Lagrangian Mechanics and Reduction on Fibered Manifolds
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2017), 019, 26 pages Lagrangian Mechanics and Reductionon Fibered Manifolds
Songhao LI, Ari STERN and Xiang TANGDepartment of Mathematics, Washington University in St. Louis,One Brookings Drive, St. Louis MO 63130-4899, USA
E-mail: [email protected], [email protected], [email protected]
Received October 05, 2016, in final form March 13, 2017; Published online March 22, 2017https://doi.org/10.3842/SIGMA.2017.019
Abstract.
This paper develops a generalized formulation of Lagrangian mechanics on fiberedmanifolds, together with a reduction theory for symmetries corresponding to Lie groupoidactions. As special cases, this theory includes not only Lagrangian reduction (includingreduction by stages) for Lie group actions, but also classical Routh reduction, which weshow is naturally posed in this fibered setting. Along the way, we also develop some newresults for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-freeformulation of the equations of motion. Finally, we extend the foregoing to include fiberedand Lie algebroid generalizations of the Hamilton–Pontryagin principle of Yoshimura andMarsden, along with the associated reduction theory.
Key words:
Lagrangian mechanics; reduction; fibered manifolds; Lie algebroids; Lie groupoids
The starting point for classical Lagrangian mechanics is a function L : T Q → R , called the Lagrangian , where
T Q is the tangent bundle of a smooth configuration manifold Q .Yet, tangent bundles are hardly the only spaces on which one may wish to study Lagrangianmechanics. When L is invariant with respect to certain symmetries, it is useful to perform Lagrangian reduction : quotienting out the symmetries and thereby passing to a smaller spacethan
T Q . For example, if a Lie group G acts freely and properly on Q , then Q → Q/G isa principal fiber bundle; if L is invariant with respect to the G -action, then one can definea reduced Lagrangian on the quotient T Q/G (cf. Marsden and Scheurle [21], Cendra et al. [3]).In particular, when Q = G , the reduced Lagrangian is defined on T G/G ∼ = g , the Lie algebraof G , and the reduction procedure is called Euler–Poincar´e reduction (cf. Marsden and Ratiu [18,Chapter 13]).Unlike
T Q , the reduced spaces
T Q/G and g are not tangent bundles – but all three areexamples of Lie algebroids . Beginning with a seminal paper of Weinstein [30], and with particularlyimportant follow-up work by Mart´ınez [23, 24, 25], this has driven the development of a moregeneral theory of Lagrangian mechanics on Lie algebroids. In this more general framework,reduction is associated with
Lie algebroid morphisms , of which the quotient map
T Q → T Q/G is a particular example. Since Lie algebroids form a category, the composition of two morphismsis again a morphism. As an important consequence, it is almost trivial to perform so-called reduction by stages – applying a sequence of morphisms one at a time rather than all at once –whereas, without this framework, reduction by stages is considerably more difficult (Cendra etal. [3], Marsden et al. [17]). a r X i v : . [ m a t h . D S ] M a r S. Li, A. Stern and X. TangIn this paper, we generalize the foregoing theory in a new direction, based on the observationthat reduction from
T Q to T Q/G is a special case of a much more general construction, involving
Lie groupoid (rather than group) actions on fibered manifolds (rather than ordinary manifolds).This includes not only Lagrangian reduction, but also the related theory of
Routh reduction ,which we show is naturally posed in the language of fibered manifolds. In the special caseof a manifold trivially fibered over a single point, i.e., an ordinary manifold, this reduces tothe previously-studied cases. Along the way, we also develop some new results on Lagrangianmechanics on Lie algebroids – most notably a new, coordinate-free formulation of the equations ofmotion, incorporating the notion of a Lie algebroid connection due to Crainic and Fernandes [6] –and extend this theory to the Hamilton–Pontryagin principle of Yoshimura and Marsden [33].The paper is organized as follows: • In Section 2, we begin by briefly reviewing the classical formulation of Lagrangian mechanicson manifolds. We then define fibered manifolds, together with appropriate spaces of verticaltangent vectors and paths, and show how Lagrangian mechanics may be generalized tothis setting. As an application, we show that Routh reduction is naturally posed in thelanguage of fibered manifolds, where the classical Routhian is understood as a Lagrangianon an appropriate vertical bundle. • In Section 3, we discuss Lagrangian mechanics on Lie algebroids. We call the associatedequations of motion the
Euler–Lagrange–Poincar´e equations , since they simultaneouslygeneralize the Euler–Lagrange equations on
T Q , Euler–Poincar´e equations on g , andLagrange–Poincar´e equations on T Q/G . We derive a new, coordinate-free formulation ofthese equations, which we show agrees with the local-coordinates expression previouslyobtained by Mart´ınez [23]. Finally, we show that, since the vertical bundle of a fiberedmanifold is a Lie algebroid, the theory of Section 2 can be interpreted in this light. • In Section 4, we employ the Lie algebroid toolkit of Section 3 to study Lagrangian reductionon fibered manifolds by Lie groupoid actions, which we call
Euler–Lagrange–Poincar´ereduction . In the special case where a Lie groupoid acts on itself by multiplication, werecover the theory of Lagrangian mechanics on its associated Lie algebroid. • Finally, in Section 5, we generalize the
Hamilton–Pontryagin variational principle ofYoshimura and Marsden [33], together with the associated reduction theory [34], to fiberedmanifolds with Lie groupoid symmetries.
Let Q be a smooth configuration manifold and L : T Q → R be a smooth function, called theLagrangian, on its tangent bundle. There are three ways in which one can use L to inducedynamics on Q .The first, which we call the symplectic approach, begins by introducing the Legendre transform (or fiber derivative) of L , which is the bundle map F L : T Q → T ∗ Q defined fiberwise by (cid:10) F q L ( v ) , w (cid:11) = dd t L ( v + tw ) (cid:12)(cid:12)(cid:12) t =0 , v, w ∈ T q Q. This is used to pull back the canonical symplectic form ω ∈ Ω ( T ∗ Q ) to the Lagrangian -form ω L = ( F L ) ∗ ω ∈ Ω ( T Q ). The Lagrangian is said to be regular if F L is a local bundle isomorphism;in this case, ω L is nondegenerate, so ( T Q, ω L ) is a symplectic manifold. The energy function E L : T Q → R associated to L is E L ( v ) = (cid:10) F L ( v ) , v (cid:11) − L ( v ) , agrangian Mechanics and Reduction on Fibered Manifolds 3and the Lagrangian vector field X L ∈ X ( T Q ) is the vector field satisfying i X L ω L = d E L , where i X L ω L = ω L ( X L , · ) is the interior product of X L with ω L . That is, X L is the Hamiltonianvector field of E L on the symplectic manifold ( T Q, ω L ). Finally, a C path q : I → Q is calleda base integral curve of X L if its tangent prolongation ( q, ˙ q ) : I → T Q is an integral curve of X L .(Here and henceforth, I denotes the closed unit interval [0 , a, b ].)The second, which we call the variational approach, begins with the action functional S : P ( Q ) → R , defined by the integral S ( q ) = (cid:90) L (cid:0) q ( t ) , ˙ q ( t ) (cid:1) d t, where P ( Q ) denotes the Banach manifold of C paths q : I → Q . A path q ∈ P ( Q ) satisfies Hamilton’s variational principle if it is a critical point of S restricted to paths with fixedendpoints q (0) and q (1), i.e., if d S ( δq ) = 0 for all variations δq ∈ T q P ( Q ) with δq (0) = 0 and δq (1) = 0.The third and final approach considers the system of differential equations that a solutionto Hamilton’s variational principle must satisfy. In local coordinates, assuming δq (0) = 0 and δq (1) = 0,d S ( δq ) = (cid:90) (cid:18) ∂L∂q i ( q, ˙ q ) δq i + ∂L∂ ˙ q i ( q, ˙ q ) δ ˙ q i (cid:19) d t = (cid:90) (cid:18) ∂L∂q i ( q, ˙ q ) − dd t ∂L∂ ˙ q i ( q, ˙ q ) (cid:19) δq i d t. (Here, we use the Einstein index convention, where there is an implicit sum on repeated indices.)Hence, this vanishes for all δq if and only if q satisfies the system of ordinary differential equations ∂L∂q i ( q, ˙ q ) − dd t ∂L∂ ˙ q i ( q, ˙ q ) = 0 , which are called the Euler–Lagrange equations .The equivalence of these three approaches for regular Lagrangians – and of the latter twofor arbitrary Lagrangians – is a standard result in geometric mechanics. We state it now asa theorem for later reference.
Theorem 2.1. If L : T Q → R is a regular Lagrangian and q ∈ P ( Q ) , then the following areequivalent: (i) q is a base integral curve of the Lagrangian vector field X L ∈ X ( T Q ) . (ii) q satisfies Hamilton’s variational principle. (iii) q satisfies the Euler–Lagrange equations.If regularity is dropped, then ( ii ) ⇔ ( iii ) still holds. Proof .
See, e.g., Marsden and Ratiu [18, Theorem 8.1.3]. (cid:4)
We begin by giving the definition of a fibered manifold, along with its vertical and coverticalbundles. These bundles generalize the tangent and cotangent bundles of an ordinary manifold,and they will play analogous roles in fibered Lagrangian mechanics. S. Li, A. Stern and X. Tang
Definition 2.2. A fibered manifold Q → M consists of a pair of smooth manifolds Q , M ,together with a surjective submersion µ : Q → M . Definition 2.3.
The vertical bundle of Q → M is V Q = ker µ ∗ , where µ ∗ : T Q → T M is thepushforward (or tangent map) of µ . The dual of V Q is denoted V ∗ Q , which we call the coverticalbundle . Remark 2.4.
Since µ is a submersion, the fiber Q x = µ − (cid:0) { x } (cid:1) is a submanifold of Q for each x ∈ M . Therefore, V q Q = T q Q µ ( q ) , V Q = (cid:71) q ∈ Q V q Q = (cid:71) x ∈ M T Q x . In other words,
V Q consists of vectors tangent to the fibers, and hence is an integrable subbundleof
T Q . Similarly, V ∗ q Q = T ∗ q Q µ ( q ) , V ∗ Q = (cid:71) q ∈ Q V ∗ q Q = (cid:71) x ∈ M T ∗ Q x , so the covertical bundle consists of covectors to the individual fibers. Example 2.5.
An ordinary smooth manifold Q can be identified with the fibered manifold Q → • , where • denotes the space with a single point. Because µ ∗ is trivial, it follows that V Q = T Q and V ∗ Q = T ∗ Q . Definition 2.6.
The space of vertical vector fields on Q is X V ( Q ) = Γ( V Q ). The space of vertical k -forms on Q is Ω kV ( Q ) = Γ( (cid:86) k V ∗ Q ).Since V Q is integrable, it follows that X V ( Q ) is closed under the Jacobi–Lie bracket [ · , · ], i.e., X V ( Q ) is a Lie subalgebra of X ( Q ). Therefore, the following vertical exterior derivative operatoron Ω • V ( Q ) is well-defined. Definition 2.7.
Given u ∈ Ω kV ( Q ), the vertical exterior derivative d V u ∈ Ω k +1 V ( Q ) is given byd V u ( X , . . . , X k ) = k (cid:88) i =0 ( − i X i (cid:2) u ( X , . . . , (cid:98) X i , . . . , X k ) (cid:3) + (cid:88) ≤ i From the characterization of V Q in Remark 2.4, it follows that X ∈ X V ( Q )restricts to an ordinary vector field X x ∈ X ( Q x ) on each fiber Q x . Likewise, u ∈ Ω kV ( Q ) restrictsto an ordinary k -form u x ∈ Ω k ( Q x ) on each fiber Q x . Moreover, by the integrability of V Q , forany X, Y ∈ X V ( Q ) and x ∈ M we have [ X, Y ] x = [ X x , Y x ] ∈ X ( Q x ). Hence, the vertical exteriorderivative d V coincides with the ordinary exterior derivative d x : Ω k ( Q x ) → Ω k +1 ( Q x ) on thefiber Q x .Note that V Q and V ∗ Q are also, themselves, fibered manifolds over M . Specifically, if τ : V Q → Q and π : V ∗ Q → Q are the bundle projections, then we have surjective submersions µ ◦ τ : V Q → M and µ ◦ π : V ∗ Q → M ; the fibers are given by ( V Q ) x = T Q x and ( V ∗ Q ) x = T ∗ Q x .Now, just as there is a tautological 1-form and canonical symplectic 2-form on T ∗ Q , there arecorresponding vertical forms on V ∗ Q , constructed as follows.agrangian Mechanics and Reduction on Fibered Manifolds 5 Definition 2.9. The tautological vertical -form θ ∈ Ω V ( V ∗ Q ) is defined by the condition θ ( v ) = (cid:104) p, π ∗ v (cid:105) for v ∈ V p V ∗ Q . The canonical vertical -form is defined by ω = − d V θ ∈ Ω V ( V ∗ Q ). Remark 2.10. Restricted to any fiber ( V ∗ Q ) x = T ∗ Q x , it follows from the preceding remarksthat θ and ω agree with the ordinary tautological 1-form θ x ∈ Ω ( T ∗ Q x ) and canonical symplectic2-form ω x ∈ Ω ( T ∗ Q x ), respectively, on the cotangent bundle of the fiber. In particular,this implies that ω is closed (with respect to d V ) and nondegenerate, since ω x is closed andnondegenerate for each x ∈ M . In this section, we show that the three approaches to Lagrangian mechanics of Section 2.1 maybe generalized to fibered manifolds, with a corresponding generalization of Theorem 2.1. Let theLagrangian be a smooth function L : V Q → R . Definition 2.11. The Legendre transform (or fiber derivative) of L is the bundle map F L : V Q → V ∗ Q , defined for each q ∈ Q by (cid:10) F q L ( v ) , w (cid:11) = dd t L ( v + tw ) (cid:12)(cid:12)(cid:12) t =0 , v, w ∈ V q Q. We say that L is regular if F L is a local bundle isomorphism. Remark 2.12. Since ( V Q ) x = T Q x , we can define a fiber-restricted Lagrangian L x : T Q x → R ,whose ordinary Legendre transform F L x : T Q x → T ∗ Q x coincides with the restriction F L | ( V Q ) x .It is therefore useful to think of L as a smoothly varying family of ordinary Lagrangians L x ,parametrized by x ∈ M .Now, F L maps fibers to fibers (i.e., it is a morphism of fibered manifolds over M ), so itspushforward maps vertical vectors to vertical vectors, and we may write ( F L ) ∗ : V V Q → V V ∗ Q .This also gives a well-defined pullback of vertical forms ( F L ) ∗ : Ω kV ( V ∗ Q ) → Ω kV ( V Q ), whichleads to the following vertical versions of the Lagrangian 2-form and Lagrangian vector field. Definition 2.13. The Lagrangian vertical -form is ω L = ( F L ) ∗ ω ∈ Ω V ( V Q ). The Lagrangianvertical vector field X L ∈ X V ( V Q ) is the vertical vector field satisfying i X L ω L = d V E L , where the energy function E L : V Q → R is given by E L ( v ) = (cid:10) F L ( v ) , v (cid:11) − L ( v ). Remark 2.14. Restricting to the fiber over x ∈ M , we have( ω L ) x = ( F L x ) ∗ ω x = ω L x ∈ Ω ( T Q x ) , i.e., the ordinary Lagrangian 2-form for L x on T Q x , and moreover( E L ) x ( v ) = (cid:10) F L x ( v ) , v (cid:11) − L x ( v ) = E L x ( v ) , v ∈ T Q x , so E L restricts to E L x . Combining these, it follows that i ( X L ) x ω L x = d x E L x , so we conclude that ( X L ) x = X L x , i.e., X L coincides with the ordinary Lagrangian vector fieldon each fiber. S. Li, A. Stern and X. TangNext, for the variational approach, we begin by defining an appropriate space of vertical pathson which the action functional will be defined, as well as an appropriate space of variations ofthese paths. Definition 2.15. The space of C vertical paths , denoted by P V ( Q ) ⊂ P ( Q ), consists of q ∈ P ( Q ) whose tangent prolongation satisfies (cid:0) q ( t ) , ˙ q ( t ) (cid:1) ∈ V Q for all t ∈ I . The actionfunctional S : P V ( Q ) → R is then S ( q ) = (cid:90) L (cid:0) q ( t ) , ˙ q ( t ) (cid:1) d t, which is well-defined for L : V Q → R since (cid:0) q ( t ) , ˙ q ( t ) (cid:1) ∈ V Q . Remark 2.16. For q ∈ P V ( Q ), the condition (cid:0) q ( t ) , ˙ q ( t ) (cid:1) ∈ V Q impliesdd t µ (cid:0) q ( t ) (cid:1) = µ ∗ (cid:0) q ( t ) , ˙ q ( t ) (cid:1) = 0 . Hence, µ (cid:0) q ( t ) (cid:1) is constant in t , so q lies in a single fiber Q x , i.e., q ∈ P ( Q x ) for some x ∈ M .It follows that S ( q ) = S x ( q ), where S x : P ( Q x ) → R is the ordinary action associated to thefiber-restricted Lagrangian L x . Moreover, since µ (cid:0) q ( t ) (cid:1) is constant in t , there is an associatedfibered (Banach) manifold structure P V ( Q ) → M , with P V ( Q ) x = P ( Q x ). Definition 2.17. An element δq ∈ V q P V ( Q ) is called a vertical variation of q ∈ P V ( Q ). Thepath q satisfies Hamilton’s variational principle for vertical paths if q is a critical point of S relative to paths with fixed endpoints, i.e., if d S ( δq ) = 0 for all vertical variations δq with δq (0) = 0 and δq (1) = 0. Remark 2.18. Since P V ( Q ) x = P ( Q x ) and V q P V ( Q ) = T q P ( Q x ), this is immediately equivalentto q ∈ P V ( Q ) x satisfying the ordinary form of Hamilton’s variational principle for the fiber-restricted Lagrangian L x .Having defined vertical versions of the symplectic and variational approaches to Lagrangianmechanics, we finally derive the corresponding Euler–Lagrange equations. Suppose that q =( x σ , y i ) are fiber-adapted local coordinates for Q . Since vertical variations satisfy δx σ = 0, bydefinition, arbitrary fixed-endpoint variations of the action functional are given byd S ( δq ) = (cid:90) (cid:18) ∂L∂y i ( q, ˙ q ) δy i + ∂L∂ ˙ y i ( q, ˙ q ) δ ˙ y i (cid:19) d t = (cid:90) (cid:18) ∂L∂y i ( q, ˙ q ) − dd t ∂L∂ ˙ y i ( q, ˙ q ) (cid:19) δy i d t. Therefore, a critical vertical path must have the integrand above vanish, in addition to thevertical path condition. This motivates the following definition. Definition 2.19. In fiber-adapted local coordinates q = ( x σ , y i ) on Q → M , the verticalEuler–Lagrange equations for L : V Q → R are˙ x σ = 0 , ∂L∂y i ( q, ˙ q ) − dd t ∂L∂ ˙ y i ( q, ˙ q ) = 0 . (2.1) Remark 2.20. Since q = ( x σ , y i ) are fiber-adapted local coordinates, y i gives local coordinatesfor the fiber Q x , and we may write L ( q, ˙ q ) = L x ( y, ˙ y ). (Note that L is defined only on verticaltangent vectors, so ˙ x is not required.) Therefore, the vertical Euler–Lagrange equations areequivalent to the ordinary Euler–Lagrange equations, ∂L x ∂y i ( y, ˙ y ) − dd t ∂L x ∂ ˙ y i ( y, ˙ y ) = 0 , for the fiber-restricted Lagrangian L x .agrangian Mechanics and Reduction on Fibered Manifolds 7We are now prepared to state the generalization of Theorem 2.1 to Lagrangian mechanics onfibered manifolds. Theorem 2.21. If L : V Q → R is a regular Lagrangian on a fibered manifold µ : Q → M , and q ∈ P V ( Q ) is a vertical C path over x ∈ M , then the following are equivalent: ( i ) q is a base integral curve of the Lagrangian vector field X L ∈ X V ( V Q ) . ( ii ) q satisfies Hamilton’s variational principle for vertical paths. ( iii ) q satisfies the vertical Euler–Lagrange equations. ( i (cid:48) ) q is a base integral curve of the fiber-restricted Lagrangian vector field X L x ∈ X ( T Q x ) . ( ii (cid:48) ) q satisfies Hamilton’s variational principle with respect to the fiber-restricted Lagrangian L x . ( iii (cid:48) ) q satisfies the Euler–Lagrange equations with respect to the fiber-restricted Lagrangian L x .If regularity is dropped, then ( ii ) ⇔ ( iii ) ⇔ ( ii (cid:48) ) ⇔ ( iii (cid:48) ) still holds. Proof . We have seen, in the foregoing discussion, that ( i ) ⇔ ( i (cid:48) ) for regular Lagrangians, while( ii ) ⇔ ( ii (cid:48) ) and ( iii ) ⇔ ( iii (cid:48) ) hold in general. Hence, it suffices to show ( i (cid:48) ) ⇔ ( ii (cid:48) ) ⇔ ( iii (cid:48) ) forthe regular case and ( ii (cid:48) ) ⇔ ( iii (cid:48) ) for the general case – but this is simply Theorem 2.1 appliedto L x . (cid:4) The technique known as Routh reduction traces its origins as far back as the 1860 treatise ofRouth [28]. Modern geometric accounts have been given by Arnold et al. [1], Marsden andScheurle [20], and Marsden et al. [19], with the latter two works developing a more general theoryof nonabelian Routh reduction .The essence of Routh reduction, as we will show, is that it passes from a Lagrangian on anordinary manifold to an equivalent Lagrangian, known as the Routhian , on a fibered manifold.Since the resulting dynamics are confined to the vertical components (i.e., restricted to individualfibers), this reduces the size of the original system by eliminating the horizontal components.Consider a configuration manifold of the form T n × S , where T n denotes the n -torus and S isa manifold called the shape space . Let θ σ and y i be local coordinates for T n and S , respectively,and suppose the Lagrangian L : T ( T n × S ) → R is cyclic in the variables θ σ , i.e., L = L ( ˙ θ, y, ˙ y )depends only on ˙ θ but not on θ itself. Then the θ σ components of the Euler–Lagrange equationsimply thatdd t ∂L∂ ˙ θ σ = ∂L∂θ σ = 0 , so x σ = ∂L/∂ ˙ θ σ is constant in t .Now, define the fibered manifold Q = R n × S with M = R n , where µ : Q → M is simply theprojection onto the R n component, so that V Q = R n × T S . The classical Routhian R : V Q → R is R ( x, y, ˙ y ) = (cid:2) L ( ˙ θ, y, ˙ y ) − x σ ˙ θ σ (cid:3) x σ = ∂L/∂ ˙ θ σ , (2.2)where each ˙ θ σ is determined implicitly by the constraint x σ = ∂L/∂ ˙ θ σ . Considering R asa Lagrangian in the sense of the previous section, the vertical Euler–Lagrange equations consistof the vertical path condition,0 = ˙ x σ = dd t ∂L∂ ˙ θ σ , S. Li, A. Stern and X. Tangand 0 = ∂R∂y i ( x, y, ˙ y ) − dd t ∂R∂ ˙ y i ( x, y, ˙ y )= (cid:32) ∂L∂y i + ∂L∂ ˙ θ σ ∂ ˙ θ σ ∂y i − x σ ∂ ˙ θ σ ∂y i (cid:33) − dd t (cid:32) ∂L∂ ˙ y i + ∂L∂ ˙ θ σ ∂ ˙ θ σ ∂ ˙ y i − x σ ∂ ˙ θ∂ ˙ y i (cid:33) = ∂L∂y i − dd t ∂L∂ ˙ y i , where the last step uses x σ = ∂L/∂ ˙ θ σ to eliminate the last two terms from each parentheticalexpression.Thus, the ordinary Euler–Lagrange equations for L are precisely equivalent to the verticalEuler–Lagrange equations for R . This reduces the dynamics from T n × S to those on theindividual fibers Q x ∼ = S , thereby eliminating the cyclic variables θ ∈ T n . We now summarizethis result as a theorem. Theorem 2.22. Suppose L : T ( T n × S ) → R is an ordinary Lagrangian that is cyclic in the T n components, and let the classical Routhian R : V ( R n × S ) → R be the fibered Lagrangian definedin (2.2) . Then ( θ, y ) ∈ P ( T n × S ) is a solution path for L if and only if ( x, y ) ∈ P V ( R n × S ) isa vertical solution path for R . Proof . This follows from Theorem 2.21, together with the foregoing calculations. (cid:4) In this section, we lay the groundwork for reduction theory on fibered manifolds, which will bediscussed in Section 4. In ordinary Lagrangian reduction, we pass from the tangent bundle T Q to the quotient T Q/G , which is generally not a tangent bundle. Likewise, in Section 4, we willpass from vertical bundles to quotients that are generally not vertical bundles. However, T Q and T Q/G – as well as their vertical analogs, as we will show – are all examples of more generalobjects called Lie algebroids , on which Lagrangian mechanics can be studied. The study ofLagrangian mechanics on Lie algebroids was largely pioneered by Weinstein [30], and importantfollow-up work was done by Mart´ınez [23, 24, 25] and several others in more recent years; seealso Cort´es et al. [4], Cort´es and Mart´ınez [5], Grabowska and Grabowski [9], Grabowska etal. [10], Iglesias et al. [12, 13].In addition to recalling some of the key results (particularly of Weinstein [30] and Mart´ınez [23])that we will need for the subsequent reduction theory, we also develop a new, coordinate-freeformulation of the equations of motion, which we call the Euler–Lagrange–Poincar´e equations (since they simultaneously generalize the Euler–Lagrange, Euler–Poincar´e, and Lagrange–Poincar´eequations). This new formulation is based on the work of Crainic and Fernandes [6], particularlythe notion of a Lie algebroid connection and its use in describing variations of paths. A -paths We begin by recalling the definition of a Lie algebroid A and an appropriate class of pathsin A , called A -paths. This review will necessarily be very brief, but for more information on Liealgebroids, we refer the reader to the comprehensive work by Mackenzie [14]. Definition 3.1. A Lie algebroid is a real vector bundle τ : A → Q equipped with a Lie bracket[ · , · ] : Γ( A ) × Γ( A ) → Γ( A ) on its space of sections and a bundle map ρ : A → T Q , called the anchor map , satisfying the following Leibniz rule-like compatibility condition:[ X, f Y ] = f [ X, Y ] + ρ ( X )[ f ] Y, for all X, Y ∈ Γ( A ), f ∈ C ∞ ( Q ) . agrangian Mechanics and Reduction on Fibered Manifolds 9 Example 3.2. The tangent bundle T Q is a Lie algebroid over Q , where τ : T Q → Q is the usualbundle projection, [ · , · ] : X ( Q ) × X ( Q ) → X ( Q ) is the Jacobi–Lie bracket of vector fields, and ρ : T Q → T Q is the identity.Furthermore, any integrable distribution D ⊂ T Q is also a Lie algebroid over Q , where τ , [ · , · ],and ρ are just the restrictions to D of the corresponding maps for T Q . We say that D is a Liesubalgebroid of T Q .In particular, if Q → M is a fibered manifold, then V Q ⊂ T Q is integrable and hence a Liealgebroid over Q . (Note that V Q is generally not a Lie algebroid over M , since it may not evenbe a vector bundle over M .) Example 3.3. Any Lie algebra g is a Lie algebroid over • (the space with one point), where themaps τ and ρ are trivial and [ · , · ] is the Lie bracket.More generally, if Q → Q/G is a principal G -bundle for some Lie group G , then T Q/G definesan algebroid over Q/G called the Atiyah algebroid . The algebroid g → • can be identified withthe special case Q = G , where G is the Lie group integrating g (which exists by Lie’s thirdtheorem). Definition 3.4. A path a ∈ P ( A ) over the base path q = τ ◦ a ∈ P ( Q ) is called an A -path if˙ q ( t ) = ρ (cid:0) a ( t ) (cid:1) for all t ∈ I . The space of A -paths is denoted by P ρ ( A ). Remark 3.5. Equivalently, a is an A -path if and only if a d t : T I → A is a morphism of Liealgebroids, where T I → I has the tangent bundle Lie algebroid structure of Example 3.2. Hence, A -paths can be seen as “paths in the category of Lie algebroids”. A -paths We now turn to discussing an appropriate class of variations on the space of A -paths, P ρ ( A ).Crainic and Fernandes [6, Lemma 4.6] show that P ρ ( A ) ⊂ P ( A ) is a Banach submanifold.However, we do not want to take arbitrary variations δa ∈ T a P ρ ( A ), just as we did not want totake arbitrary paths in P ( A ).To illustrate the reasoning behind this, consider the case of a Lie algebra g . Since this is a Liealgebroid over • , where τ and ρ are trivial, it follows that every path ξ ∈ P ( g ) is a g -path, i.e., P ρ ( g ) = P ( g ). However, the variational principle for the Euler–Poincar´e equations on g considersonly variations of the form δξ = [ ξ, η ] + ˙ η = ad ξ η + ˙ η, where η ∈ P ( g ) is an arbitrary path vanishing at the endpoints (cf. Marsden and Ratiu [18,Chapter 13]). These constraints on admissible variations are known as Lin constraints .To generalize these constrained variations to an arbitrary Lie algebroid A → Q , we firstdiscuss the notion of a connection on a Lie algebroid, of which the adjoint action ( ξ, η ) (cid:55)→ ad ξ η of g on itself will be a special case. Definition 3.6. If A → Q is a Lie algebroid and E → Q is a vector bundle, then an A -connectionon E is a bilinear map ∇ : Γ( A ) × Γ( E ) → Γ( E ), ( X, u ) (cid:55)→ ∇ X u , satisfying the conditions ∇ fX u = f ∇ X u, ∇ X ( f u ) = f ∇ X u + ρ ( X )[ f ] u, for all X ∈ Γ( A ), u ∈ Γ( E ), and f ∈ C ∞ ( Q ). Remark 3.7. A T Q -connection is just an ordinary connection. Given a T Q -connection ∇ on A ,there are two naturally-induced A -connections on A , which we write as ∇ and ∇ : ∇ X Y = ∇ ρ ( X ) Y, ∇ X Y = ∇ ρ ( Y ) X + [ X, Y ] . A = g → • , the trivial T • -connection induces two g -connections on g : ∇ X Y = 0 , ∇ X Y = [ X, Y ] = ad X Y. Hence, the induced connection ∇ can be seen as a generalization of the adjoint action of a Liealgebra. Definition 3.8. Let a ∈ P ρ ( A ) be an A -path over q ∈ P ( Q ) and ξ ∈ P (cid:0) Γ( A ) (cid:1) be a time-dependent section such that a ( t ) = ξ (cid:0) q ( t ) (cid:1) . Suppose u ∈ P ( E ) has the same base path q , alongwith a time-dependent section η ∈ P (cid:0) Γ( E ) (cid:1) satisfying u ( t ) = η (cid:0) q ( t ) (cid:1) . Then we define ∇ a u ( t ) = ∇ ξ η (cid:0) t, q ( t ) (cid:1) + ˙ η (cid:0) t, q ( t ) (cid:1) , which is independent of the choice of ξ , η . Definition 3.9. Let a ∈ P ρ ( A ) be an A -path over q ∈ P ( Q ). An admissible variation of a isa variation of the form X b,a ∈ T a P ρ ( A ), where b ∈ P ( A ) is a path in A (but not necessarilyan A -path!) over q such that b (0) = 0 and b (1) = 0. Relative to a T Q -connection ∇ , thevariation X b,a has vertical component ∇ a b and horizontal component ρ ( b ). Remark 3.10. Crainic and Fernandes [6, Proposition 4.7] show that these admissible variationsform an integrable subbundle F ( A ) ⊂ T P ρ ( A ), and the tangent subspaces F a ( A ) ⊂ T a P ρ ( A ) areindependent of the choice of connection ∇ in the above definition. Now that we have appropriate paths and variations, we are prepared to discuss the variationalapproach to Lagrangian mechanics on Lie algebroids. Definition 3.11. Given a Lagrangian L : A → R , the action functional S : P ρ ( A ) → R is definedto be S ( a ) = (cid:90) L (cid:0) a ( t ) (cid:1) d t. We say that a ∈ P ρ ( A ) satisfies Hamilton’s variational principle for A -paths if d S ( X b,a ) = 0 forall admissible variations X b,a ∈ F a ( A ).We next use the notion of admissible variation from Definition 3.9, and its expression interms of a connection on A , to give a new, coordinate-free characterization of the solutions toHamilton’s variational principle for A -paths. Theorem 3.12. An A -path a ∈ P ρ ( A ) satisfies Hamilton’s principle if and only if, given a T Q -connection ∇ on A , it satisfies the differential equation ρ ∗ d L hor ( a ) + ∇ ∗ a d L ver ( a ) = 0 , (3.1) where d L hor and d L ver are the horizontal and vertical components of d L relative to ∇ , andwhere ρ ∗ and ∇ ∗ a are the formal adjoints of ρ and ∇ a . Proof . Given an admissible variation X b,a ∈ F a ( A ), we haved S ( X b,a ) = d S (cid:0) X hor b,a (cid:1) + d S (cid:0) X ver b,a (cid:1) = (cid:90) (cid:16)(cid:10) d L hor ( a ) , ρ ( b ) (cid:11) + (cid:10) d L ver ( a ) , ∇ a b (cid:11)(cid:17) d t = (cid:90) (cid:10) ρ ∗ d L hor ( a ) + ∇ ∗ a d L ver ( a ) , b (cid:11) d t Since b is arbitrary, it follows that d S vanishes for all X b,a ∈ F a ( A ) if and only if ρ ∗ d L hor ( a ) + ∇ ∗ a d L ver ( a ) vanishes for all t . (cid:4) agrangian Mechanics and Reduction on Fibered Manifolds 11 Example 3.13. Let A = g → • , where g is a Lie algebra. Any a ∈ P ρ ( g ) = P ( g ) can beidentified with its unique time-dependent section ξ ( t ) = ξ ( t, • ) = a ( t ). Since ρ and ∇ are trivial,it follows that (3.1) becomes0 = ∇ ∗ a d L ( a ) = (cid:18) ad ξ + dd t (cid:19) ∗ δLδξ = (cid:18) ad ∗ ξ − dd t (cid:19) δLδξ , which are precisely the Euler–Poincar´e equations (cf. Marsden and Ratiu [18, Chapter 13]).Next, we show that this coordinate-free formulation agrees with the local-coordinate expressionobtained by Weinstein [30] for regular Lagrangians and by Mart´ınez [23, 24, 25] in the moregeneral case. Theorem 3.14. Let q i be local coordinates for Q , { e I } be a local basis of sections of A , and ∇ the locally trivial T Q -connection defined by ∇ ∂/∂q i e I ≡ . Let ρ iI and C KIJ be the local-coordinaterepresentations of ρ and [ · , · ] , where ρ ( e I ) = ρ iI ∂∂q i , [ e I , e J ] = C KIJ e K . If a ∈ P ( A ) has the local-coordinate representation a ( t ) = ξ I ( t ) e I (cid:0) q ( t ) (cid:1) , then a is an A -path ifand only if ˙ q i = ρ iI ξ I , and a satisfies (3.1) if and only if ρ iI ∂L∂q i − C KIJ ξ J ∂L∂ξ K − dd t ∂L∂ξ I = 0 . (3.2) Proof . For the A -path condition, we have˙ q = ˙ q i ∂∂q i , ρ ( a ) = ρ (cid:0) ξ I e I (cid:1) = ρ iI ξ I ∂∂q i , so these are equal if and only the ∂/∂q i coefficients are equal. Next, the horizontal and verticalcomponents of d L ared L hor = ∂L∂q i d q i , d L ver = ∂L∂ξ I e I , where, as usual, e I is the dual basis element satisfying e I e J = δ IJ . Moreover, extending a to thetime-dependent section ξ ( t ) = ξ J ( t ) e J , we have ∇ a η I e I = ∇ ξ J e J η I e I + ˙ η I e I = (cid:2) ξ J e J , η I e I (cid:3) + ˙ η I e I = − C KIJ ξ J η I e K + ˙ η I e I , so ∇ a = − C KIJ ξ J e I e K + d / d t . Finally, ρ ∗ d L hor + ∇ ∗ a d L ver = ρ iI ∂L∂q i e I − C KIJ ξ J ∂L∂ξ K e I − dd t ∂L∂ξ I e I , so the left side vanishes if and only if all the e I coefficients on the right side vanish, i.e., (3.1)holds if and only if (3.2) holds. (cid:4) Example 3.15. Suppose A = T Q → Q . Local coordinates q i on Q yield corresponding localsections ∂/∂q i of T Q , i.e., e i = ∂/∂q i . It follows that [ e i , e j ] ≡ C kij ≡ i , j , k .Since ρ is the identity map, we have ρ ij = δ ij , so the T Q -path condition is ˙ q i = ξ i . Putting thisall together, it follows that (3.2) yields ∂L∂q i − dd t ∂L∂ ˙ q i = 0 , i.e., the ordinary Euler–Lagrange equations.2 S. Li, A. Stern and X. Tang Remark 3.16. There is also an equivalent symplectic/pre-symplectic/Poisson approach toLagrangian mechanics on Lie algebroids, which has already been well studied in previous workon the subject.Mart´ınez [23] shows that one can define a Lie algebroid notion of differential forms (just aswe did for the vertical formalism in Section 2.2), as well as a version of the tautological 1-formand canonical 2-form on A ∗ . The Legendre transform F L = d L ver : A → A ∗ is then used to pullthis back to a Lagrangian 2-form on A (in the sense of forms on Lie algebroids) and to define anenergy function E L on A , which Mart´ınez [23] uses to obtain Lagrangian dynamics on A .Weinstein [30], on the other hand, uses the canonical Poisson structure on A ∗ (which generalizesthe Lie–Poisson structure on the dual of a Lie algebra), which can be pulled back along F L to A when L is a regular Lagrangian. In this case, the Poisson structure on A induces a Lagrangianvector field associated to E L in the usual way.The approach of Grabowska et al. [10], Grabowska and Grabowski [9] extends Weinstein’sapproach in a different direction: instead of using the canonical Poisson structure on A ∗ ,which maps T ∗ A ∗ → T A ∗ , they use a related map (cid:15) : T ∗ A → T A ∗ to define the Tulczyjewdifferential Λ L = (cid:15) ◦ d L : A → T A ∗ . (The map (cid:15) is related to the canonical Poisson map bythe Tulczyjew isomorphism T ∗ A ∗ ∼ = −→ T ∗ A .) Using this framework, one requires that a ∈ P ( A )satisfy dd t F L ( a ) = Λ L ( a ), which contains the Euler–Lagrange–Poincar´e equations together withthe A -path condition. We remark that Grabowska et al. [10], Grabowska and Grabowski [9]apply this approach both to Lie algebroids and to so-called “general algebroids,” for which themap (cid:15) is taken as primitive, and where there is generally no canonical Poisson structure on thedual. The Lagrange–Poincar´e equations on a principal bundle Q → Q/G are typically derived by theprocedure of Lagrangian reduction (cf. Marsden and Scheurle [21], Cendra et al. [3]), relative toa particular choice of principal connection. We now discuss how these equations may instead beobtained directly on the Atiyah algebroid A = T Q/G → Q/G , using the framework presentedabove, and how the choice of principal connection is related to the connection ∇ on A . (Notethat Q/G , not Q , is the base of this algebroid.) In particular, Example 3.13 corresponds to thecase Q = G , while Example 3.15 corresponds to the case where G is trivial.Let L : T Q/G → R be a Lagrangian on the Atiyah algebroid. A principal connectioncorresponds to a section of the anchor ρ : T Q/G → T ( Q/G ), i.e., a right splitting of the Atiyahsequence ,0 → (cid:101) g → T Q/G ρ −→ T ( Q/G ) → . (3.3)Here, following Cendra et al. [3], we use (cid:101) g to denote the adjoint bundle Q × G g , so a left splittingis a principal connection 1-form (cf. Mackenzie [14, Chapter 5]). This splitting lets us write T Q/G ∼ = T ( Q/G ) ⊕ (cid:101) g ; the anchor ρ is just projection onto the first component, and the bracketof two sections ξ = ( X, ξ ) and η = ( Y, η ) is (cid:2) ( X, ξ ) , ( Y, η ) (cid:3) = (cid:0) [ X, Y ] , (cid:101) ∇ X η − (cid:101) ∇ Y ξ + [ ξ, η ] − (cid:101) R ( X, Y ) (cid:1) , (3.4)where (cid:101) ∇ is the covariant derivative and (cid:101) R the curvature form of the principal connection (cf.Cendra et al. [3, Theorem 5.2.4] in this particular case and Mackenzie [14, Theorem 7.3.7] ina more general setting).Relative to the splitting induced by the principal connection, A -paths have the form a =( x, ˙ x, v ), where x is the base path in Q/G . As before, we extend a to a time-dependent section ξ = ( X, ξ ), and likewise, we extend an arbitrary path b = ( x, δx, w ) to a time-dependent sectionagrangian Mechanics and Reduction on Fibered Manifolds 13 η = ( Y, η ). To find the corresponding admissible variation δa , we calculate ρ ( b ) = δx anduse (3.4) to obtain ∇ a b = ∇ ξ η + ˙ η = ∇ ( X,ξ ) ( Y, η ) + ( ˙ Y , ˙ η ) = ∇ Y ( X, ξ ) + (cid:2) ( X, ξ ) , ( Y, η ) (cid:3) + ( ˙ Y , ˙ η )= (cid:0) ∇ Y X + [ X, Y ] + ˙ Y , (cid:101) ∇ X η + [ ξ, η ] − (cid:101) R ( X, Y ) + ˙ η (cid:1) = (cid:0) ∇ X Y + ˙ Y , ( (cid:101) ∇ X η + ˙ η ) + [ ξ, η ] − (cid:101) R ( X, Y ) (cid:1) = (cid:0) ∇ ˙ x ( δx ) , (cid:101) ∇ ˙ x w + [ v, w ] − (cid:101) R ( ˙ x, δx ) (cid:1) . (Here, we chose ∇ to be compatible with (cid:101) ∇ , so that the ∇ Y ξ and (cid:101) ∇ Y ξ terms cancel.) Therefore,admissible variations have the form δa = ( δx, δ ˙ x, δv ), where δv = (cid:101) ∇ ˙ x w + [ v, w ] − (cid:101) R ( ˙ x, δx ) , and these are precisely the admissible variations of Cendra et al. [3, Theorem 3.4.1].Furthermore, now that we have expressions for ρ and ∇ in terms of the splitting induced by theprincipal connection, it is a straightforward matter to write down the Euler–Lagrange–Poincar´eequations (3.1) in terms of their adjoints. If we write L = L ( x, ˙ x, v ), then ρ ∗ d L hor ( x, ˙ x, v ) + ∇ ∗ ( x, ˙ x,v ) d L ver ( x, ˙ x, v )= (cid:18) ∂L∂x + ∇ ∗ ˙ x ∂L∂ ˙ x − ( i ˙ x (cid:101) R ) ∗ ∂L∂v (cid:19) d x + (cid:18) (cid:101) ∇ ∗ ˙ x ∂L∂v + ad ∗ v ∂L∂v (cid:19) d v. Hence, this vanishes precisely when ∂L∂x + ∇ ∗ ˙ x ∂L∂ ˙ x − ( i ˙ x (cid:101) R ) ∗ ∂L∂v = 0 , (cid:101) ∇ ∗ ˙ x ∂L∂v + ad ∗ v ∂L∂v = 0 , which are exactly the coordinate-free Lagrange–Poincar´e equations of Cendra et al. [3, Theo-rem 3.4.1]. (The only notable difference in notation is that Cendra et al. [3] write both covariantderivatives ∇ ˙ x and (cid:101) ∇ ˙ x as D/Dt and their adjoints as − D/Dt .) Remark 3.17. The argument above works not only for the Atiyah algebroid of a principalbundle, but also in the more general setting discussed in Mackenzie [14, Chapter 7], where onecan split a short exact sequence similar to (3.3) and obtain a bracket of the form (3.4). Thisincludes the so-called transitive Lie algebroids , of which the Atiyah algebroid is a particularexample. Example 3.18. Wong’s equations [31] for a particle in a Yang–Mills field are a classic exampleof Lagrange–Poincar´e theory. Following the presentation in Cendra et al. [3, Chapter 4], wesuppose that Q → Q/G is a principal G -bundle equipped with a Riemannian metric g on thebase Q/G and a bi-invariant Riemannian metric κ on the structure group G . Using a principalconnection to split T Q/G ∼ = T ( Q/G ) ⊕ (cid:101) g , and denoting by k the fiber metric on (cid:101) g correspondingto κ , we take the Lagrangian L ( x, ˙ x, v ) = 12 k ( v, v ) + 12 g ( ˙ x, ˙ x ) . The affine connection ∇ is then chosen to agree with (cid:101) ∇ on (cid:101) g and with the Levi-Civita connectionassociated to g on the base.With this connection in hand, we now compute the d L ver components, ∂L∂ ˙ x = g ( ˙ x, · ) = g (cid:91) ( ˙ x ) , ∂L∂v = k ( v, · ) = k (cid:91) ( v ) , k is necessarily ad-invariant,the term ad ∗ v k (cid:91) ( v ) vanishes, so the d v component of the Lagrange–Poincar´e equations is (cid:101) ∇ ˙ x k (cid:91) ( v ) = 0 . (3.5)Next, since ∇ agrees with the Levi-Civita connection on Q/G , the torsion-free property implies ∇ X Y = ∇ Y X + [ X, Y ] = ∇ X Y, so we just have ∇ ≡ ∇ . Moreover, using the metric-compatibility of ∇ along with (3.5) tocompute d L hor , it can be seen that ∂L∂x + ∇ ∗ ˙ x ∂L∂ ˙ x = g (cid:91) ( ∇ ˙ x ˙ x ) , and therefore the d x component of the Lagrange–Poincar´e equations is g (cid:91) ( ∇ ˙ x ˙ x ) = (cid:0) i ˙ x (cid:101) R (cid:1) ∗ k (cid:91) ( v ) . (3.6)The equations (3.5) and (3.6) are precisely the coordinate-free version of Wong’s equations. Forfurther discussion on Wong’s equations from the perspective of Lie algebroids, see Le´on et al. [7],Grabowska et al. [10].We conclude this example with some remarks on the relationship between Wong’s equationsand the generalized notion of geodesics on a Lie algebroid. Montgomery [26] called g ⊕ k a Kaluza–Klein metric and related Wong’s equations to Kaluza–Klein geodesics . However, a Kaluza–Kleinmetric is a particular example of a Lie algebroid metric (in this case, on A = T Q/G ), for whichthere is a unique Levi-Civita (torsion-free, metric-compatible) A -connection ∇ , and one mayconsider the corresponding geodesic equations, ∇ a a = 0 . (See Crainic and Fernandes [6], Cort´es and Mart´ınez [5], Cort´es et al. [4].) Grabowska et al. [10]pointed out that Wong’s equations may in fact be considered a special case of the generalizedgeodesic equations on a Lie algebroid; this correspondence is hidden slightly by the fact thatWong’s equations are written relative to an A -connection obtained from (cid:101) ∇ rather than theLevi-Civita A -connection. The results of Section 2 for fibered manifolds are, in fact, a special case of Lagrangian mechanicson the Lie algebroid V Q .Recall from Example 3.2 that, whenever Q → M is a fibered manifold, the vertical bundle V Q is a Lie algebroid over Q ; in particular, it is a Lie subalgebroid of T Q , from which it inheritsthe bracket [ · , · ], projection ρ , and (identity) anchor ρ . Now, by Definition 3.4, a ∈ P ( V Q )over q ∈ P ( Q ) is a V Q -path if and only if it satisfies ˙ q = a . Since a ( t ) ∈ V Q for each t ∈ I ,this means that V Q -paths are precisely the tangent prolongations of vertical paths q ∈ P V ( Q ).Hence, we may identify P ρ ( V Q ) with P V ( Q ).Suppose now that L : V Q → R is a Lagrangian in the sense of Section 3.3. If ( x σ , y i ) arefiber-adapted local coordinates for Q → M , then e i = ∂/∂y i defines a basis of local sectionsof V Q . Since an A -path is just a tangent prolongation of a vertical path, it follows that the A -path conditions are ˙ x σ = 0 and ˙ y i = ξ i . Furthermore, as in Example 3.15, we have ρ ij = δ ij , ρ iσ ≡ 0, and C kij ≡ 0, so (3.2) becomes ∂L∂y i − dd t ∂L∂ ˙ y i = 0 . Together with the A -path condition, this agrees precisely with the vertical Euler–Lagrangeequations (2.1).agrangian Mechanics and Reduction on Fibered Manifolds 15 Finally, we give a brief review of Lagrangian reduction on Lie algebroids. Weinstein [30] andMart´ınez [25] showed that, whenever Φ : A → A (cid:48) is a Lie algebroid morphism, then one can relateLagrangian dynamics on A to those on A (cid:48) .Informally, a Lie algebroid morphism is a mapping that “preserves” the Lie algebroid structurein an appropriate sense. More precisely, if A → M and A (cid:48) → M (cid:48) are Lie algebroids (possiblyover different base manifolds), then a bundle mapping Φ : A → A (cid:48) is a Lie algebroid morphism ifthe dual comorphism Φ ∗ : A (cid:48)∗ → A ∗ is a Poisson relation with respect to the canonical Poissonstructures on A ∗ and A (cid:48)∗ . (See also Remark 3.16.) Theorem 3.19. Let Φ : A → A (cid:48) be a morphism of Lie algebroids, and suppose L : A → R and L (cid:48) : A (cid:48) → R are Lagrangians such that L = L (cid:48) ◦ Φ . If a ∈ P ρ ( A ) is such that a (cid:48) = Φ ◦ a ∈ P ρ (cid:48) ( A (cid:48) ) is a solution path for L (cid:48) , then a is a solution path for L . Moreover, the following converseholds when Φ : A → A (cid:48) is fiberwise surjective: If a ∈ P ρ ( A ) is a solution path for L , then a (cid:48) = Φ ◦ a ∈ P ρ (cid:48) ( A (cid:48) ) is a solution path for L (cid:48) . Proof . See Mart´ınez [25, Theorems 5–6]. This generalized results by Weinstein [30, Theorems 4.8and 4.5, respectively] for regular Lagrangians, where the converse also required the strongerassumption that Φ be a fiberwise isomorphism. (cid:4) For example, if G is a Lie group acting freely and properly on Q , then the quotient morphism T Q → T Q/G is a Lie algebroid morphism, and the corresponding reduction theory is just classicalLagrangian reduction. However, there is a much more general class of quotient morphisms –for fibered manifolds – that bear directly on reduction theory, and this is the topic of the nextsection. In this section, we recall the definition of a Lie groupoid G ⇒ M and of a free, proper Liegroupoid action on a fibered manifold Q → M over the same base manifold. We then showthat there is a quotient morphism V Q → V Q/G , which is a Lie algebroid morphism, andhence applying Theorem 3.19 yields a reduction theory for fibered Lagrangian mechanics. Thisgeneralizes the special case M = • , in which G is a Lie group acting on an ordinary manifold Q and the quotient morphism T Q → T Q/G is the one used in ordinary Lagrangian reduction. Just as it is natural to consider Lie group actions on ordinary manifolds, it is natural to consider Lie groupoid actions on fibered manifolds. We begin by recalling the definition of a Lie groupoidand a groupoid action, as well as giving a few examples. We then prove that, just as a free andproper Lie group action on an ordinary manifold Q lifts to T Q , so, too, does a free and properLie groupoid action on a fibered manifold Q → M lift to V Q . Definition 4.1. A groupoid is a small category in which every morphism is invertible. Specifically,a groupoid denoted G ⇒ M consists of a space of morphisms G , a space of objects M , and thefollowing structure maps:(i) a source map α : G → M and target map β : G → M ;(ii) a multiplication map m : G α × β G → G , ( g, h ) (cid:55)→ gh ;(iii) an identity section (cid:15) : M → G , such that for all g ∈ G , g(cid:15) ( α ( g ) (cid:1) = g = (cid:15) (cid:0) β ( g ) (cid:1) g ;6 S. Li, A. Stern and X. Tang(iv) and an inversion map i : G → G , g (cid:55)→ g − , such that for all g ∈ G , g − g = (cid:15) (cid:0) α ( g ) (cid:1) , gg − = (cid:15) (cid:0) β ( g ) (cid:1) . A Lie groupoid is a groupoid G ⇒ M where G and M are smooth manifolds, α and β aresubmersions, and m is smooth. Remark 4.2. A few other properties of the structure maps are immediate from this definitionof a Lie groupoid: in particular, it also follows that m is a submersion, (cid:15) is an immersion, and i is a diffeomorphism. Example 4.3. A Lie group is a Lie groupoid G ⇒ • over a single point. Example 4.4. If Q is a smooth manifold, then the pair groupoid Q × Q ⇒ Q , defined by thestructure maps α ( q , q ) = q , β ( q , q ) = q , m (cid:0) ( q , q ) , ( q , q ) (cid:1) = ( q , q ) ,(cid:15) ( q ) = ( q, q ) , i ( q , q ) = ( q , q ) , is a Lie groupoid. More generally, if µ : Q → M is a fibered manifold and Q µ × µ Q = (cid:8) ( q , q ) ∈ Q × Q : µ ( q ) = µ ( q ) (cid:9) , then Q µ × µ Q ⇒ Q is also a Lie groupoid, and its structure maps are just the restrictions ofthose above for Q × Q ⇒ Q . We then say that Q µ × µ Q ⇒ Q is a Lie subgroupoid of Q × Q ⇒ Q . Example 4.5. Let G be a Lie group and Q → Q/G be a principal G -bundle, i.e., G acts freelyand properly on Q . The diagonal action of G on Q × Q is also free and proper, so we may formthe quotient ( Q × Q ) /G . Let [ q ] ∈ Q/G denote the orbit of q ∈ Q and [ q , q ] ∈ ( Q × Q ) /G denotethe orbit of ( q , q ) ∈ Q × Q . Then the gauge groupoid (or Atiyah groupoid ) ( Q × Q ) /G ⇒ Q/G of the principal bundle is defined by the structure maps α (cid:0) [ q , q ] (cid:1) = [ q ] , β (cid:0) [ q , q ] (cid:1) = [ q ] , m (cid:0) [ q , q ] , [ q , q ] (cid:1) = [ q , q ] ,(cid:15) (cid:0) [ q ] (cid:1) = [ q, q ] , i (cid:0) [ q , q ] (cid:1) = [ q , q ] . Notice that G ⇒ • is the special case where Q = G acts on itself by multiplication, while Q × Q ⇒ Q is the special case where G = { e } acts trivially on Q . Definition 4.6. A left action (or just action ) of a Lie groupoid G ⇒ M on a fibered manifold Q → M is a smooth map G α × µ Q → Q , ( g, q ) (cid:55)→ gq , such that(i) µ ( gq ) = β ( g ) for all ( g, q ) ∈ G α × µ Q ,(ii) g ( hq ) = ( gh ) q for all ( g, h, q ) ∈ G α × β G α × µ Q , and(iii) (cid:15) (cid:0) µ ( q ) (cid:1) q = q for all q ∈ Q .The action is free if gq = q implies g = (cid:15) (cid:0) µ ( q ) (cid:1) , and it is proper if its graph, G α × µ Q → Q × Q, ( g, q ) (cid:55)→ ( gq, q ) , is a proper map. A principal G -space is a fibered manifold endowed with a free and proper G -action. Remark 4.7. As with group actions, it can be shown that if G ⇒ M acts freely and properlyon Q → M , then the quotient Q/G consisting of G -orbits is a smooth manifold, and there isa smooth quotient map Q → Q/G . We refer to Dufour and Zung [8, Chapter 7] for a moredetailed discussion of this and other properties of groupoid actions.agrangian Mechanics and Reduction on Fibered Manifolds 17 Example 4.8. The action of a Lie group G on a manifold Q is precisely the action of the Liegroupoid G ⇒ • on the fibered manifold Q → • . If the action is free and proper, then theassociated principal G -space corresponds to the principal G -bundle Q → Q/G . Example 4.9. For any smooth manifold Q , the pair groupoid Q × Q ⇒ Q acts on Q by( q , q ) q = q . (In this case, we treat Q as the fibered manifold Q → Q , rather than Q → • .)Since any two points q , q lie in the same orbit, it follows that Q/ ( Q × Q ) ∼ = • , and the quotientmap is simply Q → • . Example 4.10. Let G be a Lie group acting freely and properly on Q , so that Q → Q/G isa principal G -bundle. Then the gauge groupoid ( Q × Q ) /G acts on Q → Q/G , in the sense ofDefinition 4.6, and is uniquely defined by the condition [ q , q ] q = q . (Notice that Example 4.9is the special case where G = { e } acts trivially on Q .) Again, we see that any two points q , q ∈ Q lie in the same orbit, so Q/ (cid:0) ( Q × Q ) /G (cid:1) ∼ = • , and the quotient map is Q → • . Example 4.11. For any Lie groupoid G ⇒ M , the multiplication map m is an action of G on itself, treated as the fibered manifold β : G → M . This action is free, since gh = h implies g = ( gh ) h − = hh − = (cid:15) (cid:0) β ( h ) (cid:1) . Moreover, the action is proper: ( g, h ) (cid:55)→ ( gh, h ) isa diffeomorphism, having the inverse ( g, h ) (cid:55)→ ( gh − , h ), so in particular it is a proper map.The orbit of each h ∈ G is its α -fiber α − (cid:0) { x } (cid:1) , where x = α ( h ). Identifying the fiber α − (cid:0) { x } (cid:1) with the corresponding base point x ∈ M , it follows that G/G ∼ = M , and the quotientmap is just α : G → M . Example 4.12. If G ⇒ M acts on Q → M , then it also acts on V Q → M , considered asa fibered manifold. Specifically, we have the action G α × µ ◦ τ V Q → V Q, ( g, v ) (cid:55)→ g ∗ v, where g ∗ denotes the pushforward of q (cid:55)→ gq . Lemma 4.13. Suppose G ⇒ M has a free, proper action on Q → M . Then its diagonalaction on Q µ × µ Q → M , given by g ( q , q ) = ( gq , gq ) , is also free and proper. Moreover, thequotient can be given a natural Lie group structure ( Q µ × µ Q ) /G ⇒ Q/G , and the quotient map Q µ × µ Q → ( Q µ × µ Q ) /G is a morphism of Lie groupoids over Q → Q/G . Proof . The fact that (cid:0) g, ( q , q ) (cid:1) (cid:55)→ ( gq , gq ) is a free and proper groupoid action followsimmediately from the fact that, by assumption, ( g, q ) (cid:55)→ gq is. As stated in Remark 4.7, thefreeness and properness of these actions imply that Q/G and ( Q µ × µ Q ) /G are smooth manifolds,so it suffices to specify the groupoid structure maps for ( Q µ × µ Q ) /G ⇒ Q/G . These may betaken to be formally identical to those for the gauge groupoid in Example 4.5, i.e., α (cid:0) [ q , q ] (cid:1) = [ q ] , β (cid:0) [ q , q ] (cid:1) = [ q ] , m (cid:0) [ q , q ] , [ q , q ] (cid:1) = [ q , q ] ,(cid:15) (cid:0) [ q ] (cid:1) = [ q, q ] , i (cid:0) [ q , q ] (cid:1) = [ q , q ] . As with the gauge groupoid, it is simple to check directly that these satisfy the conditions ofDefinition 4.1, so this is a Lie groupoid. Finally, using (cid:101) α , (cid:101) β, . . . to denote the structure maps on Q µ × µ Q ⇒ Q , we observe that α (cid:0) [ q , q ] (cid:1) = (cid:2)(cid:101) α ( q , q ) (cid:3) , β (cid:0) [ q , q ] (cid:1) = (cid:2) (cid:101) β ( q , q ) (cid:3) ,m (cid:0) [ q , q ] , [ q , q ] (cid:1) = (cid:2) (cid:101) m (cid:0) ( q , q ) , ( q , q ) (cid:1)(cid:3) ,(cid:15) (cid:0) [ q ] (cid:1) = (cid:2)(cid:101) (cid:15) ( q ) (cid:3) , i (cid:0) [ q , q ] (cid:1) = (cid:2)(cid:101) ı ( q , q ) (cid:3) , so the quotient map preserves the structure maps and hence is a Lie groupoid morphism. (cid:4) Lemma 4.14. The action of a Lie groupoid G ⇒ M on Q → M is free ( resp., proper ) if andonly if the induced action on V Q → M is free ( resp., proper ) . Proof . If G acts freely on Q , then g ∗ v = v implies g (cid:0) τ ( v ) (cid:1) = τ ( v ), so g = (cid:15) (cid:0) µ (cid:0) τ ( v ) (cid:1)(cid:1) = (cid:15) (cid:0) ( µ ◦ τ )( v ) (cid:1) , and hence G acts freely on V Q . Conversely, if G acts freely on V Q , then gq = q implies g ∗ q = 0 q so g = (cid:15) (cid:0) ( µ ◦ τ )(0 q ) (cid:1) = (cid:15) (cid:0) µ ( q ) (cid:1) , and hence G acts freely on Q .The proof of properness essentially amounts to chasing compact sets around the followingdiagram: G α × µ ◦ τ V Q V Q µ ◦ τ × µ ◦ τ V QG α × µ Q Q µ × µ Q. id × τ τ × τ id × × First, suppose G acts properly on Q . If K ⊂ V Q µ ◦ τ × µ ◦ τ V Q is compact, then we wish to showthat the preimage, (cid:8) ( g, v ) ∈ G α × µ ◦ τ V Q : ( v, g ∗ v ) ∈ K (cid:9) , is also compact. Observe that (cid:8) v ∈ V Q : ( v, g ∗ v ) ∈ K (cid:9) is compact by the continuity of( v, g ∗ v ) (cid:55)→ v , and (cid:8) g ∈ G : ( v, g ∗ v ) ∈ K (cid:9) is compact by the continuity of ( v, g ∗ v ) (cid:55)→ ( q, gq ), with q = τ ( v ), the properness of ( g, q ) (cid:55)→ ( q, gq ), and the continuity of ( g, q ) (cid:55)→ g . Hence, the preimagein question is also compact, so G acts properly on V Q .Conversely, suppose G acts properly on V Q . If K ⊂ Q µ × µ Q is compact, then so is (cid:8) (0 q , g ∗ q ) ∈ V Q µ ◦ τ × µ ◦ τ V Q : ( q, gq ) ∈ K (cid:9) , and by properness, so is (cid:8) ( g, q ) ∈ G α × µ ◦ τ V Q : ( q, gq ) ∈ K (cid:9) .Finally, the preimage, (cid:8) ( g, q ) ∈ G α × µ Q : ( g, q ) ∈ K (cid:9) , is compact by the continuity of ( g, q ) (cid:55)→ ( g, q ), so G acts properly on Q . (cid:4) Before discussing reduction by an arbitrary free and proper groupoid action, we first considerthe important special case where a groupoid acts on itself by left multiplication. (This can bethought of as the “groupoid version” of Euler–Poincar´e reduction, which is the special case ofLagrange–Poincar´e reduction where Q = G is a Lie group.)Recall from Example 4.11 that a Lie groupoid G ⇒ M acts freely and properly on itself (as thefibered manifold β : G → M ) by left multiplication. Lemma 4.14 implies that this induces a freeand proper action of G on the β -vertical bundle V β G → M . (Since G can be seen as a fiberedmanifold in two different ways, α : G → M and β : G → M , we denote the corresponding verticalbundles by V α G and V β G to avoid any possible confusion.) Since the orbit of v ∈ V βg G isuniquely determined by its representative at the identity section, ( g − ) ∗ v ∈ V β(cid:15) ( α ( g )) G , we canidentify the quotient V β G/G with the vector bundle AG = V β(cid:15) ( M ) G over M .This vector bundle AG → M is in fact a Lie algebroid, called the Lie algebroid of G . Theanchor map is given by the restriction of α ∗ : T G → T M to AG . Furthermore, the identificationof AG with V β G/G implies that sections X ∈ Γ( AG ) correspond to G -invariant, β -verticalvector fields ←− X ∈ X β ( G ), with ←− X ( g ) = g ∗ X (cid:0) α ( g ) (cid:1) . The bracket [ X, Y ] of X, Y ∈ Γ( AG ) is thendefined so that ←−−− [ X, Y ] = [ ←− X , ←− Y ], where the bracket on the right-hand side of this expression isjust the Jacobi–Lie bracket of vector fields on G . (See Mackenzie [14].)agrangian Mechanics and Reduction on Fibered Manifolds 19 Example 4.15. Let G be a Lie group, so that G ⇒ • is a Lie groupoid. Since β is trivial,we have V β G = T G , and hence AG = T e G = g → • , where g is the Lie algebra of G and e = (cid:15) ( • ) ∈ G is the identity element of G . Example 4.16. For the pair groupoid Q × Q ⇒ Q , we have V β ( Q × Q ) = T Q × Q , and hence A ( Q × Q ) = T Q τ × Q ∼ = T Q → Q .More generally, if we consider the groupoid Q µ × µ Q ⇒ Q for a fibered manifold Q → M ,then V β ( Q µ × µ Q ) = V Q µ ◦ τ × µ Q , and hence A ( Q µ × µ Q ) = V Q τ × Q ∼ = V Q → Q . Example 4.17. For the gauge groupoid ( Q × Q ) /G ⇒ Q/G of a principal bundle Q → Q/G , wehave V β (cid:0) ( Q × Q ) /G (cid:1) = ( T Q × Q ) /G , and hence A (cid:0) ( Q × Q ) /G (cid:1) = ( T Q τ × Q ) /G ∼ = T Q/G → Q/G .This is called the gauge algebroid (or Atiyah algebroid ) of the principal bundle.More generally, considering the groupoid ( Q µ × µ Q ) /G ⇒ G of a principal G -space, we have V β (cid:0) ( Q µ × µ Q ) /G (cid:1) = ( V Q µ ◦ τ × µ Q ) /G , and hence A (cid:0) ( Q µ × µ Q ) /G (cid:1) = ( V Q τ × Q ) /G ∼ = V Q/G → Q/G . Remark 4.18. The relationship between a groupoid G and its algebroid AG has an interestingapplication to the discretization of Lagrangian mechanics, which can be used to develop structure-preserving numerical integrators. In this approach, pioneered by Weinstein [30] (see also Marreroet al. [15, 16], Stern [29]), one replaces the Lagrangian L : AG → R by a discrete Lagrangian L h : G → R , replaces AG -paths by sequences of composable arrows in G , and uses a variationalprinciple to derive discrete equations of motion. In particular, using G = Q × Q ⇒ Q to discretize AG = T Q → Q gives the framework of variational integrators (cf. Moser and Veselov [27],Marsden and West [22]). Recall from Lemma 4.14 that if G ⇒ M acts freely and properly on Q → M , then it also actsfreely and properly on V Q → M . In other words, V Q is also a principal G -space, equipped witha quotient map V Q → V Q/G . We have seen that V Q is also a Lie algebroid, and moreover, inExample 4.16, that it is the Lie algebroid of the Lie groupoid Q µ × µ Q ⇒ Q . Similarly, fromExample 4.17, we have that V Q/G is the Lie algebroid of the Lie groupoid ( Q µ × µ Q ) /G ⇒ Q/G .Therefore, in order to perform reduction using Theorem 3.19, it suffices to show that thequotient map V Q → V Q/G is in fact a Lie algebroid morphism. Lemma 4.19. Let G ⇒ M be a Lie groupoid and Q → M a principal G -space. Then thequotient map V Q → V Q/G is a Lie algebroid morphism covering Q → Q/G . Proof . We can use a result stated in Mackenzie [14, Proposition 4.3.4], which says that a mor-phism of Lie groupoids G → G (cid:48) induces a corresponding morphism of Lie algebroids AG → AG (cid:48) .This defines the so-called Lie functor between the categories of Lie groupoids and Lie algebroids,taking objects G (cid:55)→ AG and morphisms ( G → G (cid:48) ) (cid:55)→ ( AG → AG (cid:48) ).Now, we have already proved in Lemma 4.13 that the quotient map Q µ × µ Q → ( Q µ × µ Q ) /G isa morphism of Lie groupoids, so applying the Lie functor to this morphism proves the result. (cid:4) Theorem 4.20. Let G ⇒ M be a Lie groupoid and Q → M a principal G -space. Supposethe Lagrangian L : V Q → R is G -invariant, i.e., that it factors through the quotient morphism Φ : V Q → V Q/G as L = (cid:96) ◦ Φ , where (cid:96) : V Q/G → R is called the reduced Lagrangian. Then a ∈ P V ( Q ) is a solution path for L if and only if Φ ◦ a ∈ P ρ ( V Q/G ) is a solution path for (cid:96) . Proof . Apply Theorem 3.19 to the (fiberwise-surjective) Lie algebroid morphism defined inLemma 4.19. (cid:4) Example 4.21. When G ⇒ • is a Lie group acting freely and properly on Q → • , Theorem 4.20corresponds to ordinary Lagrangian reduction from T Q to T Q/G , yielding the Lagrange–Poincar´eequations of Section 3.4. In the special case where Q = G acts on itself by multiplication, thisgives Euler–Poincar´e reduction from T G to T G/G ∼ = g . Example 4.22. Suppose G ⇒ M is a Lie groupoid acting on itself by multiplication, so thatthe quotient morphism is Φ : V β G → V β G/G = AG . If L : V β G → R and (cid:96) : AG → R areLagrangians satisfying L = (cid:96) ◦ Φ, then Theorem 4.20 implies that the vertical Euler–Lagrangeequations (Section 2) on V β G reduce to the Euler–Lagrange–Poincar´e equations (Section 3) forthe Lie algebroid AG . (This special case appears in Weinstein [30, Theorem 5.3].) The evenmore special case where G ⇒ • is a Lie group again gives Euler–Poincar´e reduction on the Liealgebra g . In this section, we extend the foregoing theory to the Hamilton–Pontryagin variational principleintroduced by Yoshimura and Marsden [33] as a generalization of Hamilton’s variational principle.This principle is especially useful for the study of “implicit Lagrangian systems” that arise inmechanical and control systems with nonholonomic or Dirac constraints. (See also Yoshimuraand Marsden [32] for the non-variational approach to such systems, as well as Yoshimura andMarsden [34] for the associated reduction theory.)We begin, in Section 5.1, with a brief review of the Hamilton–Pontryagin principle for ordinarymanifolds. We then generalize it, in Section 5.2, to fibered manifolds and their (co)verticalbundles, as we did for Hamilton’s principle in Section 2. In Section 5.3, we generalize theHamilton–Pontryagin principle even further to mechanics on Lie algebroids and their duals, aswas done for Hamilton’s principle in Section 3. Finally, in Section 5.4, we discuss reduction ofthe Hamilton–Pontryagin principle by Lie algebroid morphisms, as in the Weinstein–Mart´ınezreduction theorem (Theorem 3.19), and apply this to the special case of groupoid symmetries fora fibered manifold, as in Theorem 4.20. We begin with a quick review of the Hamilton–Pontryagin principle for ordinary (non-fibered)manifolds, as introduced in Yoshimura and Marsden [33].Let L : T Q → R be a Lagrangian. The Hamilton–Pontryagin action is the functional S : P ( T Q ⊕ T ∗ Q ) → R defined, in fiber coordinates, by S ( q, v, p ) = (cid:90) (cid:0) L (cid:0) q ( t ) , v ( t ) (cid:1) + (cid:10) p ( t ) , ˙ q ( t ) − v ( t ) (cid:11)(cid:1) d t. Here, ( q, v, p ) is an arbitrary path in the Pontryagin bundle T Q ⊕ T ∗ Q . We emphasize that norestrictions are placed on this path – in particular, the second-order curve condition ˙ q = v is not a priori required.The path ( q, v, p ) satisfies the Hamilton–Pontryagin principle if d S ( δq, δv, δp ) = 0 for allvariations ( δq, δv, δp ) ∈ T ( q,v,p ) P ( T Q ⊕ T ∗ Q ) such that δq (0) = 0 and δq (1) = 0. (That is, theendpoints of q are fixed, while the endpoints of v and p are unrestricted.) In local coordinates,we haved S ( δq, δv, δp ) = (cid:90) (cid:18) ∂L∂q i ( q, v ) δq i + ∂L∂v i ( q, v ) δv i + p i ( δ ˙ q i − δv i ) + δp i ( ˙ q i − v i ) (cid:19) d t = (cid:90) (cid:20)(cid:18) ∂L∂q i ( q, v ) − ˙ p i (cid:19) δq i + (cid:18) ∂L∂v i ( q, v ) − p i (cid:19) δv i + δp i ( ˙ q i − v i ) (cid:21) d t. agrangian Mechanics and Reduction on Fibered Manifolds 21Hence, this vanishes when ( q, v, p ) satisfies the differential-algebraic equations ∂L∂q i ( q, v ) − ˙ p i = 0 , ∂L∂v i ( q, v ) − p i = 0 , ˙ q i − v i = 0 , which Yoshimura and Marsden [33] call the implicit Euler–Lagrange equations . The three systemsof equations correspond, respectively, to the Euler–Lagrange equations, the Legendre transform,and the second-order curve condition. (Note that the conjugate momentum p acts like a “Lagrangemultiplier” enforcing the second-order curve condition ˙ q = v .)In this sense, the Hamilton–Pontryagin approach generalizes and unifies the symplectic andvariational approaches to Lagrangian mechanics. Suppose, more generally, that L : V Q → R is a Lagrangian on the vertical bundle of a fiberedmanifold Q → M . Recall that V Q and V ∗ Q can both be viewed as fibered manifolds over M ,and thus so can V Q ⊕ V ∗ Q , which we call the vertical Pontryagin bundle . It follows that we maydefine a Banach manifold of vertical paths P V ( V Q ⊕ V ∗ Q ) and its bundle of vertical variations V P V ( V Q ⊕ V ∗ Q ). Definition 5.1. Given a Lagrangian L : V Q → R , the Hamilton–Pontryagin action S : P V ( V Q ⊕ V ∗ Q ) → R is defined, in fiber coordinates, by S ( q, v, p ) = (cid:90) (cid:0) L (cid:0) q ( t ) , v ( t ) (cid:1) + (cid:10) p ( t ) , ˙ q ( t ) − v ( t ) (cid:11)(cid:1) d t. A vertical path ( q, v, p ) ∈ P V ( V Q ⊕ V ∗ Q ) is said to satisfy the Hamilton–Pontryagin principle if d S ( δq, δv, δp ) = 0 for all vertical variations ( δq, δv, δp ) ∈ V ( q,v,p ) P V ( V Q ⊕ V ∗ Q ) such that δq (0) = 0 and δq (1) = 0. Theorem 5.2. A vertical path ( q, v, p ) ∈ P V ( V Q ⊕ V ∗ Q ) satisfies the Hamilton–Pontryaginprinciple if and only if, in fiber-adapted local coordinates q = ( x σ , y i ) , it satisfies the implicitvertical Euler–Lagrange equations , ˙ x σ = 0 , ˙ p i = ∂L∂y i ( q, v ) , p i = ∂L∂v i ( q, v ) , ˙ y i = v i . (5.1) Proof . The equations ˙ x σ = 0 are simply the vertical path condition. Given a vertical variation( δq, δv, δp ) ∈ V ( q,v,p ) P V ( V Q ⊕ V ∗ Q ) satisfying δq (0) = 0 and δq (1) = 0,d S ( δq, δv, δp ) = (cid:90) (cid:18) ∂L∂y i ( q, v ) δy i + ∂L∂v i ( q, v ) δv i + p i ( δ ˙ y i − δv i ) + δp i ( ˙ y i − v i ) (cid:19) d t = (cid:90) (cid:20)(cid:18) ∂L∂y i ( q, v ) − ˙ p i (cid:19) δy i + (cid:18) ∂L∂v i ( q, v ) − p i (cid:19) δv i + δp i ( ˙ y i − v i ) (cid:21) d t. This vanishes for arbitrary ( δq, δv, δp ) if and only if each of the components in the integrandvanishes, which completes the proof. (cid:4) We next generalize the Hamilton–Pontryagin principle to a Lagrangian L : A → R , where A → Q is an arbitrary Lie algebroid. The previous subsections will then correspond to the special cases A = T Q and A = V Q , respectively.One might expect that the appropriate generalization of paths in T Q ⊕ T ∗ Q or V Q ⊕ V ∗ Q would be paths in A ⊕ A ∗ . However, these generally do not contain sufficient information torecover the A -path condition (the generalization of the second-order curve condition). Instead,we consider an alternative class of paths that we call ( A, A ∗ )-paths.2 S. Li, A. Stern and X. Tang Definition 5.3. An ( A, A ∗ ) -path consists of the following components:(i) an A -path a ∈ P ρ ( A ) over some base path q ∈ P ( Q );(ii) a path v ∈ P ( A ), not necessarily an A -path, over q ;(iii) a path p ∈ P ( A ∗ ) over q .We denote this by ( a, v, p ) ∈ P ( A, A ∗ ). Example 5.4. Any path ( q, v, p ) ∈ P ( T Q ⊕ T ∗ Q ) can be identified with the ( T Q, T ∗ Q )-path( ˙ q, v, p ) ∈ P ( T Q, T ∗ Q ). More generally, ( q, v, p ) ∈ P V ( V Q ⊕ V ∗ Q ) can be identified with( ˙ q, v, p ) ∈ P ( V Q, V ∗ Q ). Thus, P ( V Q, V ∗ Q ) ∼ = P V ( V Q ⊕ V ∗ Q ).In this special case, the base path has a unique A -path prolongation, so it suffices to considerpaths in A ⊕ A ∗ – but this is not the case in general. Example 5.5. Let g be a Lie algebra. Since all paths in g → • are g -paths, it follows thata ( g , g ∗ ) path ( a, v, p ) ∈ P ( g , g ∗ ) consists of two (generally distinct) paths a, v ∈ P ( g ) and a path p ∈ P ( g ∗ ). Thus, P ( g , g ∗ ) ∼ = P ( g ⊕ g ⊕ g ∗ ). Definition 5.6. An admissible variation of ( a, v, p ) ∈ P ( A, A ∗ ) consists of an admissible variation X b,a ∈ F a ( A ) of the A -path a , together with arbitrary variations δv ∈ T v P ( A ) and δp ∈ T p P ( A ∗ ),such that all agree on the horizontal component δq = ρ ( b ) ∈ P q ( Q ). That is, if τ : A → Q and π : A ∗ → Q are the bundle projections, we require τ ∗ ( v ) = π ∗ ( p ) = ρ ( b ). Following Remark 3.10,we denote this subbundle of admissible variations by F ( A, A ∗ ) ⊂ T P ( A, A ∗ ). Remark 5.7. Given a T Q -connection ∇ , the admissible variation ( X b,a , δv, δp ) ∈ F ( a,v,p ) ( A, A ∗ )has components X ver b,a = ∇ a b and X hor b,a = δv hor = δp hor = ρ ( b ), while δv ver and δp ver are arbitrarypaths in A and A ∗ , respectively. Example 5.8. Continuing Example 5.5, let us consider the case where g is a Lie algebra. Givena ( g , g ∗ ) path ( a, v, p ), we identify a with its time-dependent section ξ ( t ) = ξ ( t, • ) = a ( t ). Then anadmissible variation of ( a, v, p ) has the form ( ˙ ξ + [ ξ, η ] , δv, δp ), where η , δv , and δp are arbitrarypaths in g .Equivalently, assuming g is the Lie algebra of a Lie group G , let g ∈ P ( G ) be a pathintegrating ξ , i.e., g ( t ) = g (0) exp( tξ ), so that ξ = ( g − ) ∗ ˙ g . It follows that arbitrary variations( δg, δv, δp ) ∈ T ( g,v,p ) ∈ P ( G × g × g ∗ ) correspond precisely to admissible variations ( ˙ ξ +[ ξ, η ] , δv, δp )of ( ξ, v, p ) ∈ P ( g , g ∗ ), where η = ( g − ) ∗ δg . This special case corresponds to the approach ofYoshimura and Marsden [34] and Bou-Rabee and Marsden [2] for Hamilton–Pontryagin mechanicson Lie algebras, where one considers paths in P ( G × g × g ∗ ) and then left-reduces by G . Definition 5.9. Given a Lagrangian L : A → R , the Hamilton–Pontryagin action S : P ( A, A ∗ ) → R is defined by S ( a, v, p ) = (cid:90) (cid:0) L (cid:0) v ( t ) (cid:1) + (cid:10) p ( t ) , a ( t ) − v ( t ) (cid:11)(cid:1) d t, and ( a, v, p ) ∈ P ( A, A ∗ ) is said to satisfy the Hamilton–Pontryagin principle if d S ( X b,a , δv, δp ) = 0for all admissible variations ( X b,a , δv, δp ) ∈ F ( a,v,p ) ( A, A ∗ ). Theorem 5.10. An ( A, A ∗ ) -path ( a, v, p ) ∈ P ( A, A ∗ ) satisfies the Hamilton–Pontryagin principleif and only if, given a T Q -connection ∇ on A , it satisfies the differential-algebraic equations, ρ ∗ d L hor ( v ) + ∇ ∗ a p = 0 , d L ver ( v ) − p = 0 , a − v = 0 . (5.2)agrangian Mechanics and Reduction on Fibered Manifolds 23 Proof . Given ( X b,a , δv, δp ) ∈ F ( a,v,p ) ( A, A ∗ ), we computed S ( X b,a , δv, δp ) = (cid:90) (cid:16)(cid:10) d L hor ( v ) , ρ ( b ) (cid:11) + (cid:10) d L ver ( v ) , δv ver (cid:11) + (cid:104) p, ∇ a b − δv ver (cid:105) + (cid:104) δp ver , a − v (cid:105) (cid:17) d t = (cid:90) (cid:16)(cid:10) ρ ∗ d L hor ( v ) + ∇ ∗ a p, b (cid:11) + (cid:10) d L ver ( v ) − p, δv ver (cid:11) + (cid:104) δp ver , a − v (cid:105) (cid:17) d t. The Hamilton–Pontryagin principle is satisfied if and only if each term in the integrand vanishes,and since b , δv ver , and δp ver are arbitrary, the result follows. (cid:4) We call the differential-algebraic equations (5.2) the implicit Euler–Lagrange–Poincar´e equa-tions . As we did in Theorem 3.14 we can give an equivalent expression for (5.2) in localcoordinates. Theorem 5.11. Let q i be local coordinates for Q , { e I } be a local basis of sections of A , { e I } bethe dual basis of local sections of A ∗ , ∇ be the locally trivial T Q -connection, and ρ iI and C KIJ bethe local-coordinate representations of ρ and [ · , · ] . Let ( a, v, p ) ∈ P ( A ⊕ A ⊕ A ∗ ) have the local-coordinate representations a ( t ) = ξ I ( t ) e I (cid:0) q ( t ) (cid:1) , v ( t ) = v I ( t ) e I (cid:0) q ( t ) (cid:1) , and p ( t ) = p I ( t ) e I (cid:0) q ( t ) (cid:1) .Then ( a, v, p ) ∈ P ( A, A ∗ ) if and only if ˙ q i = ρ iI ξ I , and ( a, v, p ) satisfies the implicit Euler–Lagrange–Poincar´e equations (5.2) if and only if ρ iI ∂L∂q i − C KIJ ξ J p K − ˙ p I = 0 , ∂L∂ξ I − p I = 0 , ξ I − v I = 0 . Proof . The proof is a straightforward computation, following Theorem 3.14. (cid:4) Finally, we consider the reduction of Hamilton–Pontryagin mechanics by a Lie algebroid morphismΦ : A → A (cid:48) , as in Theorem 3.19. Here, though, we will require the slightly stronger assumptionthat Φ be a fiberwise isomorphism. (This was actually assumed in the original Lie algebroidreduction theorem of Weinstein [30], although Mart´ınez [25] showed that it could be relaxed.)This stronger assumption is needed since Φ ∗ : A (cid:48)∗ → A ∗ points in the “wrong direction” forreduction from ( A, A ∗ ) to ( A (cid:48) , A (cid:48)∗ ), so we need fiberwise invertibility to map A ∗ → A (cid:48)∗ . Theorem 5.12. Let Φ : A → A (cid:48) be a morphism of Lie algebroids, and suppose L : A → R and L (cid:48) : A (cid:48) → R are Lagrangians such that L = L (cid:48) ◦ Φ . If Φ is a fiberwise isomorphism, then ( a, v, p ) ∈ P ( A, A ∗ ) satisfies the Hamilton–Pontryagin principle for L if and only if ( a (cid:48) , v (cid:48) , p (cid:48) ) ∈P ( A (cid:48) , A (cid:48)∗ ) satisfies the Hamilton–Pontryagin principle for L (cid:48) , where a (cid:48) = Φ ◦ a , v (cid:48) = Φ ◦ v , and p (cid:48) = (Φ ∗ ) − ◦ p . Proof . This can be shown directly from the variational principle – observing that admissiblevariations in F ( a,v,p ) ( A, A ∗ ) map to those in F ( a (cid:48) ,v (cid:48) ,p (cid:48) ) ( A (cid:48) , A (cid:48)∗ ), and vice versa – but we givean equivalent proof using the implicit Euler–Lagrange–Poincar´e equations together with theWeinstein–Mart´ınez reduction theorem (Theorem 3.19).First, since Φ is a fiberwise isomorphism, we have a = v if and only if a (cid:48) = v (cid:48) . Moreover, since L = L (cid:48) ◦ Φ, the following diagram commutes: A A (cid:48) A ∗ A (cid:48)∗ . Φ ∼ =d L ver d L (cid:48) ver Φ ∗ ∼ = p = d L ver ( v ) if and only if p (cid:48) = d L (cid:48) ver ( v (cid:48) ). Finally, substituting theseexpressions for v and p into the first equation in (5.2), we have ρ ∗ d L hor ( a ) + ∇ ∗ a d L ver ( a ) = 0 , ρ (cid:48)∗ d L (cid:48) hor ( a (cid:48) ) + ∇ (cid:48) ∗ a (cid:48) d L (cid:48) ver ( a (cid:48) ) = 0 . But these are just the Euler–Lagrange–Poincar´e equations (3.1) for L and L (cid:48) , respectively. SoTheorem 3.19 implies that one holds if and only if the other does. (cid:4) Fortunately, the fiberwise isomorphism assumption is still sufficient to perform reductionwhen A = V Q → Q and A (cid:48) = V Q/G → Q/G , since the quotient map for the groupoid actionin Lemma 4.19 is a fiberwise isomorphism. (Indeed, Higgins and Mackenzie [11] refer to Liealgebroid morphisms with this property as action morphisms .) Intuitively, this is because thequotient is taken both on the total space and on the base, so the dimension of the fibers remainsthe same. Theorem 5.13. Let G ⇒ M be a Lie groupoid and Q → M a principal G -space. Supposethe Lagrangian L : V Q → R is G -invariant, i.e., that it factors through the quotient mor-phism Φ : V Q → V Q/G as L = (cid:96) ◦ Φ , where (cid:96) : V Q/G → R is called the reduced Lagrangian.Then ( a, v, p ) ∈ P ( V Q, V ∗ Q ) satisfies the Hamilton–Pontryagin principle for L if and only if ( a (cid:48) , v (cid:48) , p (cid:48) ) ∈ P ( V Q/G, V ∗ Q/G ) satisfies the Hamilton–Pontryagin principle for (cid:96) , where a (cid:48) = Φ ◦ a , v (cid:48) = Φ ◦ v , and p (cid:48) = (Φ ∗ ) − ◦ p . Proof . Apply Theorem 5.12 to the quotient morphism Φ, which is a fiberwise-isomorphic Liealgebroid morphism from V Q to V Q/G . (cid:4) Example 5.14. As in Example 4.21, when G ⇒ • is a Lie group acting freely and properlyon Q → • , this corresponds to the case of ordinary Lagrangian reduction for the Hamilton–Pontryagin principle. In the special case where Q = G acts on itself by multiplication, thisgives Euler–Poincar´e-type reduction for the Hamilton–Pontryagin principle, as in Yoshimura andMarsden [34], Bou-Rabee and Marsden [2]. Example 5.15. As in Example 4.22, suppose G ⇒ M is a Lie groupoid acting on itself bymultiplication, so that the quotient morphism is Φ : V β G → V β G/G = AG . If L : V β G → R and (cid:96) : AG → R are Lagrangians satisfying L = (cid:96) ◦ Φ, then Theorem 5.13 implies that the implicitvertical Euler–Lagrange equations (5.1) on V β G reduce to the implicit Euler–Lagrange–Poincar´eequations (5.2) on AG . The even more special case where G ⇒ • is a Lie group again givesHamilton–Pontryagin reduction from G to g , as in Yoshimura and Marsden [34], Bou-Rabee andMarsden [2]. Acknowledgments and disclosures The authors wish to thank Rui Loja Fernandes for his helpful feedback on this work. 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