Lagrangian mechanics on centered semi-direct product
aa r X i v : . [ m a t h - ph ] O c t LAGRANGIAN MECHANICS ON CENTERED SEMI-DIRECTPRODUCTS
LEONARDO COLOMBO & HENRY O. JACOBS
Abstract.
There exists two types of semi-direct products between a Lie group G and a vector space V . The left semi-direct product, G ⋉ V , can be con-structed when G is equipped with a left action on V . Similarly, the rightsemi-direct product, G ⋊ V , can be constructed when G is equipped with aright action on V . In this paper, we will construct a new type of semi-directproduct, G ✶ V , which can be seen as the ‘sum’ of a right and left semi-direct product. We then parallel existing semi-direct product Euler-Poincar´etheory. We find that the group multiplication, the Lie bracket, and the dia-mond operator can each be seen as a sum of the associated concepts in rightand left semi-direct product theory. Finally, we conclude with a toy exampleand the group of 2-jets of diffeomorphisms above a fixed point. This finalexample has potential use in the creation of particle methods for problems ondiffeomorphism groups. Introduction
It is no secret that the use of symmetry and a preference for algebraic simplicitypervaded much (if not all) of Jerry’s intellectual endeavours. Certainly one of thesealgebraic structures would be semi-direct products, which pepper his research inthe form of rigid bodies, complex fluids, plasmas [13, 15], the KdV equation [14],and the heavy top [9].In this paper we will investigate a new semi-direct product which is inspired bya careful analysis of the second order jet groupoid. To begin, let G be a Lie groupand V be a vector space on which G acts by a left action. Given these ingredients,we may form the Lie group G ⋉ V , which is isomorphic to G × V as a set, butequipped with the composition( g, v ) · ⋉ ( h, w ) = ( g · h, g · w + v ) , ∀ ( g, v ) , ( h, w ) ∈ G ⋉ V. A standard example of a system which evolves on a left semi-direct product is theheavy top, where G = SO(3) and V = R . In contrast, if G acts on V by a rightaction, we may form the right semi-direct product G ⋊ V defined by the composition( g, v ) · ⋊ ( h, w ) = ( g · h, w + v · h ) . A standard example of a system whos configurations describe a right semi-directproduct is a fluid with a vector-valued advected parameter [9]. In any case, it seemsnatural to surmise that the composition law( g, v ) · ✶ ( h, w ) = ( g · h, g · w + v · h )(1)yields a new type of semi-direct product. The first result of this article is that (1)is a valid composition law in some circumstances, and we call the correspondingLie group a centered semi-direct product .The second result is that the second order Taylor expansions (or second orderjets) of diffeomorphisms over a fixed point form a centered semi-direct product.The main motivation behind understanding this example is to allow us to develop Date : 25 June 2013. particle-based methods for complex fluid simulation and image registration algo-rithms.1.1.
Background.
The semi-direct product is a standard tool used in the construc-tion of new Lie groups and plays an interesting role in geometric mechanics whenthe normal subgroup is interpreted as an advected parameter. A standard exampleis the modeling of the ‘heavy-top’, wherein the the axis of rotation is described by R and is advected by the action of SO(3). In other words, the configuration spacefor the heavy top can be described as the left semi-direct product SO(3) ⋉ R [9].Another standard example is the modeling of liquid crystals, in which we considerthe right semi-direct product SDiff( M ) ⋊ V . In this case, SDiff( M ) is the set ofvolume-preserving diffeomorphisms of a volume manifold M , and V is a vectorspace of maps from M into some Lie algebra and SDiff( M ) acts on V by pullback[7, 5]. Of course, the tangent bundle of a Lie group, T G , is isomorphic to a leftsemi-direct product G ⋉ g by left-trivializing the group structure of T G . Addi-tionally,
T G is isomorphic to a right semi-direct product G ⋊ g when the groupstructure of T G is right trivialized [1, section 5.3]. Thus, we see that this methodof constructing groups can be found in a number of instances. In this article, weintroduce a new type of semi-direct product which extends the existing semi-directproduct theory.A motivating example will be a desire to understand the second order jet-groupoid of a manifold M [11, section 12]. As will be illustrated in section 1.4,an isotropy group of the second order jet groupoid exhibits a group structure whichcan be written as a centered semi-direct product. A thorough understanding of thejet groupoid can be useful for the creation of new particle-based methods whereinthe particles carry jet data in addition to position and velocity data. One advantageof such a particle method is the possibility for a discrete form of Kelvin’s circulationtheorem [10]. Building such particle methods can be useful in scenarios in whichone desires to work with the material representation of a fluid. For example, thefree energy of liquid crystal is a function of the gradient of a director field advectedby the fluid. Computing this advection requires the use of second order jet dataand therefore a small portion of the material representation of the fluid is invoked[7, 5]. Additionally, the use of jet data can be useful in the realm of image registra-tion algorithms in the field of medical imaging. In particular, it is common to usethe material representation of the EPDiff equations to implement the Large Defor-mation Diffeomorphic Metric Mapping (LDDMM) framework [2, 4]. In particular,“Landmark LDDMM” discretizes the EPDiff equation using particle methods [16].A version of Landmark LDDMM wherein the particles can carry higher order jetdata is described in [17]. Thus, keeping track of jet data may play a significantrole in the construction of particle-based integrators for fluid modeling and medicalimaging algorithms.1.2. Main Contributions.
In this paper, we accomplish a sequence of goals, eachbuilding upon the previous. In particular:(1) In section 2, we define a new type of semi-direct product that we dub a centered semi-direct product .(2) In proposition 2.2, we derive the Lie algebra of a centered semi-direct prod-uct and its associated structures.(3) In section 3, we develop the Euler-Poincar´e theory of centered semi-directproducts in parallel with the existing theory of semi-direct product reduc-tion [9].
AGRANGIAN MECHANICS ON CENTERED SEMI-DIRECT PRODUCTS 3 (4) In section 4, we describe the centered semi-direct product Euler-Poincar´eequations for a few examples. We present one toy example before presentingthe theory for an isotropy group of the second order jet groupoid.Combined, these items allow for a computationally tractable algebraic under-standing of second order jets and perhaps open the door to applications which werepreviously overlooked by geometric mechanicians.1.3.
Acknowledgements.
We would like to thank Darryl D. Holm for provid-ing the initial stimulus for this project. The work of L.C has been supportedby MICINN (Spain) Grant MTM2010-21186-C02-01, MTM 2011-15725-E, ICMATSevero Ochoa Project SEV-2011-0087 and IRSES-project ”Geomech-246981”. L.Cowes additional thanks to CSIC and the JAE program for a JAE-Pre grant. Thework of H.O.J. was supported by European Research Council Advanced Grant267382 FCCA.1.4.
A motivating example.
Let Diff( M ) denote the diffeomorphisms group ofa manifold M . For a fixed x ∈ M we may define the isotropy subgroupIso( x ) = { ϕ ∈ Diff( M ) | ϕ ( x ) = x } . Let ϕ ∈ Iso( x ) and note that T x ϕ is a linear automorphism of the vector-space T x M . In particular: Proposition 1.1.
The functor “ T x ” is a group homomorphism from Iso( x ) to GL( T x M ) .Proof. Clearly Iso( x ) and GL( T x M ) are both Lie groups. Let ϕ, ψ ∈ Iso( x ). Then T x ϕ ◦ T x ψ = T x ( ϕ ◦ ψ ). (cid:3) This observation has implications for computation for the following reason: Bydefinition, T x ϕ approximates ϕ in a neighborhood of x ∈ M . Thus, if one desiredto model a continuum with activity at x , then T x ϕ carries some of the crucial datato do this task. In particular, this is computationally tractable as the dimension ofGL( T x M ) is equal to (dim M ) .However, the group GL( n ) only captures the linearization of a diffeomorphism.If we desire to capture some of the nonlinearity then we might consider lookinginto the second jet of these diffeomorphisms. We can do so by considering thefunctor T T x . Let ϕ ∈ Iso( x ) so that T T x ϕ is a map from T ( T x M ) to T ( T x M ).However, T x M is a vector-space so that T ( T x M ) ≈ T x M × T x M . The secondcomponent represents the vertical component and the isomorphism between T T x M and T x M × T x M is given by the vertical lift v ↑ ( v , v ) = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 ( v + ǫv ) . We can therefore represent
T T x ϕ as ( T x ϕ, A ϕ ) where A ϕ : T x M × T x M → T x M isthe symmetric (1 ,
2) tensor A kij = ∂ ϕ k ∂x i ∂x j ( x )(2)where ϕ k is the k th component of ϕ . In other words, upon choosing a Riemannianmetric to induce an coordinate system at x we obtain the 1-1 correspondence T T x ϕ ↔ ( A , A )where A = ∂ϕ i ∂x j and A is given by (2). If we denote the set of rank (1 , T x M which are symmetric in the covariant indices by S ( x ), then this LEONARDO COLOMBO & HENRY O. JACOBS correspondence is given by a mapΨ : J (cid:12)(cid:12) xx (Diff( M )) → GL( T x M ) × S ( x )where J (cid:12)(cid:12) xx (Diff( M )) is the group of second order taylor expansions about x ofdiffeomorphisms which send x to itself (these are called second order jets). Thisallows us to write the Lie group structure of J (cid:12)(cid:12) xx (Diff( M )) as a type of semi-directproduct. In particular: Proposition 1.2.
If we represent
T T x ϕ and T T x ψ as ( A , A ) and ( B , B ) where A = T x ϕ, B = T x ψ, A = ∂ ϕ k ∂x i ∂x j , and B = ∂ ψ k ∂x i ∂x j , then T T x ϕ ◦ T T x ψ ≡ T T x ( ϕ ◦ ψ ) is given by the composition ( A , A ) ◦ ( B , B ) = ( A ◦ B , A ◦ B + A ◦ ( B × B )) . Proof.
We find that ∂∂x i ( ϕ k ◦ ψ ) = ∂ϕ k ∂x l · ∂ψ l ∂x i ◦ ψ and the second derivative is ∂∂x j ∂∂x i ( ϕ k ◦ ψ ) = ∂∂x j (cid:18) ∂ϕ k ∂x l · ∂ψ l ∂x i ◦ ψ (cid:19) = (cid:18) ∂ ϕ k ∂x l ∂x m ∂ψ l ∂x i ∂ψ m ∂x j + ∂ϕ k ∂x l ∂ ψ l ∂x i ∂x j (cid:19) ◦ ψ. Noting that ψ ( x ) = x we can set A = ∂ϕ k ∂x l (cid:12)(cid:12)(cid:12)(cid:12) x , A = ∂ ϕ k ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) x B = ∂ψ k ∂x l (cid:12)(cid:12)(cid:12)(cid:12) x , B = ∂ ψ k ∂x i ∂x j (cid:12)(cid:12)(cid:12)(cid:12) x and rewrite the equations in the form ∂∂x i ( ϕ k ◦ ψ ) = A · B ∂∂x j ∂∂x i ( ϕ k ◦ ψ ) = A · B + A ◦ ( B × B ) . Therefore, if we define the composition( A , A ) · ( B , B ) := ( A · B , A · B + A ◦ ( B × B ))on the manifold GL( T x M ) × S , then Ψ : J (cid:12)(cid:12) xx (Diff( M )) → GL( T x M ) × S is aLie group isomorphism by construction. (cid:3) We see that the composition law of Proposition 1.2 is of the form described inequation (1). In this paper, we will condense the composition law for second orderjets to the algebraic level and study (1) in the abstract Lie group setting. Of course,one would naturally like to consider diffeomorphisms which are not contained inIso( x ). However, this extension brings us into the realm of Lie groupoid theory andwill need to be addressed in future work.2. A centered semi-direct product theory
In this section, we will discover a new type of semi-direct product. We willoutline the necessary ingredients for the construction of such a Lie group and wewill derive the corresponding structures on the Lie algebra.
AGRANGIAN MECHANICS ON CENTERED SEMI-DIRECT PRODUCTS 5
Preliminary material on Lie groups.
Let G be a Lie group with identity e ∈ G and Lie algebra g . In this subsection we will establish notation and recallrelevant notions related to Lie groups and Lie algebras.2.1.1. Group actions:
Let V be a vector space. A left action of G on V is a smoothmap ρ L : G × V → V for which: ρ L ( e, v ) = v and ρ L ( g, ρ L ( h, v )) = ρ L ( gh, v ) , ∀ g, h ∈ G, ∀ v ∈ V. As using the symbol ‘ ρ L ’ can become cumbersome and since we will only needa one left Lie group action in a given context, we will opt to use the notation g · v := ρ L ( g, v ) . Finally, the induced infinitesimal left action of g on V is ξ · v := ddǫ (cid:12)(cid:12)(cid:12) ǫ =0 exp( ǫ · ξ ) · v , ∀ ξ ∈ g , v ∈ V. Similarly, a right action of G on V is the smooth map ρ R : V × G → V for which: ρ R ( v, e ) = v and ρ L ( ρ L ( v, g ) , h ) = ρ L ( v, gh ) , ∀ g, h ∈ G, ∀ v ∈ V. Again, we will primarily use the notation v · g := ρ R ( v, g ) for right actions. The induced infinitesimal right action of g on V is given by v · ξ = ddǫ (cid:12)(cid:12)(cid:12) ǫ =0 v · exp( ǫ · ξ ) , ∀ ξ ∈ g , v ∈ V Lastly, we say that the left action and the right action commute if( g · v ) · h = g · ( v · h )for any g, h ∈ G and v ∈ V .2.1.2. Adjoint and coadjoint operators:
In this section we will recall the “AD , Ad , ad”-notation used in [8]. For g ∈ G we define the inner automorphism AD : G × G → G as AD( g, h ) ≡ AD g ( h ) = ghg − . Differentiating AD with respect to the secondargument along curves through the identity produces the Adjoint representation of G on g denoted Ad : G × g → g and given byAd g ( η ) = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 (AD g (exp( ǫη ))) = g · η · g − , for g ∈ G and ξ ∈ g . Differentiating Ad with respect to the first argument alongcurves through the identity produces the adjoint operator ad : g × g → g given byad ξ ( η ) = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 (Ad exp( ǫξ ) ( η )) = ξ · η − η · ξ. The ad-map is an alternative notation for the Lie bracket of g in the sense thatad( ξ, η ) ≡ ad ξ ( η ) ≡ [ ξ, η ] . For each ξ ∈ g the map ad ξ : g → g is linear and therefore has a formal dualad ∗ ξ : g ∗ → g ∗ which we call the coadjoint operator . Explicitly, ad ∗ ξ is defined by therelation h ad ∗ ξ ( µ ) , η i = h µ, ad ξ ( η ) i (3)for each η ∈ g and µ ∈ g ∗ . LEONARDO COLOMBO & HENRY O. JACOBS
Centered semi-direct products.
In this subsection, we will construct asemi-direct product which can be thought of as a ‘sum’ of a right semi-direct productand a left semi-direct product.
Proposition 2.1.
Let G be a Lie group which acts on a vector-space V via left andright group actions. Then, the product G × V with the composition law (4) ( g , v ) · ( g , v ) := ( g g , g · v + v · g ) is a Lie group if and only if the left and right actions of G commute.Proof. It is clear that G × V is a smooth manifold and that the composition law(4) is a smooth map. We must prove that this composition makes G × V a group. • That the composition map (4) produces another element of G × V can beobserved directly. Thus ‘closure’ is satisfied. • The identity element is given by ( e, ∈ G × V where e ∈ G is the identityof G . • The inverse element of an arbitrary ( g, v ) ∈ G × V is ( g − , − g − vg − ) where g − is the inverse of g ∈ G. • Given three elements of G × V we find( g , v ) · (( g , v ) · ( g , v )) = ( g , v ) · ( g g , g · v + v · g )= ( g g g , g · ( g · v + v · g ) + v · ( g g ))= (( g g ) g , ( g g ) · v + g · ( v · g ) + ( v · g ) · g ) . By the commutativity of the group actions we may equate the above linewith: = (( g g ) g , ( g g ) · v + ( g · v ) · g + ( v · g ) · g )= (( g g ) g , ( g g ) · v + ( g · v + v · g ) · g )= (( g g ) , g · v + v · g ) · ( g , v )= (( g , v ) · ( g , v )) · ( g , v ) . Thus, the associative property is satisfied.Moreover, all maps in sight including the inverse map are smooth. In conclusion wesee that G × V with the composition (4) defines a Lie group. Moreover, if the leftand right actions of G on V do not commute, then we can observe that associativityis violated. (cid:3) Definition 2.1.
Given commuting left and right representations of a group G ona vector space V , the Lie group G × V with the composition (4) is denoted G ✶ V and called the centered semi-direct product of G and V. It customary to denote the left semi-direct product using the symbol ⋉ and theright semi-direct product via the symbol ⋊ . We justify our use of the symbol ✶ in that the concept of centered semi-direct product is merely a ‘sum’ of a left anda right semi-direct product. The formula ✶ = ⋊ + ⋉ can be used as a heuristicthroughout the paper. In particular, this heuristic applies to the Lie algebra. Proposition 2.2.
Let G ✶ V be a centered-semi direct product Lie group. The Liealgebra g ✶ V is given by the set g × V with the Lie bracket (5) [( ξ , v ) , ( ξ , v )] ✶ = ([ ξ , ξ ] g , ( ξ · v + v · ξ ) − ( ξ · v + v · ξ )) , for ξ , ξ ∈ g , v , v ∈ V . AGRANGIAN MECHANICS ON CENTERED SEMI-DIRECT PRODUCTS 7
Proof.
Firstly, it is simple to verify that the tangent space at the identity, ( e, ∈ G × V , is g × V . To derive the Lie bracket, we will derive the the ad-map via theAd and AD-maps. For ( g, v ) , ( h, w ) ∈ G ✶ V we findAD ( g,h ) ( h, w ) = ( gh, v · h + g · w ) · ( g − , − g − · v · g − )= (AD g ( h ) , v · hg − + g · w · g − − AD g ( h ) · v · g − ) . If we substitute ( h, w ) with the ǫ -dependent curve (exp( ǫ · ξ ) , ǫ · v ) we cancalculate the adjoint operator , Ad : ( G ✶ V ) × ( g ✶ V ) → g ✶ V. Given byAd ( g,v ) ( ξ , v ) = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 AD ( g,v ) (exp( ǫ · ξ ) , ǫ · v )= (Ad g ( ξ ); v · ξ g − + g · v · g − − Ad g ( ξ ) · v · g − ) . If we substitute ( g, v ) with the t -dependent curve (exp( tξ ) , tv ) we can differentiatewith respect to t to produce the adjoint operator ad : ( g ✶ V ) × ( g ✶ V ) → g ✶ V .Specifically, the adjoint operator is given byad ( ξ ,v ) ( ξ , v ) = ddt (cid:12)(cid:12)(cid:12) t =0 (Ad (exp( t · ξ ) ,t · v ) ( ξ , v ))= ddt (cid:12)(cid:12)(cid:12) t =0 ( gξ g − , v · ξ g − − gξ g − · v · g − + g · v · g − )= (ad ξ ( ξ ) , ξ · v + v · ξ − ξ · v − v · ξ )= ([ ξ , ξ ] g , ( ξ · v + v · ξ ) − ( ξ · v + v · ξ )) . Noting that the ad-map is merely an alternative notation for the Lie bracketcompletes the proof. (cid:3)
We complete this section by defining operations designed to express interactionterms between momenta in V and momenta in G in mechanical systems. Definition 2.2.
The heart operator ♥ : g × V ∗ → V ∗ is defined by (6) h ξ ♥ α, v i V := h α, ξ · v − v · ξ i V . The diamond operator , ♦ : V × V ∗ → g ∗ , is defined as h v ♦ α, ξ i g := h α, v · ξ − ξ · v i V . (7)The diamond operator can be seen as the sum of a diamond operator of a leftsemi-direct product and that of a right semi-direct product [9]. If we view G ⊲⊳ V as a Lie group and take the corresponding Line variations then the heart operatorand diamond operator comes into play. However, if we restrict the variations sothat V acts as an advected parameter, only the diamond operator is present. Wewill elaborate on both these options in the next section.3. Euler-Poincar´e theory
The Euler-Lagrange equations on a Lie group, ˜ G , can be expressed by a vectorfield over T ˜ G . If the Lagrangian is ˜ G -invariant then the equations of motion are ˜ G -invariant as well and the evolution equations can be reduced. While the unreducedsystem evolves by the Euler-Lagrange equations on T ˜ G , the reduced dynamicsevolve on the quotient T ˜ G/ ˜ G . However, T ˜ G/ ˜ G is just an alternative descriptionof the Lie algebra ˜ g and so the reduced equations of motion can be described on˜ g where we call them the Euler-Poincar´e equations.
This reduction procedure issummarized by the commutative diagram:
LEONARDO COLOMBO & HENRY O. JACOBS T ˜ G T ˜ G ˜ g ˜ g flow by ‘EL’flow by ‘EP’ / ˜ G / ˜ G To be even more specific. A Lagrangian L : T ˜ G → R is said to be (right) ˜ G -invariant if L ((˜ g, ˙˜ g ) · h ) = L (˜ g, ˙˜ g )for all h ∈ ˜ G . If L is ˜ G -invariant, then L is uniquely specified by its restriction ℓ = L | ˜ g : ˜ g → R . The Euler-Poincar´e theorem states that the Euler-Lagrangeequations ddt (cid:18) δLδ ˙˜ g (cid:19) − δLδ ˜ g = 0on T ˜ G are equivalent to the Euler-Poincar´e equations and reconstruction formula ddt (cid:18) δℓδ ˜ ξ (cid:19) = − ad ∗ ˜ ξ (cid:18) δℓδ ˜ ξ (cid:19) , ˜ ξ := ˙˜ g · ˜ g − . A review of Euler-Poincar´e reduction is given in [12, Ch 13] while a specializationto the case of semidirect products with advected parameters is described in [9]. Inthis section we will specialize the Euler-Poincar´e theorem to the case of centeredsemi-direct products by setting ˜ G = G ✶ V .To begin let us compute how variations of curves in the group induce variationson the trivializations of the velocities to the Lie algebra. Studying such varia-tions will allow us to transfer the variational principles on the group to variationalprinciples on the Lie algebra. Proposition 3.1.
Let G ✶ V be a centered semi-direct product and consider acurve ( g, v )( t ) ∈ G ✶ V . Let ( ξ g ( t ) , ξ v ( t )) := ( ˙ g ( t ) , ˙ v ( t )) · ( g ( t ) , v ( t )) − ∈ g ✶ V bethe right trivialization of ( ˙ g, ˙ v )( t ) . An arbitrary variation of ( g, v )( t ) is given by ( δg, δv )( t ) = ( η g , η v )( t ) · ( g, v )( t ) ∈ T ( g,v )( t ) ( G ✶ V ) , where ( η g , η v )( t ) ∈ g ✶ V . Given such a variation, the induced variation on ( ξ g , ξ v ) is given by ( δξ g , δξ v ) = ( ˙ η g − ad ξ g η g , ˙ η v + ( η g ξ v + η v ξ g ) − ( ξ g η v + ξ v η g ))(8) = ddt ( η v , η v ) − [( ξ g , ξ v ) , ( η g , η v )] ✶ . Proof.
For any Lie group, ˜ G , and any curve ˜ g ( t ) ∈ ˜ G , the variation of ˜ ξ ( t ) :=˙˜ g ( t ) · ˜ g − ( t ) induced by the variation δ ˜ g ( t ) = ˜ η ( t ) · ˜ g ( t ) is δ ˜ ξ = ˙˜ η − [ ˜ ξ, ˜ η ]. For matrixgroups see [12, Theorem 13.5.3] and [3] for the general case. If we set ˜ G = G ✶ V and use the bracket derived in Proposition 2.2 then the theorem follows. (cid:3) Now that we understand the relationship between variations of curves in G ✶ V and the induced variations in g ✶ V we can state the Euler-Poincar´e theorem forcentered semi-direct products. Theorem 3.1.
Let L : T ( G ✶ V ) → R be (right) G ✶ V -invariant, and let ℓ : g ✶ V → R be its reduced Lagrangian. Let ( g, v )( t ) ∈ G ✶ V and denote theright trivialized velocity by ( ξ g , ξ v )( t ) := ( ˙ g, ˙ v )( t ) · ( g, v )( t ) − . Then the followingstatements are equivalent: AGRANGIAN MECHANICS ON CENTERED SEMI-DIRECT PRODUCTS 9 (i) Hamilton’s principle holds. That is, (9) δ Z t t L ( g ( t ) , ˙ g ( t ) , v ( t )) dt = 0 for variations of ( g, v )( t ) with fixed endpoints.(ii) ( g, v )( t ) satisfies the Euler-Lagrange equations for L .(iii) The constrained variational principle (10) δ Z t t ℓ ( ξ g ( t ) , ξ v ( t )) dt = 0 holds on g × V for variations of the form (11) ( δξ g , δξ v ) = ( ˙ η g − ad ξ g η g , ˙ η v + η g ξ v − ξ v η g + η v ξ g − ξ g η v ) , where ( η g , η v )( t ) is an arbitrary curve in g ✶ V which vanishes at the end-points.(iv) The Euler-Poincar´e equations ddt (cid:18) δℓδξ g (cid:19) + ad ∗ ξ g (cid:18) δℓδξ g (cid:19) + ξ v ♦ δℓδξ v = 0 ,ddt (cid:18) δℓδξ v (cid:19) + ξ g ♥ δℓδξ v = 0 hold on g ✶ V .Proof. The equivalence (i) and (ii) holds for any configuration manifold and so,in particular it holds in this case.Next we show the equivalence (iii) and (iv) . We compute the variations of theaction integral to be δ Z t t ℓ ( ξ g ( t ) , ξ v ( t )) dt = Z t t D δℓδξ g , δξ g E + D δℓδξ v , δξ v E dt = Z t t D δℓδξ g , ˙ η g − ad ξ g η g E + D δℓδξ v , ˙ η v + η g ξ v − ξ v η g + η v ξ g − ξ g η v E dt and applying integration by parts and equation (3) we find= Z t t D − ddt (cid:18) δℓδξ g (cid:19) − ad ∗ ξ g (cid:18) δℓδξ g (cid:19) , η g E + D − ddt δℓδξ v , η v E + D δℓδξ v , η g ξ v − ξ v η g E + D δℓδξ v , η v ξ g − ξ g η v E dt + D δℓδξ g , η g E(cid:12)(cid:12)(cid:12) t t + D δℓδξ v , η v E(cid:12)(cid:12)(cid:12) t t = Z t t D − ddt (cid:18) δlδξ g (cid:19) − ad ∗ ξ g (cid:18) δℓδξ g (cid:19) − (cid:18) ξ v ♦ δℓδξ v (cid:19) , η g E + D − ddt (cid:18) δℓδξ v (cid:19) − ξ g ♥ δℓδξ v , η v E dt. By noting that ( η g , η v )( t ) is arbitrary on the interior of the integration domain, theresult follows.Finally, we show that (i) and (iii) are equivalent. The G − invariance of L impliesthat the integrands in (9) and (10) are equal. However, by Proposition 3.1 all thevariations of ( g, v )( t ) with fixed endpoints induce, and are induced by, variations ( δξ g , δξ v )( t ) ∈ g ✶ V of the form given in equation (11). Conversely if (i) holdswith respect to arbitrary variations ( δg, δv ), we define( η g , η v )( t ) = ( δg, δv ) · ( g, v ) − , to produce the variation of ( ξ g , ξ v ) given in equation (11). (cid:3) Remark 3.1.
There is a left invariant version of theorem (3.1) in which ( ξ g , ξ v ) :=( g, v ) − · ( ˙ g, ˙ v ) and L is left G ✶ V -invariant. In this case the Euler-Poincar´eequations take the form ddt (cid:18) δℓδξ g (cid:19) − ad ∗ ξ g (cid:18) δℓδξ g (cid:19) − ξ v ♦ δℓδξ v = 0 ,ddt (cid:18) δℓδξ v (cid:19) − ξ g ♥ δℓδξ v = 0 . Remark 3.2.
There is a version of semi-direct product mechanics wherein thevector-space V is a set of advected parameters as in [9] . In this case we imposethe holonomic constraint ˙ v = ˙ g · v + v · ˙ g and the set of admissible variations in g ✶ V become δξ g = ˙ η g − [ ξ g , η g ] , δv = η g · v + v · η g . If we do this, the ♥ -term is removed and δℓδv equation is replaced with a holonomicconstraint. In particular we find that ddt (cid:18) δℓδξ g (cid:19) ± ad ∗ ξ g (cid:18) δℓδξ g (cid:19) ± ξ v ♦ δℓδξ v = 0 dvdt = ξ g · v + v · ξ g . where we use a plus sign for right trivialization and a minus sign for left trivializa-tion. Examples
In this section we will present two examples of Euler-Poincar´e equations oncentered semidirect products. This first is a toy example designed to illustrate howcomputations of the diamond and heart operators can be done in practice. Thesecond example is concerns second order jets as described in subsection 1.4.4.1.
A toy example.
Consider the group GL( n ) and let Mat( n ) denote the vectorspace of n × n real matrices. Noting that GL( n ) acts on Mat( n ) by left and rightmultiplication, we can define the composition law on the Lie group GL ( n ) ✶ Mat( n )by: ( A, v ) · ( B, w ) = (
AB, Aw + vB ) . Moreover, we can identify gl ∗ ( n ) with gl ( n ) and Mat( n ) ∗ with Mat( n ) by the matrixtrace pairing h A, B i = trace( A T B ). This allows us to calculate the heart operator ♥ : gl ( n ) × Mat( n ) ∗ → Mat( n ) as AGRANGIAN MECHANICS ON CENTERED SEMI-DIRECT PRODUCTS 11 h A ♥ w, v i = h w, A · v − v · A i = trace (cid:0) w T ( A · v − v · A ) (cid:1) = trace (cid:0) w T · ( A · v ) − w T ( v · A ) (cid:1) = trace (cid:0) ( w T · A ) v − ( A · w T ) · v (cid:1) = trace (cid:0) ( w T · A − A · w T ) · v (cid:1) = trace (cid:0) ( A T w − w · A T ) T · v (cid:1) = h A T w − wA T , v i Therefore, A ♥ w = A T w − wA T . By a similar calculation, diamond operator is found to be v ♦ w = v T w − wv T , and the coadjoint action on GL( n ) is given byad ∗ A ( α A ) = A T · α A − α A · A T . Now, we have all the ingredients to write the Euler-Poincar´e equations. Given areduced Lagrangian ℓ : gl ( n ) ✶ Mat( n ) → R we may denote the reduced momentaby µ = δℓδξ , γ = δℓδv . where ( ξ, v ) ∈ gl ( n ) ✶ Mat( n ). The Euler-Poincar´e equations can be written as˙ µ = ( ξ T µ − µξ T ) + v T γ − γv T ˙ γ = ξ T γ − γξ T . An isotropy group of a second order jet groupoid.
In proposition 1.1we illustrated how the second order jets of diffeomorphisms of the stabilizer groupof a point x ∈ M is identifiable with a centered semidirect product. In particular,if dim( M ) = n we can consider the group GL( n ) ✶ S , where S is the set of (1 , , T . If we let e , . . . , e n ∈ R n be a basis with dual basis e , . . . , e n ∈ ( R n ) ∗ we can write an arbitrary element of T as T = T ijk e i ⊗ e j ⊗ e k . The left action of GL( n ) on T is g · T := T ijk ( g · e i ) ⊗ e j ⊗ e k ≡ T ijk g li e l ⊗ e j ⊗ e k while the right action is T · g := T ijk e i ⊗ ( g T · e j ) ⊗ ( g T · e k ) . Clearly these actions commute, and so we may form the centered semidirect productLie group GL( n ) ✶ T .Let us now focus on the Lie algebra. The Lie algebra gl ( n ) is equivalent to T and the Lie bracket is then given in the bases e i ⊗ e j by[ ξ, η ] = ( ξ ik η kj − η ik ξ kj ) e i ⊗ e j , where ξ = ξ ij e i ⊗ e j and η = η ij e i ⊗ e j . We can use the dual basis e i ⊗ e j to seethat the coadjoint action of ξ on µ = µ ji e i ⊗ e j is given byad ∗ ξ µ = ( µ jk ξ ki − µ ki ξ jk ) e i ⊗ e j . By differentiation we see that the infinitesimal left and right actions of gl ( n ) on T are given by ξ · T = T ijk ξ li e l ⊗ e j ⊗ e k T · ξ = T ilk h e i ⊗ ( ξ jl · e l ) ⊗ e k + e i ⊗ e j ⊗ ( ξ kl · e l ) i = ( T ilk ξ lj + T ijl ξ lk ) e i ⊗ e j ⊗ e k . If we choose an arbitrary element α ∈ ( T ) ∗ ≡ T given by α = α jki e i ⊗ e j ⊗ e k we find that h α, ξ · T i = ( α jkl ξ li ) T ijk = ( α lki T jlk ) ξ ij h α, T · ξ i = ( α lki ξ jl + α jli ξ kl ) T ijk = ( α jkl T lik + α kjl T lki ) ξ ij . Therefore the heart operator is given by ξ ♥ α = ( ξ li α jkl − α lki ξ jl − α jli ξ kl ) e i ⊗ e j ⊗ e k and the diamond operator is α ♦ T = ( α jkl T lik + α kjl T lki − α lki T jlk ) e i ⊗ e j . Given a reduced Lagrangian ℓ : gl ( n ) ✶ T → R we can denote µ = δℓδξ and γ = δℓδT .In terms of the basis e i ⊗ e j and e i ⊗ e j ⊗ e k we may write the (right) Euler-Poincar´eequations as: ˙ µ ji = α lki T jlk + µ jk ξ ki − µ ki ξ jk − α jkl T lik − α kjl T lki ˙ T ijk = ξ li α jkl − α lki ξ jl − α jli ξ kl . By restricting T to the subspace S , we can obtain a Lie group which models sec-ond order jets of diffeomorphisms as demonstrated in proposition 1.2. This exampleprovides a first step towards the creation of higher-order, spatially accurate particlemethods [10, section 4]. Moreover, the data of second order jets is necessary forthe advection of quantities seen in complex fluids in which the advected parametersdepend on gradients of the flow [5, 7]. Therefore, the structures described here mayprove useful in the construction of particle-based integrators for complex fluids aswell. 5. Conclusion
In this paper, we have presented a variant of traditional semi-direct products,dubbed centered semi-direct products, and we have illustrated the associated Euler-Poincar´e theory. The diamond operator, the group multiplication, and the Liebracket can all be seen as sums of the associated concepts for left and right semi-direct products. As a result, the Euler-Poincar´e theory associated with centeredsemi-direct products can also be seen as a sum of the left and right invariant Euler-Poincar´e theories for semi-direct products. Presently, many of these constructionsremain fairly theoretical. However, an isotropy group of the second order jetgroupoid can be seen as a centered semi-direct product. This has potential ap-plications in simulation of complex fluids. We hope this paper provides a steppingstone towards realizing this application.
AGRANGIAN MECHANICS ON CENTERED SEMI-DIRECT PRODUCTS 13
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