangent lifted Gaction ΦT?invariant, for μ ∈g?, the regular value of the momentum map JQ: T?Q →g?, we have the associated Rp-reduced Hamiltonian hμ(ν,θ,l):(T?Q)μ～=Oμ×V ×V?→R, defined by hμ·πμ= H·iμ, and the Rp-reduced Hamiltonian vector field Xhμgiven by Xhμ(Kμ) ={Kμ,hμ}?|Oμ×V×V?, where Kμ(ν,θ,l):Oμ×V ×V?→R.

Thus,we consider that the fiber-preserving map F :T?Q →T?Q and the fiber submanifold W of T?Q are all left cotangent lifted G-action ΦT?invariant, then, for u ∈W, the 6-tuple(T?Q,G,ωQ,H,F,u) is a regular point reducible RCH system with a control law u, and its dynamical vector field can be expressed by

(i) If the one-form γ :Q →T?Q is closed with respect to T πQ:TT?Q →T Q, then ˉγ is a solution of the Type I Hamilton-Jacobi equation Tˉγ·?Xγ=Xhμ·ˉγ.

(ii) The ε and ˉε satisfy the Type II Hamilton-Jacobi equation Tˉγ·?Xε= Xhμ·ˉε if and only if they satisfy the equation Tˉε·(Xhμ·ˉε)=Tˉλ·?X·ε.Here Xhμand Xhμ·ˉε∈TT?Q are the Hamiltonian vector fields of the Rp-reduced Hamiltonian functions hμand hμ·ˉε : T?Q →R,respectively.

5.2 Hamilton-Jacobi Equations of a Rigid Body with Internal Rotors

In what follows,we regard the rigid body with three symmetric internal rotors as a regular point reducible RCH system on the generalization of the rotation group SO(3)×R3, and give its two types of Hamilton-Jacobi equations by calculations in detail.Note that our description of the motion and the equations of the rigid body with internal rotors in this subsection follow some of the notations and conventions in Marsden [13], Marsden and Ratiu [4], Marsden et al.[8], and Wang [11].

We consider a rigid body carrying three“non-mass”internal rotors,which is called a carrier body,where“non-mass”means that the mass of a rotor is very small compared to the mass of the rigid body.Denote the system center of mass by O in the carrier body frame and at O place a set of(orthonormal)body axes,and assume that the rotor and the body coordinate axes are aligned with principal axes of the carrier body.Translations are ignored and only rotations of the rigid body-rotor system are considered, and the rotor spins under the influence of a torque u acting on the rotor.In this case, the configuration space is Q=SO(3)×V, where V =S1×S1×S1,with the first factor being the carrier body attitude and the second factor being the angles of the rotors.The corresponding phase space is the cotangent bundle T?Q and, locally, T?Q ～=T?SO(3)×T?V, where T?V =T?(S1×S1×S1)～=T?R3locally,with the canonical symplectic form ωQ.By using the local left trivialization, locally, T?SO(3)～=SO(3)×so?(3) and T?R3～=R3×R3?,so we have that,locally,T?Q ～=SO(3)×so?(3)×R3×R3?.For convenience,we denote uniformly that,locally,Q=SO(3)×R3and T?Q=T?(SO(3)×R3)～=SO(3)×so?(3)×R3×R3?.Assume that the Lie group G = SO(3) acts freely and properly on Q by the left translation on the first factor SO(3) and the trivial action on the second factor R3.Then the action of SO(3) on the phase space T?Q is by the cotangent lift of the left SO(3) action on Q; that is,ΦT?:SO(3)×T?Q ～=SO(3)×SO(3)×so?(3)×R3×R3?→T?Q ～=SO(3)×so?(3)×R3×R3?,given by ΦT?(B,(A,Π,α,l)) = (BA,Π,α,l) for any A,B ∈SO(3), Π ∈so?(3), α ∈R3, l ∈R3?,which is also free and proper.Assume that the left SO(3) action ΦT?is symplectic and admits an associated Ad?-equivariant momentum map JQ:T?Q ～=SO(3)×so?(3)×R3×R3?→so?(3)for the cotangent lift of the left SO(3) action.If Π ∈so?(3) is a regular value of JQ, then the regular point reduced space (T?Q)Π=(Π)/SO(3)Πis symplectically diffeomorphic to the coadjoint orbit OΠ×R3×R3??so?(3)×R3×R3?.

Let I = diag(I1,I2,I3) be the matrix of the inertia moment of the rigid body in the body fixed frame,which is a principal body frame,and let Ji, i=1,2,3 be the moments of inertia of the rotors around their rotation axes.Let Jik, i=1,2,3, k =1,2,3 be the moments of inertia of the ith rotor with i=1,2,3 around the kth principal axis with k =1,2,3, respectively, and denote that ˉIi= Ii+J1i+J2i+J3i?Jii, i = 1,2,3.Let ? = (?1,?2,?3) be the angular velocity vector of the rigid body-rotors computed with respect to the axes fixed in the carrier body and (?1,?2,?3) ∈so(3).Let αi, i = 1,2,3 be the relative angles of the rotors and let ˙α = ( ˙α1, ˙α2, ˙α3) be the relative angular velocity vector of the rotors about the principal axes with respect to a carrier body fixed frame.For convenience, we assume that the total mass of the system m=1.

Now, by using the local left trivialization, locally, T SO(3) ～= SO(3)×so(3) and T R3～=R3×R3, so we have that, locally, TQ ～=SO(3)×so(3)×R3×R3.We consider the Lagrangian of the system L(A,?,α, ˙α) : TQ ～= SO(3)×so(3)×R3×R3→R, which is the total kinetic energy of the rigid body plus the total kinetic energy of the rotor, given by

and the Legendre transformation FL:TQ ～=SO(3)×so(3)×R3×R3→T?Q ～=SO(3)×so?(3)×R3×R3?, (A,?,α, ˙α) →(A,Π,α,l), where Π = (Π1,Π2,Π3) ∈so?(3), l = (l1,l2,l3) ∈R3?,we have the Hamiltonian H(A,Π,α,l):T?Q ～=SO(3)×so?(3)×R3×R3?→R given by

From the above expression of the Hamiltonian, we know that H(A,Π,α,l) is invariant under the cotangent lift of the left SO(3)-action ΦT?: SO(3)×T?Q →T?Q.Moreover, from the

Lie-Poisson bracket of the rigid body on so?(3); that is, for F,K : so?(3) →R, we have that{F,K}?(Π) = ?Π·(?ΠF ×?ΠK), and from the Poisson bracket on T?R3, we can get the Poisson bracket on so?(3)×R3×R3?; that is, for F,K :so?(3)×R3×R3?→R, we have that

where vlift(u)=vlift(u)·XHis the change of XHunder the action of the control torque u.

(ii) The ε and ˉε satisfy the Type II Hamilton-Jacobi equation Tˉγ·?Xε= Xhμ·ˉε if and only if they satisfy the equation Tˉε·(Xhμ·ˉε)=Tˉλ·?X·ε.

When the rigid body does not carry any internal rotor, in this case Q = G = SO(3), and the rigid body is a regular point reducible Hamiltonian system (T?SO(3),SO(3),ω,H), and hence it is also a regular point reducible RCH system without the external force and control.For a point μ ∈so?(3), the regular value of the momentum map J : T?SO(3) →so?(3), the Marsden-Weinstein reduced rigid body system is 3-tuple (Oμ,ωOμ,hOμ), where Oμ?so?(3) is the co-adjoint orbit, and ωOμis the orbit symplectic form on Oμ, which is induced by the rigid body Lie-Poisson bracket on so?(3), and hOμ(Π)·πOμ=H(A,v,Π)|Oμ.From the Proposition 5.2, we can obtain the Proposition 5.3 given in Wang [11]; that is, we give the two types of Lie-Poisson Hamilton-Jacobi equations for the Marsden-Weinstein reduced rigid body system(Oμ,ωOμ,hOμ) (see Marsden and Ratiu [4], Ge and Marsden [6], and Wang [11]).

5.3 Hamilton-Jacobi Equations of Heavy Top with Internal Rotors

In what follows, we regard the heavy top with two pairs of symmetric internal rotors as a regular point reducible RCH system on the generalization of the Euclidean group SE(3)×R2,and give its two types of Hamilton-Jacobi equations by calculations performed in detail.Note that our description of the motion and the equations of the heavy top with internal rotors in this subsection follows some of the notations and conventions in Marsden [13], Marsden and Ratiu [4], Marsden et al.[8], and Wang [11].

We first describe a heavy top with two pairs of symmetric,“non-mass”rotors;this is called a carrier body, where“non-mass”means that the mass of a rotor is very small compared to the mass of the heavy top.We mount two pairs of rotors within the top so that each pair’s rotation axis is parallel to first and second principal axes of the heavy top.The heavy top is moving in a gravitational field, and the rotor spins under the influence of a torque u acting on the rotor.Then the motion of the controlled heavy top-rotor system is just the rotation motion with drift,in this case, the configuration space is Q=SO(3)?R3×V ～=SE(3)×V, where V =S1×S1,with the first factor being the position of the carrier body and the second factor being the angles of rotors.The corresponding phase space is the cotangent bundle T?Q and, locally,T?Q ～=T?SE(3)×T?V,where T?V =T?(S1×S1)～=T?R2locally,with the canonical symplectic form ωQ.By using the local left trivialization, locally, T?SE(3)～=SE(3)×se?(3) and T?R2～=R2×R2?,so we have that,locally,T?Q ～=SE(3)×se?(3)×R2×R2?.For convenience,we denote uniformly that,locally,Q=SE(3)×R2and T?Q=T?(SE(3)×R2)～=SE(3)×se?(3)×R2×R2?.Assume that the Lie group G=SE(3) acts freely and properly on Q by the left translation on the first factor SE(3) and the trivial action on the second factor R2.Then the action of the SE(3) on the phase space T?Q is by the cotangent lift of the left SE(3) action on Q; that is,ΦT?:SE(3)×T?Q ～=SE(3)×SE(3)×se?(3)×R2×R2?→T?Q ～=SE(3)×se?(3)×R2×R2?,given by ΦT?((B,u)((A,v),(Π,w),α,l)) = ((BA,u+Bv),(Π,w),α,l) for any A,B ∈SO(3), Π ∈so?(3), u,v,w ∈R3, α ∈R2, l ∈R2?, and where SO(3) acts on R3in the standard way.The action ΦT?is also free and proper.Moreover, assume that the action ΦT?is symplectic and admits an associated Ad?-equivariant momentum map JQ:T?Q ～=SE(3)×se?(3)×R2×R2?→se?(3) for the cotangent lift of the left SE(3) action.If (Π,w)∈se?(3) is a regular value of JQ,then the Rp-reduced space (T?Q)(Π,w)=J?1Q(Π,w)/SE(3)(Π,w)is symplectically diffeomorphic to the coadjoint orbit O(Π,w)×R2×R2??se?(3)×R2×R2?.

Let I = diag(I1,I2,I3) be the matrix of the inertia moment of the heavy top in the body fixed frame, which is a principal body frame.Let Jk, k =1,2 be the moments of inertia of the rotors around their rotation axes.Let Jki, k =1,2, i=1,2,3,be the moments of inertia of the k-th rotor with k =1,2 around the i-th principal axis with i=1,2,3, respectively, and denote that ˉIk=Ik+J1k+J2k?Jkk, k =1,2 and that ˉI3=I3+J13+J23.Let ?=(?1,?2,?3) be the angular velocity vector of heavy top-rotors computed with respect to the axes fixed in the carrier body and let (?1,?2,?3) ∈so(3).Let θk, k = 1,2 be the relative angles of the rotors and let ˙θ = ( ˙θ1, ˙θ2) be the relative angular velocity vector of the rotors about the principal axes with respect to the carrier body fixed frame.Let g be the magnitude of the gravitational acceleration and let h be the distance from the origin O to the center of mass of the system.For convenience, we assume that the total mass of the system is m=1.

Now, by the local left trivialization, locally, T SE(3) ～= SE(3)×se(3) and T R2～= R2×R2, so we have that, locally, TQ ～= SE(3)×se(3)×R2× R2.We consider the Lagrangian L(A,v,?,Γ,θ, ˙θ):TQ ～=SE(3)×se(3)×R2×R2→R, which is the total kinetic energy of the heavy top plus the total kinetic energy of the rotor minus the potential energy of the system,given by

where (A,v) ∈SE(3), (?,Γ) ∈se(3) and ? = (?1,?2,?3) ∈so(3), θ = (θ1,θ2) ∈R2, ˙θ =(˙θ1, ˙θ2) ∈R2, Γ ∈R3, and the variable Γ is regarded as a parameter with respect to the potential energy of the system.If we introduce the conjugate angular momentum, which is given by

and the Legendre transformation with the parameter Γ; that is, FL : TQ ～= SE(3)×se(3)×R2× R2→T?Q ～= SE(3)× se?(3) × R2× R2?, (A,v,?,Γ,θ, ˙θ) →(A,v,Π,Γ,θ,l), where Π = (Π1,Π2,Π3) ∈so?(3), l = (l1,l2) ∈R2?, we have the Hamiltonian H(A,v,Π,Γ,θ,l) :T?Q ～=SE(3)×se?(3)×R2×R2?→R given by

From the above expression of the Hamiltonian, we know that H(A,v,Π,Γ,θ,l) is invariant under the cotangent lift of the left SE(3)-action ΦT?: SE(3)×T?Q →T?Q.Moreover, from the semidirect product Poisson bracket (see Marsden et al.[16]), we can get the heavy top Lie-Poisson bracket on se?(3); that is, for F,K :se?(3)→R, we have that

Hence, from the heavy top Lie-Poisson bracket on se?(3) and the Poisson bracket on T?R2, we can get the Poisson bracket on se?(3)×R2×R2?; that is, for F,K :se?(3)×R2×R2?→R, we have that

When the heavy top does not carry any internal rotor,in this case Q=G=SE(3),and the heavy top is a regular point reducible Hamiltonian system(T?SE(3),SE(3),ω,H),and hence it is also a regular point reducible RCH system without the external force and control.For a point(μ,a) ∈se?(3), the regular value of the momentum mapJ: T?SE(3) →se?(3), the Marsden-Weinstein reduced heavy top system is 3-tuple (O(μ,a),ωO(μ,a),hO(μ,a)), where O(μ,a)?se?(3)is the co-adjoint orbit, and ωO(μ,a)is orbit symplectic form on O(μ,a), which is induced by the heavy top Lie-Poisson bracket on se?(3), and hO(μ,a)(Π,Γ)·πO(μ,a)=H(A,v,Π,Γ)|O(μ,a).From the Proposition 5.3 we can obtain the Proposition 5.5 given in Wang [11]; that is, we give the two types of Lie-Poisson Hamilton-Jacobi equation for the Marsden-Weinstein reduced heavy top system(O(μ,a),ωO(μ,a),hO(μ,a))(see Marsden and Ratiu[4],Ge and Marsden[6], and Wang[11]).

6 Conclusion

An RCH system is a Hamiltonian system with external force and control (see Marsden et al.[8]), which is a dynamical system closely related to a Hamiltonian system, and it can be explored and studied by extending the methods for external force and control used in the study of Hamiltonian systems.Thus, we can emphasize explicitly the impact of external force and control in the study for the RCH systems.In this paper, we first gave the geometric constraint conditions of a canonical symplectic form for the dynamical vector field of an RCH system.These conditions are the two types of Hamilton-Jacobi equations, which are a development of the classical Hamilton-Jacobi equation given by Abraham and Marsden[1](also see Wang[11]).Next, we generalized the above results for a regular reducible RCH system with symmetry and a momentum map, and derived two types of Hamilton-Jacobi equations for the regular point reduced RCH system and the regular orbit reduced RCH system.We also proved that the RCH-equivalence for the RCH system, and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries, leave the solutions of the corresponding Hamilton-Jacobi equations invariant.Finally, as an application of the theoretical results, we showed the Type I and Type II Hamilton-Jacobi equations for the Rp-reduced controlled rigid body-rotor system and the Rp-reduced controlled heavy top-rotor system on the generalizations of the rotation group SO(3) and the Euclidean group SE(3), respectively.In particular, it is worth pointing out that the motions of the controlled rigid body-rotor system and the controlled heavy top-rotor system are different,and that the configuration spaces,the Hamiltonian functions, the actions of the Lie groups, the Rp-reduced symplectic forms and the Rp-reduced systems of the controlled rigid body-rotor system and the controlled heavy top-rotor system are also different.However,the two types of Hamilton-Jacobi equations given are same;that is,the internal rules are same, as can be seen by comparing Proposition 5.2 and Proposition 5.3.This is very important! Thus, we have shown the deeply internal relationships of the geometrical structures of phase spaces, the dynamical vector fields and the controls of the RCH system,by analyzing carefully the geometrical and topological structures of the phase space and the reduced phase space of the corresponding RCH system.

The theory of controlled mechanical systems is a very important subject,gathering together some separate areas of research such as mechanics, differential geometry and nonlinear control theory,etc..The emphasis of this research on geometry is motivated by trying to understand the structure of the equations of motion of the system in a way that helps both analysis and design.Thus, it is natural to study the controlled mechanical systems in combination with the analysis of dynamical systems and the geometric reduction theory of Hamiltonian and Lagrangian systems.Following the theoretical development of geometric mechanics, a lot of important problems involving this subject are being explored and studied, see Wang [24].In mechanics,it happens very often that systems have constraints.A nonholonomic Hamiltonian system is a Hamiltonian system with nonholonomic constraint, and a nonholonomic RCH system is also a RCH system with a nonholonomic constraint.Usually, under the restrictions given by nonholonomic constraints,in general,the dynamical vector fields of the nonholonomic Hamiltonian system and the nonholonomic RCH system may not be Hamiltonian.Thus,we cannot describe the Hamilton-Jacobi equations for the nonholonomic Hamiltonian system and the nonholonomic RCH system from the viewpoint of a generating function as in the classical Hamiltonian case.Since the Hamilton-Jacobi theory is developed based on the Hamiltonian picture of the dynamics, it is a natural idea to extend the Hamilton-Jacobi theory to the nonholonomic Hamiltonian system and the nonholonomic RCH system, and to do so with symmetries and momentum maps(see Le′on and Wang[29],Wang[30]).Undertaking this work is one of our goals for future research papers.It is a natural problem to investigate if there is a practical RCH system, and to try to show the effect on controls in regular symplectic reductions of the system.Wang,in [31, 32], applied the Hamilton-Jacobi theoretical result for the regular point reducible RCH system to give explicitly the two types of Hamilton-Jacobi equations for the regular reduced controlled spacecraft-rotor system,and the regular reduced controlled underwater vehicle-rotor system.In addition,we note that there have been a lot of beautiful results of the reduction theory of Hamiltonian systems in celestial mechanics, hydrodynamics and plasma physics.Thus,it is important to study the application of the reduction theory and the Hamilton-Jacobi theory of the systems; this will be another of our goals in future research papers.

It was the key pursuit of Professor Jerrold E.Marsden to explore and reveal the deeply internal relationship between the geometrical structure of phase space and the dynamical vector field of a mechanical system;we have interested and continue pursuing and inheriting this goal.