aa r X i v : . [ m a t h - ph ] S e p A time-dependent energy-momentum method
J. de Lucas and B.M. ZaworaDepartment of Mathematical Methods in Physics, University of Warsaw,ul. Pasteura 5, 02-093 Warszawa, Poland
Abstract
We devise a generalisation of the energy momentum-method for studying thestability of non-autonomous Hamiltonian systems with a Lie group of Hamiltoniansymmetries. A generalisation of the relative equilibrium point notion to a non-autonomous realm is provided and studied. Relative equilibrium points of non-autonomous Hamiltonian systems are described via foliated Lie systems, whichopens a new field of application of such differential equations. We reduce non-autonomous Hamiltonian systems via the Marsden-Weinstein theorem and we pro-vide conditions ensuring the stability of the projection of relative equilibrium pointsto the reduced space. As an application, we study the stability of relative equi-librium points for a class of mechanical systems, which covers rigid bodies as aparticular instance.
MSC 2010:
PACS numbers:
Key words: energy-momentum method, foliated Lie system, integrable system, Lyapunovintegrability, relative equilibrium point, rigid body, rotating system
Symplectic geometry has a fruitful history of applications to classical mechanics [2, 8, 9].Its origin can be traced back to the pioneering works by Lagrange, who carefully analysedin [13] the rotational motion of mechanical systems.Toward the end of the XXth century, the Marsden-Weinstein reduction theorem [18]was devised so as to describe the reduction of Hamiltonian systems on a symplecticmanifold admitting a certain Lie group of symmetries of the Hamiltonian of the systemand the symplectic form of the manifold. This theorem, an improvement of previous ideasby Lie, Smale, and Cartan [19], led to relevant applications in classical mechanics as wellas many extensions to other types of geometric structures [3, 5, 17].Let Φ : G × P → P be a Lie group action having a family of Hamiltonian fundamentalvector fields relative to a symplectic form ω on P , i.e. a Hamiltonian Lie group action , andleaving invariant h ∈ C ∞ ( P ). Following techniques in [19], Weinstein and Marsden usedΦ and ω to define the so-called momentum-map J : P → g ∗ , where g ∗ is the dual to the Lie1lgebra g of G . By assuming J to obey certain conditions, e.g. to be equivariant , Marsdenand Weinstein reduced the Hamiltonian problem h on P to a problem on P µ := J − ( µ ) /G µ for a regular point µ ∈ g ∗ of J and an appropriate Lie subgroup G µ ⊂ G acting freelyand properly on J − ( µ ). Remarkably, P µ admits a canonically defined symplectic form, ω µ , while the Hamiltonian system h on P leads to a new one on P µ given by the uniquefunction k µ such that k µ ◦ π µ := h on J − ( µ ) for π µ : J − ( µ ) → P µ being the quotientmap.The Hamiltonian system k µ on P µ has equilibrium points , i.e. stable points relative tothe evolution given by the Hamilton equations for k µ in P µ , that are the projection of notnecessarily equilibrium points of the Hamilton equations for h on P , the referred to as relative equilibrium points of h relative to Φ [16]. One wonders which are the propertiesof the solutions to h that project onto equilibrium points of k µ . It is also interestingto study the stability of the Hamilton equations for k µ close to its equilibrium points.The energy-momentum method was developed to study these problems [16]. Instead ofanalysing straightforwardly the reduced system on P µ , the energy-momentum methodstudies the Hamiltonian problem on P µ via the properties of the initial function h on P ,which is easier as it avoids the necessity of constructing P µ and k µ explicitly (cf. [16]).There has been several generalisations of the energy-momentum method as well assome improvements and many applications of the developed theories (see [22, 23, 24, 25]and references therein). In this work, we present a time-dependent generalisation of theenergy-momentum method on symplectic manifolds. The Marsden–Weinstein theoremcan also be applied to a function h : R × P → R relative to a Hamiltonian Lie group Φrelative to the symplectic form ω on P and leaving invariant h (cf. [18]). We suggest adefinition of a relative equilibrium point for h relative to Φ. We study the structure ofthe space of relative equilibrium points in P . It is proven that the dynamics of h on thespace of relative equilibrium points for h : R × P → R can be described through foliatedLie systems [7]. The work [7] details only the potential application of the foliated Liesystem notion in integrable Hamiltonian systems, our work instead shows another, muchdifferent, potential field of applications to the t -dependent energy-momentum method.The stability of the Hamilton equations for k µ , obtained through the reduction of h : R × P → R via the Marsden-Weinstein theorem, close to its equilibrium points isaddressed by studying the properties of h . As in the standard energy-momentum method[16], this simplifies the study of the problem as one does not need to construct the spaces P µ and the functions k µ . Theoretically, our analysis requires the use of time-dependentLyapunov stability theory [12, 27], which is much more complicated than in the time-independent case (see for instance Lemma 6.1 and Theorems 6.2 and 6.5). Meanwhile,our theory retrieves quite easily the results of the time-independent energy-momentummethod. As an application, we study an orbiting mechanical system that, as a particularcase, retrieves the rigid body and the standard theory developed by Marsden and Simo inthe pioneering work [16]. Relevantly, it is simple to see that our methods can be extendedto study the energy-Casimir method and other of its generalisations quite easily. Dueto the many applications of the energy-momentum methods and their generalisations[23, 25], our results may have many potential applications.2he work goes as follows. Section 2 details some general results on Lyapunov sta-bility for non-autonomous differential equations. Section 3 details some basic notions onsymplectic manifolds and the conventions to be used hereafter. It also gives some general-isations to the t -dependent realm of results on autonomous Hamiltonian systems. Section4 generalises the notion of relative equilibrium point to time-dependent Hamiltonian sys-tems. Section 5 studies the relation between the manifold of relative equilibrium pointsand foliated Lie systems. Section 6 analyses the stability of trajectories around relativeequilibrium points of non-autonomous Hamiltonian systems. Section 7 details an exampleof our theory. Finally, our results are summarised and an outlook of further research ispresented in Section 8. From now on, and if not otherwise stated, we assume all structures to be smooth andglobally defined. This stresses the key ideas of our presentation.Let us provide a simple adaptation of the basic Lyapunov stability theory on R n tomanifolds. This will allow us to use this theory to study differential equations on manifolds(see [12, 21, 27] for details on Lyapunov stability theory on R n ). It is simple to see thatour approach retrieves the standard Lyapunov theory when restricted to problems on aEuclidean space R n . Our final aim is to apply these techniques to studying the stabilityof Hamilton equations of reduced t -dependent Hamiltonian systems by the Marsden–Weinstein theorem close to its equilibrium points.Recall that any manifold P admits a Riemannian metric [4]. Moreover, the integralof the curvature of a Riemannian metric over a manifold P by the Gauss–Bonet theoremis 2 π X ( P ), where X ( P ) stands for the Euler characteristic of P (see [11]). If X ( P ) = 0,the curvature of the Riemannian metric will not be zero everywhere. Then, not everymanifold admits a flat Riemannian metric.If we assume P to be endowed with a Riemannian metric g , one can define a distancebetween two points x , x ∈ P as the smallest length, d ( x , x ), of a curve from x to x relative to g . Let us define B r,x e to be the ball of radius r around x e relative to g , namely B r,x e := { x ∈ P : d ( x, x e ) < r } .Hereafter, t stands for the physical time. Let X : ( t, x ) ∈ R × P X ( t, x ) ∈ P be a t -dependent vector field, namely a t -parametric family of vector fields X t : x ∈ P X ( t, x ) ∈ T P with t ∈ R (see [14] for details). Let us consider the followingnon-autonomous dynamical system dxdt = X ( t, x ) , x ∈ P, t ∈ R (2.1)where X is assumed to be smooth enough for (2.1) to satisfy the conditions of the Theoremof existence and uniqueness of solutions. 3et ¯ R = R + ∪ { } be the space of non-negative real numbers. A point x e ∈ P is an equilibrium point of (2.1) if X ( t, x e ) = 0 for every t ∈ R . An equilibrium point is stable iffor every t ∈ R and ball B ǫ,x e there exists a ball B δ ( t ,ǫ ) ,x e such that every solution x ( t ) to(2.1) with x ( t ) ∈ B δ ( t ,ǫ ) ,x e satisfies that x ( t ) ∈ B ǫ,x e for all time t ≥ t . An equilibriumpoint is uniformly stable if for every ǫ >
0, one can chose δ ( t , ǫ ) to be independent of t .An equilibrium point is unstable if it is not stable.An equilibrium point x e is asymptotically stable if x e is stable and for every t ∈ R there exists an open neighbourhood B r ( t ) ,x e of x e such that every solution x ( t ) to (2.1)with x ( t ) ∈ B r ( t ) ,x e converge to x e . Moreover, x e is uniformly asymptotically stable if r ( t ) can be chosen to be independent of t . Definition 2.1.
A continuous function M : ¯ R × P → R is a positive definite function from t = t if for some r > g : ¯ R → R with lim t → g ( t ) = 0, satisfies the following M ( t, x e ) = 0 , M ( t, x ) ≥ g ( d ( x, x e )) , ∀ x ∈ B r,x e , ∀ t ≥ t . Definition 2.2.
A continuous function M : ¯ R × P → R is decrescent from t = t if for some ǫ > σ : ¯ R → R with lim t → σ ( t ) = 0,is fulfilled M ( t, x ) ≤ σ ( d ( x, x e )) , ∀ x ∈ B ǫ,x e , ∀ t ≥ t . Above definitions are significant to understand Theorem 2.3, which allows us to de-termine the stability of (2.1) by studying the properties of an appropriate function.Since all distances are locally equivalent [1], the stability of points will be independentof the metric employed. In particular, we are interested on applying Theorem 2.3 thatapplies to stability problems on linear spaces relative to the distance induced by a norm.In this regard, a much simpler version of the above constructions and definitions can beconsidered. For every point p of a manifold P , one can map an open neighbourhood U of p onto an open subset φ ( U ) of R n via a chart φ : U → R n of the differentiable structureof the manifold P . Since R n admits a Euclidean metric, the open subset φ ( U ) can beconsidered as a flat Riemannian space with a metric g . Moreover, the map φ : U → φ ( U )allows us to endow U with a flat Riemannian metric φ ∗ g . In coordinates, this approachallows us to study stability of dynamical systems as we were in R n . Theorem 2.3. (Basic Lyapunov’s Theorem [12, 21, 27])
Let M ( t, x ) be a non-negative function and let ˙ M ( t, x ) stand for the total derivative in terms of time of M along a particular solution x p ( t ) of (2.1) with initial condition x p ( t ) = x and P = R n .Then, one has the following results:1. If M ( t, x ) is locally positive definite and ˙ M ( t, x ) ≤ for x locally around x e and forall t , then x e is locally stable.2. If M ( t, x ) is locally positive definite and decrescent, and ˙ M ( t, x ) ≤ locally around x e and for all t , then x e is uniformly locally stable.3. If M ( t, x ) is locally positive definite and decrescent, and − ˙ M ( t, x ) is locally positivedefinite around x e , then x e is uniformly locally asymptotically stable. Some results on symplectic geometry
Let us review some known facts on symplectic geometry while stabilising the notionsto be used hereafter and proving some non-autonomous extensions of classical resultsconcerning autonomous Hamiltonian systems. For details on the topics and standardresults provided in this section, we refer to [2, 6].A symplectic manifold is a pair (
P, ω ), where P is a manifold and ω is a closed differ-ential two-form on P that is non-degenerate , namely the mapping b ω : T P T ∗ P suchthat b ω ( v p ) := ω p ( v p , · ) ∈ T ∗ p P for every p ∈ P and every v p ∈ T p P , is a diffeomorphism.We call ω a symplectic form .From now on, ( P, ω ) will always stand for a symplectic manifold. The symplecticorthogonal of a subspace V p ⊂ T p P is defined as V ⊥ p := { w p ∈ T p P : ω p ( w p , v p ) =0 , ∀ v p ∈ V p } . Let τ : T ∗ Q → Q be the canonical projection and let h· , ·i be the pairingbetween covectors and vectors. The canonical one-form on T ∗ Q is defined to be( θ Q ) α p ( v α p ):= h α p , T α p τ ( v α p ) i , ∀ α q ∈ T ∗ q Q, ∀ q ∈ Q. On local adapted coordinates { x i , p i } i =1 ,...,n to T ∗ Q , one has θ Q := P ni =1 p i dx i . Then, ω Q := − dθ Q = P ni =1 dx i ∧ dp i is a symplectic form, the referred to as canonical symplecticform on T ∗ Q .Let X ( P ) be the Lie algebra of vector fields on P . A vector field X ∈ X ( P ) is Hamiltonian if the contraction of ω with X is an exact differential one-form, i.e. ι X ω = df for some f ∈ C ∞ ( P ). Then, f is called a Hamiltonian function of X .Since ω is non-degenerate, every f ∈ C ∞ ( P ) is the Hamiltonian function of a uniqueHamiltonian vector field X f . Then, the Cartan’s magic formula [2] yields L X f ω = ι X f dω + dι X f ω = 0, where L X f ω is the Lie derivative of ω with respect to X f . Let us define abracket {· , ·} : ( f, g ) ∈ C ∞ ( P ) × C ∞ ( P ) ω ( X f , X g ) ∈ C ∞ ( P ). This bracket is bilinear,antisymmetric, and, since dω = 0, it obeys the Jacobi identity , which makes {· , ·} into a Lie bracket . Moreover, {· , ·} obeys the Leibniz property , i.e. { f, gh } = { f, g } h + g { f, h } for all f, g, h ∈ C ∞ ( P ). Due all such properties, {· , ·} is called a Poisson bracket . Since ι [ X f ,X g ] = L X f ι X g − ι X g L X f for every f, g ∈ C ∞ ( P ) (see [2]), we get ι [ X f ,X g ] ω = L X f ι X g ω − ι X g L X f ω = L X f ι X g ω = dX f g = d { g, f } and X { g,f } = [ X f , X g ]. It is worth noting that other conventions in previous definitions,e.g. ι X f ω = − df , may lead to different expressions differing in a sign, e.g. X f,g = [ X f , X g ](see [26]).The fundamental vector field of a Lie group action Φ : G × P → P related to ξ ∈ g isthe vector field on P given by( ξ P ) p := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Φ(exp( tξ ) , p ) , ∀ p ∈ P. Our convention in the definition of fundamental vector fields gives rise to an anti-morphismof Lie algebras ξ ∈ g ξ P ∈ X ( P ) (cf. [7]). If Φ is known from context, we write gp g, p ) for every g ∈ G, p ∈ P . By the constant rank theorem, the orbits of Φare immersed submanifolds in P (see [2]). We also defineΦ g : ˜ p ∈ P g ˜ p ∈ P, Φ p : ˜ g ∈ G ˜ gp ∈ P, ∀ g ∈ G, ∀ p ∈ P. Each Φ g has an inverse Φ g − . Thus, Φ g is a diffeomorphism. The isotropy subgroup of Φat p ∈ P is G p := { g ∈ G : gp = p } ⊂ G . Let Gp stand for the orbit of p ∈ P relative toΦ, i.e. Gp := { gp : g ∈ G } . Then T ˜ p G p = { ( ξ P ) ˜ p : ξ ∈ g } for each ˜ p ∈ Gp .Recall that each g ∈ G acts as a diffeomorphism on G in the following manners: L g : h ∈ G gh ∈ G, R g : h ∈ G hg ∈ G, Ad g : h ∈ G ghg − ∈ G. We hereafter assume that G acts on g via the adjoint action , namelyAd : ( g, ξ ) ∈ G × g Ad g ( ξ ) ∈ g , (3.1)where Ad g ( ξ ) := T e ( R g − ◦ L g ) ξ . The fundamental vector field of the adjoint action relatedto ξ ∈ g is given by( ξ g ) v = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Ad exp( tξ ) ( v ) = [ ξ, v ] =: ad ξ v, ∀ v ∈ g , where [ · , · ] denotes the Lie bracket in g . While ( ξ g ) v ∈ T v g , one has that ad ξ v ∈ g .Although belong to different spaces, ( ξ g ) v ∈ T v g and ad ξ v ∈ g are assumed to be equalbecause, for every finite-dimensional vector space V , there exists a natural isomorphism v ∈ V ≃ D v ∈ T w V , at each w ∈ V , identifying each v ∈ V to the tangent vector at w associated with the derivative at w in the direction v . Let O ξ be the orbit of the adjointaction passing through ξ ∈ g . Then, T ν O ξ = { ( ξ g ) ν : ξ ∈ g } for every ν ∈ O .The group G also acts on g ∗ through the co-adjoint action Ad ∗ : ( g, µ ) ∈ G × g ∗ Ad ∗ g − µ ∈ g ∗ , where Ad ∗ g is the transpose of Ad g , i.e. h Ad ∗ g ( µ ) , ξ i = h µ, Ad g ( ξ ) i for all ξ ∈ g , and where h· , ·i denotes the duality pairing on g ∗ × g . One has that,( ξ g ∗ ) µ = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 Ad ∗ exp( − tξ ) ( µ ) = −h µ, [ ξ, · ] i = − ad ∗ ξ µ. (3.2)Given the co-adjoint orbit of µ ∈ g ∗ , i.e. S µ := { Ad ∗ g − ( µ ) : g ∈ G } , we have T ν S µ = { ( ξ g ∗ ) ν : ξ ∈ g } at every point ν ∈ S µ . Then, ξ g and ξ g ∗ are related as follows h ( ξ g ∗ ) ν , v i = h− ad ∗ ξ ν, v i = −h ν, ( ξ g ) v i , ∀ v ∈ g , ∀ ν ∈ g ∗ . A Lie group action Φ : G × P → P is Hamiltonian if its fundamental vector fieldsare Hamiltonian relative to ω . An equivariant momentum map for a Lie group actionΦ : G × P → P is a map J : P → g ∗ such that: • J ( gp ) = Ad ∗ g − ( J ( p )) , for all g ∈ G and every p ∈ P .6
2) ( ι ξ P ω ) p = d h J ( p ) , ξ i =: ( dJ ξ ) p , for all ξ ∈ g , every p ∈ P , and J ξ : P ∋ p J ( p ) , ξ i ∈ R .We obtain that 2) gives ξ P J ν = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h J (exp( tξ ) p ) , ν i = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 h Ad ∗ exp( − tξ ) J ( p ) , ν i = J [ ν,ξ ] , for all ξ, ν ∈ g and p ∈ P . Then, { J ν , J ξ } = J [ ν,ξ ] . Hence, J gives rise to a Lie algebramorphism ν ∈ g J ν ∈ C ∞ ( P ).A Lie group action Ψ : G × Q → Q induces a new Lie group action Φ : ( g, α q ) ∈ G × T ∗ Q Φ g ( α q ) ∈ T ∗ Q such that h Φ g ( α q ) , v gq i = h α q , T gq Φ g − ( v gq ) i , ∀ q ∈ Q, ∀ v gq ∈ T gq Q, the so-called cotangent lift of Ψ. This notion is ubiquitous in physics and provides easilyderivable momentum maps. Proposition 3.1.
Every Lie group action
Ψ : G × Q → Q has a cotangent lift Φ : G × T ∗ Q → T ∗ Q , admitting an equivariant momentum map J : T ∗ Q → g ∗ such that J ξ ( α q ) =: h J ( α q ) , ξ i , J ξ ( α q ) := h α q , ( ξ Q ) q i , ∀ α q ∈ T ∗ Q, ∀ ξ ∈ g . (3.3)We hereafter assume that µ ∈ g ∗ is a regular value of J . Hence, J − ( µ ) is a submanifoldof P and T p ( J − ( µ )) = ker( T p J ) for every p ∈ J − ( µ ) Proposition 3.2. If p ∈ J − ( µ ) and G µ is the isotropy group at µ ∈ g ∗ of the coadjointaction of G , then: • a) T p ( G µ p ) = T p ( Gp ) ∩ T p ( J − ( µ )) , • b) T p ( J − ( µ )) = ( T p Gp ) ⊥ ω . Let us enunciate the Marsden-Weinstein theorem ([2, p. 300]).
Theorem 3.3.
Let
Φ : G × P → P be a Hamiltonian Lie group action of G on thesymplectic manifold ( P, ω ) admitting an equivariant momentum map J : P → g ∗ . Assumethat µ ∈ g ∗ is a regular point of J and G µ acts freely and properly on J − ( µ ) . Let ι µ : J − I ( µ ) → P denote the natural embedding and let π µ : J − ( µ ) → J − ( µ ) /G µ =: P µ bethe canonical projection. There exists a unique symplectic structure ω µ on P µ such that π ∗ µ ω µ = ι ∗ µ ω . Definition 3.4. A G -invariant Hamiltonian system is a 5-tuple ( P, ω, h, Φ , J ), whereΦ is a Lie group action of G on P with an equivariant momentum map J , and h is a realfunction on R × P satisfying h ( t, Φ( g, p )) = h ( t, p ) for every g ∈ G , t ∈ R , and p ∈ P .7rom now on, ( P, ω, h, Φ , J ) will always stand for a G -invariant Hamiltonian system.Proposition 3.5 analyses the evolution of J : P → g ∗ under the dynamics of X h for a( P, ω, h, Φ , J ). In particular, let us briefly prove that J : P → g ∗ is conserved for thedynamics of X h ; i.e. the flow F of the t -dependent vector field X leaves the set J − ( µ )invariant and commutes with the action of G µ on J − ( µ ) . Our proof is just an analogueof the t -independent case that can be found in any standard reference [2]. Proposition 3.5.
Let
Φ : G × P → P be a Lie group action on a symplectic mani-fold P and let h : R × P → R be a G -invariant t -dependent Hamiltonian function, i.e. h ( t, Φ( g, p )) = h ( t, p ) for every p ∈ P , t ∈ R , and g ∈ G . Let J : P → g ∗ be an equivari-ant momentum map for Φ . Then, J is invariant relative to the evolution of h , i.e. if F is the flow of the t -dependent vector field X h : ( t, p ) ∈ R × P X ( t, p ) ∈ T P , then J ( F ( t, p )) = J ( p ) , ∀ p ∈ P, ∀ t ∈ R . Proof.
On the one hand, ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 J ξ ( F t ) = X h ( t ) J ξ = { J ξ , h ( t ) } = − X J ξ h ( t ) = − ξ P h ( t ) = 0 , ∀ ξ ∈ g , where the last equality stems from the fact that h ( t ) is invariant by assumption relativeto the fundamental vector fields of the action of G on P , namely, the vector fields ξ P .Since the J ξ are invariant relative to the dynamics of h ( t ) for every ξ ∈ g , we get that J is invariant relative to the evolution in time of the Hamiltonian system determined by h ( t ).The G -invariance property of h also yields that F induces canonically a Hamiltonianflow on the reduced phase space P µ = J − ( µ ) /G µ associated with a Hamiltonian function h µ : R × P µ → R defined in a unique way via the equation h µ ( t, ◦ π µ ( p )) = h ( t, i µ ( p ))for every p ∈ J − ( µ ), the referred to as reduced Hamiltonian . The proof of this fact is astraightforward generalisation of the t -independent proof (cf [2, 18]). Let us prove certainfacts on the geometry of the regular elements of J for ( P, ω, h, Φ , J ) that seems to beabsent in the literature. Theorem 3.6. If µ is a regular value for the momentum map J of ( P, ω, h, Φ , J ) , thenevery µ ′ belonging to the coadjoint orbit, O µ , of µ ∈ g ∗ is also a regular value. If G µ actsproperly and freely in J − ( µ ) , then G µ ′ acts also freely and properly on J − ( µ ′ ) for every µ ′ ∈ O µ .Proof. If µ is a regular point of J , then J has maximal rank on the points of J − ( µ ).The equivariance of J yields that, for any g ∈ P and p ∈ J − ( µ ), one has that J ( gp ) =Ad ∗ g − ( J ( p )). Hence, if p ∈ J − ( µ ), then gp ∈ J − (Ad ∗ g − µ ). Since Φ g is a diffeomorphism,it follows that J − (Ad ∗ g − µ ) = Φ g J − ( µ ) , ∀ g ∈ G, ∀ µ ∈ J ( P ) . T gµ J = Ad ∗ g − T µ J and J has maximal rank on J − (Ad ∗ g − µ ) for every g ∈ G and regular value µ ∈ J ( P ) of J .Note that G Ad ∗ g − µ = Ad g G µ for every g ∈ G and µ ∈ J ( P ). Let us set µ ′ := Ad ∗ g − µ .Moreover, if Φ : G µ × J − ( µ ) → J − ( µ ) is free and proper, by the equivariance of Φ, itfollows that Φ : G µ ′ × J − ( µ ′ ) → J − ( µ ′ ) is also.To prove that J − ( O µ ) is a submanifold of P , we recall that if f : M → N , S ⊂ N is asubmanifold of N , and Im T p f + T s S = T s N for every s ∈ S and p ∈ f − ( s ), we say that f is transversal to S , then f − ( S ) is a submanifold of M (see [2]). Since µ is a regular pointof J , one has that Im T p J = T J ( p ) g ∗ for every p ∈ P . Consequently, Im T p J + T s O µ = T s g ∗ for every p ∈ J − ( O µ ). Therefore, J is transversal to O µ and J − ( O µ ) is a submanifoldof P . Let us extend Poincar´e’s terminology of a relative equilibrium point for a t -independentHamiltonian function to the realm of t -dependent Hamiltonian systems on symplecticmanifolds. Definition 4.1. A relative equilibrium point for ( P, ω, H, Φ , J ) is a point z e ∈ P suchthat, for every t ∈ R , there exists a curve ξ ( t ) in g so that( X h ( t ) ) z e = ( ξ ( t ) P ) z e , ∀ t ∈ R . (4.1)The following proposition explains why z e can be called a relative equilibrium point. Proposition 4.2.
Every solution p ( t ) to ( P, ω, h, Φ , J ) passing through a relative equilib-rium point z e ∈ P projects onto the point π µ ( z e ) .Proof. By applying Proposition 3.5, every solution p ( t ) to the Hamilton equations of h ( t )is contained in a certain submanifold J − ( µ ). Moreover, the solution projects, via π µ ,onto a curve in P µ := J − ( µ ) /G µ , where G µ is the isotropy subgroup of µ relative tothe coadjoint action, that is a solution to a Hamiltonian system ( P µ , ω µ , k µ ( t )), where k µ is the only function on P µ such that k µ ( t, π µ ( p )) = h ( t, p ) for every p ∈ J − ( µ ).Indeed, the π µ ( p ( t )) is the integral curve to the t -dependent vector field Y µ on P µ givenby ( Y µ ) t := π µ ∗ ( X h ( t ) ) for every t ∈ R . If p ( t ) passes through z e and since X h ( t ) = ξ ( t ) P fora certain curve ξ ( t ) and every t ∈ R , then (( Y µ ) t ) π µ ( z e ) π µ ∗ z e ξ ( t ) P = 0 for every t ∈ R . As aconsequence, the integral curve of the t -dependent vector field Y µ passing through π µ ( z e )is just π µ ( z e ). Hence, π µ ( p ( t )) = π µ ( z e ) for every t ∈ R . Therefore, p ( t ) ∈ π − µ ( z e ) forevery t ∈ R . Then, the projection of every solution passing through z e is just a stabilitypoint of the reduced Hamiltonian system Y µ on P µ .Proposition 4.2 yields that every solution passing through a relative equilibrium pointsatisfies that p ( t ) = g ( t ) z e for a certain curve g ( t ) in G µ for µ = J ( z e ). Let us show thatthe converse is also true. 9 roposition 4.3. If every solution p ( t ) to ( P, ω, H, Φ , J ) passing through a point z e projects onto π µ ( z e ) , then z e is a relative equilibrium point.Proof. Let p ( t ) be the solution to ( P, ω, H, Φ , J ) passing through z e at t = t . By ourassumptions, π µ ( p ( t )) projects onto π µ ( z e ). Consequently, there exists a curve g ( t ) in G µ such that p ( t ) = Φ( g ( t ) , p ( t )) and g ( t ) = e . Therefore, dpdt ( t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t = t ( g ( t ) z e ) = T e exp z e (cid:18) dgdt ( t ) (cid:19) = ( ν P ) z e , for a certain ν ( t ) ∈ g µ . Since the above holds for every t ∈ R , we obtain that z e is arelative equilibrium point.Note that if p ( t ) is a solution to ( P, ω, H, Φ , J ) and p ( t ) = g ( t ) p , Proposition 3.5ensures that J ( p ( t )) = J ( p ). Hence, the action of g ( t ) leaves invariant the value of J ( p )and it belongs to G µ e for µ e = J ( p ). From previous results, we have the following corollary. Corollary 4.4.
The following two conditions are equivalent: • The point z e ∈ P is a relative equilibrium point of ( P, ω, H, Φ , J ) , • Every particular solution to ( P, ω, H, Φ , J ) passing through z e ∈ P is of the form p ( t ) = g ( t ) z e for a curve g ( t ) in G . It is remarkable that, in t -dependent systems, the Hamiltonian need not be a constantof the motion since dhdt = ∂h∂t + { h, h } = ∂h∂t . (4.2)Meanwhile, Corollary 4.4 ensures that p ( t ) = g ( t ) z e and then h ( p ( t )) = h ( t, z e ) and h is aconstant of the motion for the Hamiltonian along solutions to h passing through relativeequilibrium points. It is remarkable that the fact that h is not a constant of motion willmake the analysis of the stability of solutions of Hamiltonian systems k µ on P µ muchmore complicated, as k µ will not be in general autonomous and much of the proceduresgiven in standard stability analysis must be changed (cf. [16]).Instead of characterising relative equilibria as critical points of h ( t, z ) for every t ∈ R subject to the constraint z ∈ J − ( J ( z e )), it is easier to remove the condition δz ∈ ker[ T z e J ] ⊂ T z e P by introducing Lagrange multipliers. Let us explain this process in thefollowing proposition. Proposition 4.5. (Time-Dependent Relative Equilibrium Theorem)
A point z e ∈ P is a relative equilibrium for ( P, ω, h, Φ , J ) if and only if there exists a curve ξ ( t ) in g such that z e is a critical point of h ξ,t : P → R given by h ξ,t := h t − [ J ξ ( t ) − h µ e , ξ ( t ) i ] = h t − h J − µ e , ξ ( t ) i for every t ∈ R and µ e := J ( z e ) . roof. Assume first that z e is a point of relative equilibrium. The definition of the mo-mentum map and Corollary 4.4 yield ( X h t ) z e − ( X J ξ ( t ) ) z e = 0 for every t ∈ R . Since P issymplectic, the latter is equivalent to z e being a critical point of h t − J ξ ( t ) for every t ∈ R ,which is the same as being a critical point of h ξ,t for every t ∈ R , namely ( dh ξ,t ) z e = 0.The conditions ( dh ξ,t ) z e = 0 are in turn equivalent to ( X h ξ,t ) z e = 0 and/or z e being astationary point of X h ξ,t .Conversely, assume z e is a critical point of h ξ,t , namely z e is a stationary point of thedynamical system X h t − J ξ ( t ) for every t ∈ R . In view of Corollary 4.4, one has that z e becomes a relative equilibrium point. This section shows that the set of relative equilibrium points for a G -invariant Hamiltoniansystem ( P, ω, h, Φ , J ) is given by a sum of immersed submanifolds. Moreover, we also showthat the restriction of the original t -dependent Hamiltonian system to such immersedsubmanifolds has the structure of a foliated Lie system [7]. Proposition 5.1. If z e is a relative equilibrium point of ( P, ω, h, Φ , J ) , then Gz e is animmersed submanifold of P consisting of relative equilibrium points.Proof. Since z e is a relative equilibrium point, every solution passing through z e is of theform p ( t ) = g ( t ) z e for a certain curve g ( t ) in G . Since h ◦ Φ g = h and Φ ∗ g ω = ω for every g ∈ G , one obtains that ι X h ( t ) ω = dh ( t ) ⇒ ( ι Y ι Φ g ∗ X h ( t ) ω )( gp ) = [(Φ ∗ g ω )( X h ( t ) , Φ g − ∗ Y )]( p )= ω ( X h ( t ) , Φ g − ∗ Y )( p ) = h dh ( t ) , Φ g − ∗ Y i ( p ) = h d Φ ∗ g − h, Y i ( gp ) = h dh ( t ) , Y i ( gp ) , for every g ∈ G and p ∈ P . Therefore, Φ g ∗ X h ( t ) = X h ( t ) . Hence, every solution z ′ ( t )passing through gz e is such that g − z ′ ( t ) is a solution p ( t ) to X h ( t ) passing through z e .Thus, p ′ ( t ) = g ( p ( t )) = gg ( t ) z = gg ( t ) g − gz . In other words, gz e is a relative equilibriumpoint for ( P, ω, h, Φ , J ). Since Gz e is an immersed submanifold of P , the result follows.A foliated Lie system on a manifold N is a first-order system of differential equationstaking the form dxdt = X ( t, x ) , so that X ( t, x ) = r X α =1 g α ( t, x ) X α ( x ) , ∀ t ∈ R , ∀ x ∈ P, where X , . . . , X r span an r -dimensional Lie algebra of vector fields, i.e.[ X α , X β ] = c γαβ X γ , α, β, γ = 1 , . . . , r, c γαβ , and the functions g α,t : x ∈ M g α ( t, x ) ∈ R , for every t ∈ R and α = 1 , . . . , r , are first integrals of X , . . . , X r .Let us show how foliated Lie systems appear while studying the relative equilibriumpoints for G -invariant Hamiltonian systems. Theorem 5.2.
Let z e be a relative equilibrium point for ( P, ω, h, Φ , J ) and let µ e := J ( z e ) .Then, X h ( t ) can be restricted to J − ( O µ e ) and it becomes on it a foliated Lie system witha Vessiot–Guldberg Lie algebra V := { ξ P | J − ( O µe ) : ξ ∈ g µ e } . Proof.
Proposition 5.1 ensures that every z e ∈ J − ( O µ e ) is a relative equilibrium point.Then, every integral curve to X h ( t ) passing through it takes the form p ( t ) = g ( t ) z e for acertain curve g ( t ) in G . Proposition 3.5 yields that J is constant on integral curves of X h ( t ) . Consequently, the integral curves of X h ( t ) are contained in J − ( µ e ) ⊂ J − ( O µ e ).Hence, 0 = ddt J ( p ( t )) = ddt J ( g ( t ) z e ) = ddt Ad ∗ g − ( t ) J ( z e ) = − [ ξ ( t ) g ∗ ] µ e . Therefore, ξ ( t ) ∈ g µ e and the theorem follows. Theorem 4.5 characterises the relative equilibria as the extremum of the Hamiltoniansubject to the constraint of the constant momentum map. Then, h ξ,t := h t − h J − µ e , ξ ( t ) i is to be optimised and ξ ( t ) ∈ g is a Lagrange multiplier depending on time.The study of the stability of equilibrium points in J − ( µ e ) for non-autonomous Hamil-tonian systems requires the use of t -dependent Lyapunov analysis. This is much morecomplicated than the autonomous case, which relies on searching a minimiun for theHamiltonian of the system (see [16]). To tackle the non-autonomous problem, we will useTheorem 2.3 and a more general approach, which easily retrieves the standard stabilitytheory for time-independent Hamiltonian systems.Let z e be a relative equilibrium point of ( P, ω, h, J , Φ). Let us analyse the function h z e : R × P → R given by h z e ( t, z ) := h ( t, z ) − h ( t, z e ) , ∀ ( t, z ) ∈ R × P. Then, h z e ( t, z e ) = 0 for every t ∈ R . If p ( t ) is the particular solutions to our G -invariantHamiltonian system ( P, ω, h, J , Φ) with initial condition z , then ddt h z e ( t, p ( t )) := ddt h ( t, p ( t )) − ddt h ( t, z e ) . Recall that the time derivative of a Hamiltonian function h along the solutions of itsHamilton equations is given by dhdt = ∂h∂t + { h ( t ) , h ( t ) } = ∂h∂t . ddt h z e ( t, p ( t )) := ∂h∂t ( t, p ( t )) − ∂h∂t ( t, z e ) = ∂h z e ∂t ( t, p ( t )) . Note that h z e ( t, gz ) = h z e ( t, z ) for every g ∈ G and ( t, z ) ∈ R × P , i.e. h z e ( t, z ) is G -invariant. Then, we can define a function H z e : R × P µ e → R of the form H z e ( t, [ z ]) := h z e ( t, z ) , ∀ z ∈ J − ( z e ) , where we recall that [ z ] is the equivalence class of z ∈ J − ( µ e ) in J − ( µ e ) /G µ e . Moreover, ddt H z e ( t, [ z ]) = ∂h z e ∂t ( t, z ) . Let us use H z e to study the stability of [ z e ] in P µ e . In particular, we will study theconditions on h to ensure that H z e gives rise to stability a stability point in [ z e ]. Withthis aim, consider a coordinate system { x , . . . , x n } on an open neighbourhood U of [ z e ]in J − ( µ e ) /G µ e such that x i ([ z e ]) = 0 for i = 1 , . . . , n . Let α = ( α , . . . , α n ), with α , . . . , α n ∈ N ∪ { } , be a multi-index with n := dim J − ( µ e ) /G µ e . Let | α | := P ni =1 α i and D α := ∂ α x · · · ∂ α n x n . Lemma 6.1.
Let us define the t -parametric family of matrices, M ( t ) , with entries [ M ( t )] ij := 12 ∂ H z e ∂x i ∂x j ( t, [ z e ]) , ∀ t ∈ R , (6.1) and let spec( M ( t )) stand for the spectrum of the matrix M ( t ) at t . Assume that thereexists a constant λ > such that λ < inf t ∈ R min spec( M ( t )) . Assume also that there existsa real constant c such that c ≥ max | α | =3 max [ y ] ∈O | D α H z e ( t, [ y ]) | for a certain open neighbourhood O of [ z e ] . Then, there exists an open neighbourhood U of [ z e ] where the function H z e : ¯ R × U → R is locally positive definite. If additionallythere exists a constant Λ such that sup t ∈ R max spec( M ( t )) ≤ Λ , then H z e : ¯ R × U → R isa decreasing function.Proof. Since z e is a point of relative equilibrium of ( M, ω, h, J , φ ) , then H z e ( t, · ) has acritic point at [ z e ] for every t ∈ R . By the Taylor expansion of H z e ( t, · ) around [ z e ] andthe fact that z e is a relative equilibrium point of each H z e ( t, · ), one has H z e ( t, [ z ]) = 12 ∂ H z e ∂x i ∂x j ( t, [ z e ]) x i x j + R t ([ z ]) , [ z ] ∈ U. Since M ( t ) is symmetric, it can diagonalised via an orthogonal transformation O t foreach t ∈ R . Let λ ( t ) , . . . , λ n ( t ) be the (possibly repeated) eigenvalues of M ( t ) andlet w = ( w , . . . , w n ) T be the coordinated vector corresponding to z = ( x , . . . , x n ) T
13n the diagonalising basis induced by O t . Then, z T M ( t ) z = w T D ( t ) w , where D ( t ) =diag( λ ( t ) , . . . , λ n ( t )). Thus, w T D ( t ) w = P ni =1 λ i ( t ) w i . Then,12 ∂ H z e ∂x i ∂x j ( t, [ z e ]) x i x j = z T M ( t ) z = w T D ( t ) w ≥ λ ( t ) k w k , where λ ( t ) := min i =1 ,...,n λ i ( t ). By our assumption on the existence of λ > O t is orthogonal, one gets that12 ∂ H z e ∂x i ∂x j ( t, [ z e ]) x i x j ≥ λ ( t ) k z k ≥ λ k z k . Recall that the third-order Taylor remainder R t around [ z e ] can be written as R t ([ z ]) = X | β | =3 B β ( t, [ z ]) z β , z β := x β · . . . · x β n n , on points [ z ] of the open coordinate subset U for certain functions B β : R × U → R . The B β are known to be bounded by | B β ( t, [ z ]) | ≤
13! max | α | =3 max y ∈O | D α H z e ( t, [ y ]) | , ∀ [ z ] ∈ O on any open subset O of [ z e ]. By our assumptions, there exists a constant c > c ≥
13! max | α | =3 max y ∈O | D α H z e ( t, [ y ]) | , ∀ t ∈ R , for a particular open neighbourhood O of [ z e ]. Let us prove that12 ∂ H z e ∂x i ∂x j ( t, [ z e ]) x i x j + R t ([ z ]) − λ k z k is bigger or equal to zero for every t ∈ R and every [ z ] ∈ U ∋ [ z e ] for a certain openneighbourhood U of [ z e ]. By our general assumptions, λ < inf t ∈ R λ ( t ). Note that λ i ( t ) − λ ≥ λ ( t ) − λ and λ ( t ) − λ is larger than a certain properly chosen λ ′ > t ∈ R .Then,12 ∂ H z e ∂x i ∂x j ( t, [ z e ]) x i x j − λ k z k = w T diag( λ ( t ) − λ, . . . , λ n ( t ) − λ ) w ≥ λ ′ k w k = λ ′ k z k . Then, the first bracket in the following expression (cid:18) ∂ H z e ∂x i ∂x j ( t, [ z e ]) x i x j − λ k z k − λ ′ k z k (cid:19) + ( λ ′ k z k + R t ([ z ])) . is larger or equal to zero. Let us prove the same for the second bracket. Note that | R t ([ z ]) | ≤ X | β | =3 | B β ( t, [ z ]) | x | β · . . . · | x n | β n ≤ c X | β | =3 | x | β · . . . · | x n | β n . λ ′ k z k − c X | β | =3 λ β x β , where the { λ β } is any set of constants such that λ β ∈ {± } for every multi-index β with | β | = 3, admits a minimum at [ z e ] as follows from standard differential calculus arguments.As a consequence, the above function is bigger or equal to zero on a neighbourhood U { λ β } .Considering the intersection of all the possible open subsets U { λ β } for every set of constants λ β , we obtain an open neighbourhood U of [ z e ]. Assume that [ z ] is such that0 > λ ′ k z k − c X | β | =3 | x | β Then, 0 > λ ′ k z k − c X | β | =3 sgn n Y i =1 x β i i ! x β , where sgn( a ) is the sign of the constant a . Then, [ z ] cannot belong to U . In other words, λ ′ k z k − c X | β | =3 | x | β ≥ U . Since | R t ([ z ]) | ≤ c P | β | =3 | x | β , then λ ′ k z k + R t ([ z ]) ≥ z ] ∈ U . Finally, one gets that H z e ( t, [ z ]) ≥ λ k z k , ∀ [ z ] ∈ U , ∀ t ∈ R . Hence, the restriction H z e : ¯ R × U → U of H z e to ¯ R × U is a locally partially definitefunction.Now, the orthogonal change of variables O t allows us to write12 ∂ H z e ∂x i ∂x j ( t, [ z e ]) x i x j = z T M h ( t ) z = w T D ( t ) w ≤ Λ( t ) k w k = Λ( t ) k z k , for Λ( t ) := max i =1 ,...,n λ i ( t ). By assumption, Λ ≥ Λ( t ). Hence,12 ∂ H z e ∂x i ∂x j ( t, [ z e ]) x i x j ≤ Λ k z k . Recall the expression (6.2) for every t ∈ ¯ R and [ z ] ∈ U . Then, one has that H z e ( t, [ z ]) ≤ Λ k z k + λ ′ k z k and H z e is decreasing on ¯ R × U . 15y using the above lemma, we obtain the following immediate theorem. Theorem 6.2.
Let assume that there exist λ, c > and an open U of [ z e ] so that λ < min(spec( M ( t ))) , c ≥
13! max | α | =3 max [ y ] ∈ U | D α H z e ( t, [ y ]) | , ∂H∂t (cid:12)(cid:12)(cid:12)(cid:12) U ≤ , for every t ≥ t and a certain t , then [ z e ] is a stability point of the Hamiltonian system k µ on J − ( µ e ) /G µ e . If there exists Λ such that max(spec( M ( t ))) < Λ , then k µ is uniformlylocally stable.Proof. Consider a coordinated open neighbourhood of [ z e ] (we assume without loss of gen-erality than it is the U of the enunciate of our theorem) with local coordinates { x , . . . , x n } .The coordinates allow us to identify U with an open subset of the Euclidean space R n and to apply Lyapunov stability theory on it. By Lemma 6.1 and our given assumptions, H z e ( t, [ z ]) is a locally positive definite function. By Theorem 2.3 and ∂H z e /∂t ≤
0, weobtain that [ z e ] is locally stable. If additionally Λ exists, then again Theorem 2.3 showsthat [ z e ] is uniformly locally stable.The main idea of the momentum-energy method is to determine properties of h on aneighbourhood of z e in J − ( µ e ) to ensure that the conditions needed for a certain typeof stability at the stability points of k µ on J − ( µ e ) /G µ e . In particular, we want to giveconditions on the matrix m ( t ) of second derivatives of h µ e : ( t, z ) ∈ R × J − ( µ e ) h ( t, z ) ∈ R and ∂h µ e /∂t to ensure that the spectrum of the matrix (6.1) is bounded frombelow or from above for every t ∈ R . Instead of checking M ( t ), which can be complicatedas it is defined on a submanifold, we will search for conditions on m ( t ), which is morepractical. The following ideas are pretty much similar to the t -independent formulationof the energy-momentum method. Proposition 6.3.
Let z e ∈ P be a relative equilibrium point for ( P, ω, h, Φ , J ) . Then, ( δ h ξ,t ) z e (( η P ) z e , v z e ) = 0 , ∀ η ∈ g , ∀ v z e ∈ T z e J − ( µ e ) , ∀ t ∈ R (6.3) Proof.
The G -invariance of h : R × P → R and the equivariance condition for J yields h ξ,t ( gp ) = h ( t, gp ) − h J ( gp ) , ξ i + h µ e , ξ i = h ( t, p ) − h Ad ∗ g − ( J ( p )) , ξ i + h µ e , ξ i and h ξ,t ( gp ) = h ( t, p ) − h J ( p ) , Ad g − ( ξ ) i + h µ e , ξ i , for any g ∈ G and p ∈ P . Substituting g := exp( sη ),with η ∈ g , and differentiating with respect to the parameter s , one obtains( ι η P dh ξ,t )( p ) = − (cid:28) J ( p ) , dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 Ad exp( − sη ) ( ξ ) (cid:29) = h J ( p ) , [ η, ξ ] i . Taking variations relative to p ∈ P above, evaluating at z e , since ( dh ξ,t ) z e = 0 because z e is a critical point, ( δ h ξ,t ) z e (( η P ) z e , v p ) = h T z e J ( v p ) , [ η, ξ ] i , which vanishes if T z e J ( v p ) = 0,i.e. if v p ∈ ker[ T z e J ] = T z e J − ( µ e ).Propositions 6.3 and 3.1 yield the following.16 orollary 6.4. ( δ h ξ,t ) z e vanishes identically on T z e ( G µ e z e ) .Proof. Proposition 3.1 shows that T z e ( G µ e z e ) = T z e ( Gz e ) ∩ ker[ T z e J ]. Since T z e ( G µ e z e ) ⊂ T z e ( Gz e ), the result follows from (6.3) by taking v z e := ( ξ P ) z e , with ξ ∈ g µ e .By Corollary 6.4, there exists a t -parametric family of bilinear mappings b g t, [ z e ] : T [ z e ] J − ( µ e ) × T [ z e ] J − ( µ e ) → R of the form b g t, [ z e ] ([ v ] , [ v ′ ]) = ( δ h ξ,t )( v, v ′ ) , ∀ v, v ′ ∈ T z e J − ( µ e )for [ v ] , [ v ′ ] being the equivalence classes of elements v, v ′ in T z e J − ( µ e ) /T z e ( G µ e z e ). Notethat the spectrum of M ( t ) is given by the eigenvalues of b g t, [ z e ] .Recall that we assume that G µ acts freely and properly on J − ( µ e ). Consider a setof coordinates { y , . . . , y s } on an open A ⊂ J − ( µ ) containing z e . Note that due to the G µ e -invariance of h µ e , one has that there exists c such that c ≥
13! max | ϑ | =3 max y ∈A | D ϑ h µ e ( t, y ) | where ϑ is a multi-index ϑ = ( ϑ , . . . , ϑ s ) if and only if c ≥
13! max | α | =3 max y ∈O | D α H z e ( t, y ) | where O is an open neighbourhood of [ z e ].Therefore, we obtain the following theorem. Theorem 6.5.
Let assume that there exist λ, c > and an open U of [ z e ] so that λ < min(spec( b g t, [ z e ] )) , c ≥
13! max | ϑ | =3 max y ∈A | D ϑ h µ e ( t, y ) | , ∂h∂t (cid:12)(cid:12)(cid:12)(cid:12) A ≤ , (6.4) for every t ≥ t for a certain t , then [ z e ] is a stability point of the Hamiltonian system k µ on J − ( µ e ) /G µ e for t ≥ t . If there exists Λ such that max(spec( M ( t ))) < Λ , then [ z e ] is uniformly locally stable. Recall that in the case of an autonomous Hamiltonian, the third condition in (6.4) isimmediately satisfied. Moreover, assuming h to be smooth enough, there always existsthe required c for a certain open neighbourhood A of z e . Finally, the condition on λ boilsdown to the standard condition on the positiveness of the eigenvalues of b g t, [ z e ] (cf. [16]). Let us illustrate our t -dependent energy-momentum method via a generalisation of thestandard example of the freely spinning rigid body [16]. Let SO be the Lie group of allorthogonal unimodular linear automorphisms on the Euclidean space R . The Lie algebra17f SO , let us say so , consists of all the 3 × R via the isomorphism φ : R → so , Ω b Ω := − Ω Ω Ω − Ω − Ω Ω , (7.1) where Ω := (Ω , Ω , Ω ) T . Let × be the vector product in R . Then, b Ω r = Ω × r , [ b Ω , b Θ] = \ Ω × Θ, and Λ b ΘΛ T = c ΛΘ for every Λ ∈ SO , and every Θ , Ω ∈ R . Hence, φ is a Liealgebra isomorphism between R (which is a Lie algebra relative to the vector product)and so with the commutator of matrices.The adjoint action Ad : SO × so → so defined geometrically in (3.1) reduces to theexpression Ad Λ b Θ = Λ b ΘΛ T , as Λ − =Λ T , for all Λ ∈ SO and Θ ∈ R . Moreover, \ Λ( r × s ) = Λ d r × s Λ T = Λ[ b r , b s ]Λ T = [Λ b r Λ T , Λ b s Λ T ] = [ c Λ r , c Λ s ] = \ Λ r × Λ s , ∀ r , s ∈ R . One can identify T Λ SO with so via two isomorphisms. Recall that L Λ : Θ ∈ SO ΛΘ ∈ SO is a diffeomorphism for every Λ ∈ SO . Then, T Id L Λ : T Id SO ≃ so T Λ SO , where Id is the 3 × b Θ Λ :=( T Id L Λ ) b Θ =: (Λ , Λ b Θ), for every Θ ∈ R . Then, b Θ Λ is called the left invariant extension of b Θ. Meanwhile, we set b θ Λ := ( T Id R Λ ) b θ := (Λ , b θ Λ), for every θ ∈ R . It is said that b θ Λ is the right invariant extension of b θ . We omit the base point, if it is known from context.We write Λ b Θ and b ΘΛ for b Θ Λ and b θ Λ , respectively.Since so is a simple Lie algebra, its Killing metric, κ , is non-degenerate, which givesan isomorphism b Θ ∈ so κ ( b Θ , · ) ∈ so ∗ . (7.2)In paricular, κ reads, up to a non-zero optional proportional constant, as κ ( b Θ , b Ω) = tr( b Θ T b Ω), for all Θ , Ω ∈ R . Moreover, Π · Υ = κ ( b Π , b Υ), for all Π , Υ ∈ R . This extendsto h b Π Λ , b Θ Λ i := 12 tr( b Π T Λ b Θ Λ ) = 12 tr( b Π T b Θ) = Π · Θ , ∀ Θ , Π ∈ R . We will denote the elements of so ∗ by b Π, where Π ∈ R , (or b π with π ∈ R ) andelements of T ∗ Λ SO by b π Λ = (Λ , b π Λ) and b Π Λ = (Λ , Λ b Π) . If b π Λ = b Π Λ , then b π = Λ b ΠΛ T ,which matches the coadjoint action. Indeed, h Ad ∗ Λ T b Π , ·i = 12 Tr( b Π T Ad Λ T ( · )) = 12 Tr( b Π T Λ T ( · )Λ)= 12 Tr(Λ b Π T Λ T ( · )) = 12 Tr((Λ b ΠΛ T ) T ( · )) = h b π, ·i . Using (7.1), we get π = ΛΠ. The mechanical framework to be hereafter studied goesas follows: the configuration manifold is SO , whilst T ∗ SO is endowed with its canonicalsymplectic structure. It is remarkable that our framework retrieves the dynamics of a18olid rigid under no exterior forces as a particular case. Moreover, we have the followingelements:i) A t -dependent Hamiltonian h : R × T ∗ SO → R of the form h t := 12 π · I − t π, I t := Λ J t Λ T . (7.3)where I t is the time-dependent inertia tensor (in spatial coordinates) and J t is the inertiadyadic given by J t = R R ̺ ν ( t, X )[ k X k − X ⊗ X ] d X. Here, ̺ ν : R × B → R is thereference density. Note that J t can be understood as a matrix depending only on time.We understand h in (7.3) as a function h : R × SO × so ∗ → R , with so ∗ ≃ R ∗ . Thisamounts to regarding h as a function on R × T ∗ SO via the isomorphism(Λ , π ) ∈ SO × R ∗ (Λ , b π Λ) =: b π Λ ∈ T ∗ Λ SO . (7.4)However, h ( t, Λ , π ) is more appropriate for calculations. Note that h is the kinetic energyof the mechanical system, which we call a quasi-rigid body (cf. [16]).ii) Invariance properties - Since b π = Λ b ΠΛ T , the t -dependent Hamiltonian (7.3) be-comes h t = 14 tr( b π T Λ J − t Λ T b π ) = 14 tr((Λ T b π ) T J − t Λ T b π ) =14 tr(( b ΠΛ T ) T J − t b ΠΛ T ) = 14 tr( b Π T J − t b Π) = 12 Π · J − t Π , (7.5)which illustrates the left invariance of h relative to the action of SO . Thus, the leftreduction by SO induces a function on the quotient R × T ∗ SO /SO ≃ R × so ∗ .As a consequence, h t is only a quadratic function on the momenta b π . Consequently,the second condition in (6.4) is immediatelly satisfied sinceiii) Momentum map - We consider G = SO to act on Q = SO by left translations,i.e. Ψ : ( A, Λ) ∈ G × Q L A Λ := A Λ ∈ Q . Hence, the cotangent lift of Ψ, let us say b Ψ,is by left translations. In particular b Ψ(Λ ′ , (Λ , b π Λ)) = (Λ ′ Λ , d Λ ′ π Λ) , ∀ Λ ′ , Λ ∈ SO , ∀ π ∈ ( R ) ∗ . Since ( b ξ so ) Λ = ddt (cid:12)(cid:12) t =0 exp( t b ξ )Λ = b ξ Λ, for every ξ ∈ so , Proposition 3.1 yields that J b ξ ( b π Λ ) = 12 tr[ b π T Λ ξ so ] = 12 tr[Λ T b π T b ξ Λ] = 12 tr[ b π T b ξ ] = π · ξ. (7.6)Thus, J ( b π Λ ) = b π , J b ξ ( b π Λ ) = π · ξ . Then, every π ∈ so ∗ is a regular value of J . Moreover, G π is given by the elements of SO that leave invariant π . Hence, G π ≃ SO for π = 0and G = SO . Moreover J − ( π ) = SO for every π ∈ R . Since each G π are alwayscompact, it acts properly on J − ( π ). Moreover, the action of G π on J − ( π ) is always free.Hence, J − ( π ) /G π is always a well-defined two-dimensional manifold for π = 0, a sphereindeed, and a zero-dimensional manifold for π = 0.19et us study h ξ,t = h t − [ J ξ − π e · ξ ] = 12 π · I − t π − ξ · ( π − π e ) , and look into its critical points. To derive the first variation, it is appropriate to consider h ξ,t as a function of (Λ , π ) ∈ SO × R ∗ . If b π Λ e := (Λ e , b π e Λ e ) ∈ T ∗ SO is a relativeequilibrium point, then, for any δθ ∈ R , we can build the curve ǫ Λ ǫ := exp[ ǫ b δθ ]Λ e in SO . Let δπ ∈ R ∗ and consider the curve in R ∗ defined as ǫ π ǫ := π e + ǫδπ ∈ R ∗ . These constructions induce a curve ǫ b π Λ ǫ ∈ T ∗ SO through the isomorphism (7.4),that is b π Λ ǫ := (Λ ǫ , b π ǫ Λ ǫ ). Let us compute the first variation.i) First variation - By using the chain rule, we can establish0 = δh ξ,t (cid:12)(cid:12) e := ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 h ξ,t,ǫ = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 (cid:18) π ǫ · I − t,ǫ π ǫ − ξ · ( π ǫ − π e ) (cid:19) , (7.7)where I − t,ǫ := Λ ǫ J − t Λ Tǫ . At equilibrium, ( π − π e ) · η = 0 for all η ∈ R , from varying theLagrange multiplier. Recall that π e · ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 I − t,ǫ π e = 12 π e · [ b δθ I − t,e − I − t,e b δθ ] π e =12 [ π e · δθ × I − t,e π e − I − t,e π e · δθ × π e ] = δθ × ( I − I t t,e π e × π e ) , (7.8) by using elementary vector product identities. By (7.8), expression (7.7) reduces to δh ξ,t (cid:12)(cid:12) e = δπ · [ I − t,e π e − ξ ] + δθ · [ I − t,e π e × π e ] = 0 . (7.9)Thus, ξ × π e = 0 , I − t,e ξ = λ t ξ, (7.10)where λ t > I t,e = Λ e J t Λ Te . These conditions yieldthat π e lays along a principal axis, and that the rotation is around this axis. Moreover, π e = I t,e ω e and π e = I t,e ξ .ii) Second variation - By (7.9), we obtain at equilibrium( δ h ξ,t ) (cid:12)(cid:12) e := ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 [ δπ · ( I − t,ǫ π ǫ − ξ ) + δθ ( I − t.ǫ π ǫ × π ǫ )] . Proceeding as to obtain (7.9) and using (7.10), we obtain each( δ h ξ,t ) (cid:12)(cid:12) e = [ δπ T δθ T ] (cid:20) I − t,e ( I − t,e − λ t b π e − b π e ( I − t,e − λ t − b π e ( I − t,e − λ t b π e (cid:21) (cid:20) δπδθ (cid:21) . Let us restrict the ( δπ, δθ ) ∈ R ∗ × R . We already know that J ( b π Λ ) = b π . Hence, µ e = b π e and T z e ( G µ e z e ) are the generators of infinitesimal rotations around the axis π e .Moreover, we can choose S = { ( δπ, δθ ) : δπ = 0 , δθ ⊥ π e } . Since δθ is a variation that20nfinitesimally rotates π e on the sphere O π e := { π ∈ R : k π k = k π e k } , which is theco-adjoint orbit passing through π e . Checking the positive definiteness of δ h ξ,µ reducesto checking δ h ξ,µ (cid:12)(cid:12) e = δθ · ( b π Te ( I − t,e − λ b π e ) δθ = ( π e × δθ ) · ( I − t,e − λ π e × δθ ) . If λ t is the largest or smallest eigenvalue of I t , then δ h ξ,µ | e will be definite as the nullspace of I − t − λ
11 consists of vectors parallel to π e , which have been excluded. Moreover, S is a 2-dimensional space and δ h ξ,µ | e represents a 2 × ∂h/∂t < δ h are bounded as in our previous assumptions,one can ensure the stability of the projection of relative equilibrium points of J − ( π ) to J − ( π ) /G π . This work has extended the formalism for the energy-momentum method on symplecticmanifolds to the t -dependent realm. This has required the use of more complicate tech-niques to study the stability of t -dependent problems. A simple example concerning amodification of a rotating quasi-solid rigid has been used to illustrate our techniques.Note that the energy-momentum method has extensions to concern problems on Pois-son manifolds [16]. Our techniques should be easily extended to such a new realm. Weplan to study the topic in the future. We additionally search for new applications of ourtechniques in physics. In particular, we are interested in the study of foliated Lie systemsappearing in the study of relative equilibrium points of mechanical systems. J. de Lucas acknowledges funding from the research project HARMONIA (grant number:2016/22/M/ST1/00542) financed by the Polish National Science Center (POLAND).
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