Leonid Ryvkin
email: leonid.ryvkin@rub.de.

Tilmann Wurzbacher
email: tilmann.wurzbacher@univ-lorraine.fr.

Marco Zambon
email: marco.zambon@kuleuven.be.

Conserved quantities on multisymplectic manifolds

Keywords:

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Introduction

Conserved quantities (or "conservation laws") play a large rôle in variational problems and related areas, as continuum mechanics.

Their mathematical formulation as well as application is most transparent in the case of classical point mechanics, in its symplectic (or Hamiltonian) presentation. Given a symplectic manifold (M, ω) and a Hamiltonian function H in C ∞ (M, R) = Ω 0 (M ), a function f on M is a "conserved quantity" if the Lie derivative L v H (f ) = -{H, f } vanishes, where v H is the Hamiltonian vector field associated to H (i.e., fulfilling ι v H ω = -dH) and { , } is the Poisson bracket of (M, ω). If df is different from zero, the dimension of the phase space (M, ω) can be reduced by two and the associated Hamilton equation descends to the reduction. Iterating this process leads -at least locally -to the essentially trivial problem of solving a Hamilton equation on the real plane with its standard symplectic form. A typical source of conserved quantities is given by the Noether mechanism, here very simple: if a finite-dimensional Lie algebra g acts on (M, ω) with a (co-)momentum map and the Hamiltonian function is g-invariant, then the image of every element of g under the co-momentum is a conserved quantity.

The advent of a mathematical rigorous framework for observables and symmetries on a multisymplectic manifold (M, ω) -i.e., a manifold with a closed, non-degenerate n+1-form for n ≥ 1 (cf. [START_REF] Rogers | L ∞ -algebras from multisymplectic geometry[END_REF][START_REF] Rogers | 2-plectic geometry, Courant algebroids, and categorified prequantization[END_REF]) -raises the question whether the above generalizes from symplectic to multisymplectic geometry. Accordingly, we consider the set-up of a multisymplectic manifold (M, ω) and a "Hamiltonian form" H ∈ Ω n-1 (M ), allowing for a vector field v H such that ι v H ω = -dH. We call a differential form α ∈ Ω • (M ) "strictly conserved by H (or under v H )" if L v H α = 0. Working with forms rather than functions, we immediately have two natural weakened notions: "global conservation" resp. "local conservation" in case L v H α is exact resp. closed. Since conserved quantities are typically considered in integrated form, it is often enough that a "quantity is preserved up to a total divergence", which corresponds to these two weakened notions, that are less interesting in the symplectic case. Among these three kinds of conserved quantities, the one we consider most useful are the globally conserved quantities.

Our main goal is to understand to which extent a homotopy co-momentum associated to a multisymplectic action of a finite-dimensional Lie algebra g (cf. [START_REF] Callies | Homotopy moment maps[END_REF], [START_REF] Ryvkin | Existence and unicity of co-moments in multisymplectic geometry[END_REF]) furnishes conserved quantities if g keeps the Hamiltonian form H invariant. Given the more involved algebraic structure of the observables (and of the homotopy co-momentum), we find the following as the "correct" generalization of the above conservation law on symplectic manifolds (see Prop. 2.21):

Proposition. Let (M, ω) be a multisymplectic manifold, H ∈ Ω n-1 Ham (M ) and (f ) a homotopy co-momentum map for an infinitesimal action g → X(M ) which leaves H invariant. Let p be a k-cycle in the complex defining the Lie algebra homology of g. Then f k (p) is a globally conserved quantity. In fact, denoting the cycles of Lie algebra homology by Z k (g) and the boundaries by B k (g), we have the following table, whose last column reflects the above proposition (see Definition 2.7 for the three notions of preservation of H by the action of the Lie algebra g):

H locally g-preserved H globally g-preserved H strictly g-preserved f k (Z k (g))
locally conserved locally conserved globally conserved

f k (B k (g))
globally conserved globally conserved globally conserved

We underline that all implications in the table are sharp, as we show by explicit examples. Further, our results hold even relaxing the assumption that ω be multisymplectic, allowing ω to be any closed n + 1-form.

In discussing conserved quantities (in the flavors strong/global/local), it turns out to be useful to first work in the more general situation of a manifold M together with a vector field v and to discuss differential forms "preserved by this continuous dynamical system". In this general context, we associate "integral invariants" to conserved quantities by integrating the conserved forms over manifolds that are smoothly mapped to M . We obtain a very general form of Kelvin's classical circulation theorem, that should be of use in continuum mechanics beyond the case of isentropic, incompressible fluids. More precisely we have (compare with Proposition 4.1):

Proposition. Let Σ be a compact, oriented d-dimensional manifold (without boundary), v a vector field on M with flow φ t , and σ 0 : Σ → M a smooth map. Consider

σ t := φ t • σ 0 : Σ → M . If α ∈ Ω d (M ) is a differential form, then the number Σ (σ t ) * α is independent of the time parameter t if α is globally conserved by v.
The above proposition makes apparent, in a geometric way, the usefulness of conserved quantities.

Let us now describe the content of the different sections in more detail. In §1 we introduce multisymplectic manifolds and define the various notions of conserved quantities. The heart of this note is §2: given an action on multisymplectic manifold that preserves (in one of the ways we make precise) a Hamiltonian form, we show that certain components of the homotopy co-momentum map are conserved quantities. Further, in §2.5 we provide an alternative, homological approach to prove these statements. In §3 we explain how the set-up needed in the previous section arises naturally, we make remarks on conserved quantities, and provide in §3.

Conserved quantities in multisymplectic geometry

The purpose of this section is to address conserved quantities associated to a vector field on a manifold. We will be mainly interested in the case that the manifold carries a multisymplectic structure and the vector field is Hamiltonian. We introduce these notions in §1.1 and §1.2 , and display some algebraic properties of conserved quantities in §1.3.

Multisymplectic manifolds

Definition 1.1. A manifold M equipped with a closed n+1-form ω ∈ Ω n+1 (M ) is called a pre-n-plectic manifold. It is called an n-plectic or multisymplectic manifold if the following map is injective for all p ∈ M ,

T p M → Λ n T * p M, v → ι v ω p .
Definition 1.2. Let (M, ω) be a pre-n-plectic manifold. An (n-1)-form α is called Hamiltonian if there exists a vector field v α ∈ X(M ) such that dα = -ι vα ω.

We say that v α is a Hamiltonian vector field for α. In the n-plectic case v α is unique. The set of Hamiltonian (n-1)-forms is denoted as Ω n-1 Ham (M ). Remark 1.3. Observe that if v α is a Hamiltonian vector field corresponding to α, then L vα ω = 0 by Cartan's formula.

The following L ∞ -algebra was constructed for n-plectic manifolds in [START_REF] Rogers | L ∞ -algebras from multisymplectic geometry[END_REF]Thm. 5.2] and generalized to the pre-n-plectic case in [START_REF] Zambon | L ∞ -algebras and higher analogues of Dirac structures and Courant algebroids[END_REF]Thm. 6.7] Definition 1.4. Given a pre-n-plectic manifold (M, ω), the Lie n-algebra of observables L ∞ (M, ω) = (L, {l k }) is the graded vector space given by

L i = Ω n-1 Ham (M ) i = 0, Ω n-1-i (M ) 0 < i ≤ n -1,
together with the maps l k :

L ⊗k → L | 1 ≤ k ≤ n+1 given by l 1 (α) = dα if deg α > 0,
l 1 (α) = 0 for deg α = 0, and for all k > 1

l k (α 1 , . . . , α k ) = 0 if deg α 1 ⊗ • • • ⊗ α k > 0, ς(k)ι(v α 1 ∧ • • • ∧ v α k )ω if deg α 1 ⊗ • • • ⊗ α k = 0,
where v α i is a Hamiltonian vector field associated to α i ∈ Ω n-1

Ham (M ) and ς(k) = -(-1) k(k+1)/2 . Here the contraction with multivector fields is defined by

ι(v α 1 ∧ • • • ∧ v α k ) = ι vα k . . . ι vα 1 .

Conserved quantities

Definition 1.5. Let M be a manifold and v a vector field on M . A form α ∈ Ω • (M ) is called a (a) locally conserved quantity if L v α is a closed form, (b) globally conserved quantity if L v α is an exact form, (c) strictly conserved quantity if L v α = 0. We denote the graded vector spaces of those quantities by C loc (v) resp. C(v) and C str (v).

The following inclusions follow directly from Cartan's formula. Lemma 1.6. Let M be a manifold and v a vector field on M . Then we have

(i) C str (v) ⊂ C(v) ⊂ C loc (v), (ii) Ω • cl (M ) ⊂ C(v), (iii) d(C loc (v)) ⊂ C str (v).
We will be especially interested in the case where (M, ω) is pre-n-plectic and v preserves ω. In this case additional results hold. Lemma 1.7. Let (M, ω) be a pre-n-plectic manifold and v a vector field on M such that

L v ω = 0. Then we have (i) α ∈ Ω n-1
Ham (M ) is locally conserved by v if and only if ι [vα,v] ω = 0, for some (or equivalently for every) Hamiltonian vector field v α for α.

If moreover v = v H is a Hamiltonian vector field for H ∈ Ω n-1 Ham (M ), then (ii) α ∈ Ω n-1
Ham (M ) is locally conserved by v H if and only if L vα H is closed for some (or equivalently for every) Hamiltonian vector field v α for α.

(iii) α ∈ Ω n-1
Ham (M ) is globally conserved by v H if and only if L vα H is exact for some (or equivalently for every) Hamiltonian vector field v α for α.

(iv) H ∈ C(v H ). Proof. Assertion (i) follows from the identity L X • ι Y = ι Y • L X + ι [X,Y ] applied to ω. Assertions (ii) -(iv) follow from Cartan's formula.
Remark 1.8. We observe that the closeness resp. exactness of L vα H is equivalent to the closeness resp. exactness of l 2 (α, H).

As the following example illustrates, even in the n-plectic case, in general

L v H H = 0. Example 1.9. Let M = R 3 , ω = dx ∧ dy ∧ dz and H = xdy + zdz. Then v H = -∂ ∂z , so ι v H H = -z and L v H H = -dz.
Remark 1.10. In the symplectic (i.e. the 1-plectic) case with

H ∈ C ∞ (M ) = Ω 0 Ham (M ) we have the following statements for f ∈ C ∞ (M ) and v = v H :
(1) f is globally conserved if and only if f is strictly conserved and this is the case, if and only if {H, f } = 0, (2) f is locally conserved if and only if {H, f } is locally constant. As the following example shows, in the symplectic situation local conservedness does not suffice to formulate a "conservation law".

Example 1.11. Let M = R 2 with coordinates q, p, ω = dp ∧ dq and H = p. Taking f = q, the Hamiltonian vector field is given by v H = ∂ ∂q and thus L v H f = 1 i.e. f is locally but not globally conserved. Then for any integral curve γ(t) = (q 0 + (t -t 0 ), p 0 ) of v H we have f (γ(t)) = f (γ(t 0 )) + (t -t 0 ), i.e., f is not a constant of motion.

The algebraic structure of conserved quantities

Throughout this subsection M will denote a manifold and v a vector field on M . We present here elementary methods to construct new conserved quantities from known ones.

Lemma 1.12. The space C str (v) is a graded subalgebra of Ω • (M ).

As the following example illustrates the spaces C(v) and C loc (v), unlike C str (v), are not closed under wedge-multiplication.

Example 1.13. Let M = R 3 , ω = dx ∧ dy ∧ dz and H = -xdy. We observe that dH = -dx ∧ dy and consequently v H = ∂ ∂z . We set α = zdx and β = zdy.

Then L v H α = dx and L v H β = dy are exact but L v H (α ∧ β) = 2zdx ∧ dy is not even closed.
However stability under multiplication with elements from the following graded-commutative subalgebra of Ω • (M ) is assured:

A(v) := {β ∈ Ω(M ) | dβ = 0 and L v β = 0} ⊂ C str (v).
Lemma 1.14. The spaces C(v) and C loc (v) are graded modules over A(v).

Proof. We prove the statement for C(v), the proof for

C loc (v) being identical. Let α ∈ C(v) (that is, there is a form γ with L v α = dγ) and β ∈ A(v). Then L v (α ∧ β) = L v α ∧ β + α ∧ L v β = dγ ∧ β = d(γ ∧ β).
Again, more can be said if (M, ω) is pre-n-plectic and v preserves ω.

Proposition 1.15. Let (M, ω) be pre-n-plectic and v a vector field satisfying L v ω = 0. The graded vector spaces

L ∞ (M, ω) ∩ C loc (v), L ∞ (M, ω) ∩ C(v) and L ∞ (M, ω) ∩ C str (v) are L ∞ -subalgebras of L ∞ (M, ω). Moreover L v (l k (β 1 , ..., β k )) = 0 for k ≥ 1 and β 1 , ..., β k ∈ L ∞ (M, ω) ∩ C loc (v).
Proof. We claim that brackets of locally conserved quantities in L ∞ (M, ω) are strictly conserved. The only bracket which is nontrivial on components other than

Ω n-1 Ham (M ) is l 1 = d. It follows from part (iii) of Lemma 1.6 that l 1 = d applied to a locally conserved quantity is strictly conserved. Now for k ≥ 2 consider β 1 , ..., β k ∈ Ω n-1
Ham (M ), such that L v β i is closed for all i. We want to show that

L v (l k (β 1 , ..., β k )) = 0. As l k (β 1 , ..., β k ) = ±ι(v β 1 ∧ • • • ∧ v β k )ω, for any collection of Hamiltonian vector fields {v β i } for {β i } this is equivalent to showing L v ι v β k ...ι v β 1 ω = 0. Using the identity L X • ι Y = ι Y • L X + ι [X,Y ] we can move L v past the ι v β i since ι [v,v β i ] ω = 0 by part (i) of Lemma 1.7. We find L v ι v β k ...ι v β 1 ω = ι v β k ...ι v β 1 L v ω = 0, proving our claim.

Conserved quantities from homotopy co-momentum maps

In this section we consider (infinitesimal) actions. More precisely, (M, ω) will always denote a pre-n-plectic manifold and G a Lie group (resp. g a Lie algebra) acting on M . Given a Hamiltonian form H on M , we define three notions of "preservedness" of H with respect to the action, see Definition 2.7. Suitable conserved quantities are constructed from co-momentum maps for each of these three notions of preservedness, respectively in §2.2, §2.3 and §2.4. Finally, in §2.5 we propose a less intuitive homological approach that has the advantage of being rather concise.

Actions on multisymplectic manifolds

Definition 2.1. Let (M, ω) be a pre-n-plectic manifold. A right action ϑ of a Lie group G on M is called multisymplectic if ϑ * g ω = ω for all g ∈ G, where ϑ g = ϑ(•, g). An infinitesimal right action of a Lie algebra g on M , i.e. a Lie algebra homomorphism g → X(M ), x → v x , is called multisymplectic if L vx ω = 0 for all x ∈ g. For a connected Lie group G, a right action ϑ is multisymplectic iff the corresponding infinitesimal right action (given by x → v x where v x (m) = d dt | 0 ϑ(m, exp(tx)) at all m ∈ M ) is multisymplectic. A multisymplectic infinitesimal action is thus a Lie algebra homomorphism from g to X(M, ω) = {X ∈ X(M )|L X ω = 0}. One may ask, whether such an action admits an "L ∞ -lift" to L ∞ (M, ω). For an explicit description of the equations fulfilled by such a lift, the following definition is useful. Definition 2.2. Let g be a Lie algebra. We define the Lie algebra homology differential ∂ by setting

∂ k = ∂| Λ k g : Λ k g → Λ k-1 g, x 1 ∧• • •∧x k → 1≤i<j≤k (-1) i+j [x i , x j ]∧x 1 ∧• • •∧x i ∧• • •∧x j ∧• • •∧x k ,
for k ≥ 1. We put Λ -1 g = {0} and ∂ 0 to be the zero map.

Definition 2.3. An L ∞ -morphism (f ) from g to L ∞ (M, ω) is a collection of maps (f ) = {f i : Λ i g → Ω n-i (M )|1 ≤ i ≤ n} satisfying im(f 1 ) ⊂ Ω n-1
Ham (M ) and the equation

-f k-1 (∂(p)) = df k (p) + ς(k)ι(v p )ω (1)
for all k = 1, . . . , n + 1 and p ∈ Λ k g (setting f 0 and f n+1 to be zero). Here we use the short hand notation

v p := v x 1 ∧ • • • ∧ v x k whenever p = x 1 ∧ • • • ∧ x k for x i ∈ g.
Remark 2.4. This is, of course, the general definition of an L ∞ -algebra morphism specialized to the case at hand.

We recall from [2, §5] the higher analogue of momentum map in symplectic geometry: Definition 2.5. A (homotopy) co-momentum map for a multisymplectic infinitesimal action

v : g → X(M ) on (M, ω) is an L ∞ -morphism (f ) : g → L ∞ (M, ω) such that for all x ∈ g d(f 1 (x)) = -ι vx ω. (2)
Remark 2.6.

(1) In [START_REF] Frégier | A cohomological framework for homotopy moment maps[END_REF] and [START_REF] Ryvkin | Existence and unicity of co-moments in multisymplectic geometry[END_REF] equations ( 1) and ( 2) are interpreted as a coboundary condition on a certain chain complex.

(2) A co-momentum map (f ) is G-equivariant if the components f i : Λ i g → Ω n-i (M ) are
equivariant for all i ∈ {1, ..., n}. When G is connected, this can be expressed infinitesimally: ∀q ∈ Λ i g and for all x ∈ g = T e G the equality

L vx (f i (q)) = f i ([x, q]) holds. Here [x, •] is ad(x) acting on Λ • g.
Now we turn to infinitesimal actions preserving a Hamiltonian n-1-form H on a pre-nplectic manifold (M, ω). As in the case of the conserved quantities, one has to distinguish to which extent the action preserves the Hamiltonian form. In the following we will investigate the conserved quantities arising from co-momentum maps separately for these three cases.

Conserved quantities from locally H-preserving actions

In this subsection we assume that (M, ω) is a pre-n-plectic manifold,

H ∈ Ω n-1 Ham (M ) and that (f ) : g → L ∞ (M, ω) is the co-momentum of a locally H-preserving infinitesimal action g → X(M, ω), x → v x .
By the definition of a co-momentum map, the generator of the infinitesimal action associated to x in g is a Hamiltonian vector field of f 1 (x). As earlier, for p

= x 1 ∧ ... ∧ x k ∈ Λ k g we write v p := v x 1 ∧ ... ∧ v x k and ι(v p ) = ι vx k ...ι vx 1 . Lemma 2.9. Let (f ) = {f i |1 ≤ i ≤ n} be a co-momentum for v : g → X(M, ω) and H ∈ Ω n-1
Ham (M ). Then for any Hamiltonian vector field v H of H we have

(i) f 1 (x) ∈ C loc (v H ) for all x ∈ g, (ii) ι [v H ,vx] ω = 0 for all x ∈ g, (iii) ι(v p )ω ∈ C str (v H ) for all p ∈ Λ k g.
Proof. (i) follows from Lemma 1.7 (ii) and (ii) from Lemma 1.7 (i). Further, (iii) follows upon recalling that [L v , ι w ] = ι [v,w] and part (ii):

L v H (ι(v p )ω) = L v H ι vx k ...ι vx 1 ω = -ι vx k L v H ...ι vx 1 ω = ... = ±ι(v p )(L v H ω) = 0.
It turns out, that certain subspaces of the image of the higher components of the comomentum map constitute locally conserved quantities. To specify this we recall the definition of Lie algebra homology. Definition 2.10. Let g be a Lie algebra, k ≥ 1 and ∂ k the k-th Lie algebra homology differential. We define (a) the cycles

Z k (g) = ker(∂ k ) ⊂ Λ k g, (b) the boundaries B k (g) = im(∂ k+1 ) ⊂ Λ k g and (c) the k-th Lie algebra homology space H k (g) = Z k (g) B k (g) .
Remark 2.11. The space Z k (g) is denoted by P g,k and called the k-th Lie kernel of g in [START_REF] Madsen | Closed forms and multi-moment maps[END_REF].

Proposition 2.12. Let p ∈ Z k (g). Then f k (p) is locally conserved by any Hamiltonian vector field v H of H.

Proof. The case k = 1 is part (i) of Lemma 2.9. Assume now k > 1. We have to show that

L v H f k (p) is closed. We have dL v H f k (p) = L v H df k (p) = -ς(k)L v H ι(v p )ω = 0,
where the first equality holds because the Lie derivative commutes with the exterior derivative, the second one, because of Equation [START_REF] Audin | Torus actions on symplectic manifolds[END_REF], and the last one because of Lemma 2.9 (iii).

Proposition 2.12 states that L v H f k (p) is a closed (n-k)-form, hence we obtain: Proof. Let q be a potential for p, i.e. ∂ k+1 q = p. Then

Corollary 2.13. Let p ∈ Z k (g). If H n-k dR (M ) is zero, then f k (p)
L v H (f k (p)) = L v H (f k (∂q)) = L v H (-df k+1 (q) -ς(k + 1)ι(v q )ω) = -dL v H f k+1 (q) -ς(k + 1)L v H ι(v q )ω),
using Equation [START_REF] Audin | Torus actions on symplectic manifolds[END_REF]. The statement then follows by Lemma 2.9 (iii).

The following example shows sharpness of the statement of Proposition 2.12, i.e. for p ∈ Λ k g the condition ∂p = 0, in general, does not imply that f k (p) is globally conserved.

Example 2.15. Let M = R 3 , ω = dx ∧ dy ∧ dz and H = -xdy. We already observed dH = -dx ∧ dy and v H = ∂ ∂z . We consider the two-dimensional abelian Lie algebra g = a, b R and the homomorphism v : g → X(M ) given by v a = ∂ ∂x and v b = ∂ ∂y . We have that L va H = -dy is exact and L v b H = 0. We construct a co-momentum map for this action by f 1 (a) = -ydz, f 1 (b) = xdz and f 2 (a ∧ b) = -z. Then a ∧ b ∈ Z 2 (g) is a cycle and -z is locally conserved, as predicted by Proposition 2.12, but not globally:

L v H (-z) = ι ∂ ∂z d(-z) = -1 = 0.

Conserved quantities from globally H-preserving actions

In this subsection we assume that (M, ω) is a pre-n-plectic manifold, H ∈ Ω n-1

Ham (M ) and that (f

) : g → L ∞ (M, ω) is the co-momentum of a globally H-preserving infinitesimal action g → X(M, ω), x → v x .
As Example 2.15 indicates, no significant improvements of the above results are to be expected upon passing from locally to globally H-preserving actions. There is only a slight improvement of Lemma 2.9 (i) with essentially the same proof: Lemma 2.16. Let (f ) = {f i |1 ≤ i ≤ n} be a co-momentum for v : g → X(M ). Then f 1 (x) ∈ C(v H ) for all x ∈ g and for any Hamiltonian vector field v H of H. Remark 2.17. Notice that Lemma 2.9 and Lemma 2.16 hold for any element Ω n-1

Ham (M ) whose Hamiltonian vector field is v x .

Conserved quantities from strictly H-preserving actions

In this subsection we assume that (M, ω) is a pre-n-plectic manifold, H ∈ Ω n-1

Ham (M ) and that (f ) : g → L ∞ (M, ω) is the co-momentum of a strictly H-preserving infinitesimal action g → X(M, ω), x → v x . The assumption that the action is strictly H-preserving can be easily realized if there is a connected compact Lie group G with a smooth action on M whose differential is the infinitesimal action of g on M , see Lemma 3.1.

To prove a stronger result than Proposition 2.12 in this situation we need the following observation (cf., e.g., [START_REF] Madsen | Closed forms and multi-moment maps[END_REF]Lemma 3.4]).

Lemma 2.18. Let M be a manifold and let Ω be a not necessarily closed differential form on M . For all m ≥ 1 and all vector fields v 1 , . . . , v m in the Lie algebra X(M ) we have:

(-1) m dι(v 1 ∧ • • • ∧ v m )Ω = ι(∂(v 1 ∧ . . . ∧ v m ))Ω + m 1=1 (-1) i ι(v 1 ∧ • • • ∧ vi ∧ • • • ∧ v m )L v i Ω + ι(v 1 ∧ • • • ∧ v m )dΩ.
Definition 2.19. Given a differential form Ω ∈ Ω • (M ) and a multi-vector field Y ∈ Γ(Λ m T M ), the Lie derivative of Ω along Y is defined as a graded commutator, by

L Y Ω := dι Y Ω -(-1) m ι Y dΩ.
Remark 2.20. This definition allows to combine the first and last term in the above formula into a Lie derivative. Hence the above formula can be written

L v 1 ∧•••∧vm Ω = (-1) m [ι(∂(v 1 ∧ . . . ∧ v m ))Ω + m 1=1 (-1) i ι(v 1 ∧ • • • ∧ vi ∧ • • • ∧ v m )L v i Ω].
Proposition 2.21. Let p ∈ Z k (g). Then f k (p) is a globally conserved quantity.

Proof. We have

ι v H df k (p) = -ς(k)ι v H ι(v p )ω = (-1) k ς(k)ι(v p )dH = ς(k)d(ι(v p )H)
where we used Equation (1) in the first equality, and in the last equality Lemma 2.18 applied to the form H, as well as the g-invariance of H and once more the assumption p ∈ Z k (g). Therefore we conclude from Cartan's formula that

L v H f k (p) = d ι v H f k (p) + ς(k)ι(v p )H .
Remark 2.22. In the symplectic case, a homotopy co-momentum map boils down to its first component, f 1 : g → Ω 0 (M ), a classical co-momentum map. Upon observing that Z 1 (g) = g, the preceding proposition then reduces to the well-known obvious but important fact that if a Hamiltonian function H is g-invariant, then for all x in g we have that

{f 1 (x), H} = L v H f 1 (x) = 0.
In particular, by Proposition 2.21, f 1 (x) is a globally conserved quantity for all x ∈ g. Even in the case at hand of strictly H-preserving actions, f 1 (x) is not strictly conserved in general, as the following example shows.

Example 2.23. Let M = R 3 , ω = dx∧dy∧dz and H = -xdy. Then v H = ∂ ∂z . Furthermore we consider α = zdx and the R-action given by g = R → X(M ), 1 → v α = -∂ ∂y . This action clearly admits a co-momentum map determined by

f 1 (1) = α. Then L vα H = 0 but L v H α = dx = 0.
More is true: even if one assumes that x ∈ B 1 (g) is a boundary, f 1 (x) is still not strictly conserved in general, as Example 3.19 below shows. Specializing k to n in Proposition 2.21, we obtain scalar functions f n (x 1 , . . . , x n ) on M . Assembling these functions we obtain a map M → Z n (g) * , very similar to the multi-momentum maps of Madsen-Swann, except that it is not equivariant in general. As in the symplectic case, it satisfies: Corollary 2.24. The vector field v H is tangent to the level sets of the map

M φ → Z n (g) * given by φ(m)(p) = f n (p)(m). Proof. We have to show that v H (m) ∈ ker(T m φ : T m M → T φ(m) Z n (g) * = Z n (g) * ). We have (T m φ)(v H (m)) (p) = (df n (p))(v H (m)) = (ι v H df n (p))(m) = (L v H f n (p))(m) = 0,
where the last equation uses the fact that L v H f n (p) is exact and an exact function is necessarily 0.

Remark 2.25. An analogue of this result, where Z n (g) is substituted by B n (g) holds in the setting of Proposition 2.14.

We close this subsection by showing how a co-momentum map yields elements of the algebra A(v H ) from Lemma 1.14. Lemma 2.26. For a strictly H-preserving infinitesimal action x → v x we have:

(i) Let p ∈ Z k (g) for k ≥ 1. Then ι(v p )ω ∈ A(v H ).
(ii) Let (f ) be a co-momentum map for the g-action.

Let p ∈ Z k-1 (g), for k ≥ 2. Then l k (f 1 (x 1 ), . . . , f 1 (x k-1 ), H) ∈ A(v H ). Proof. Let us first observe that if α ∈ C(v H ), then dα ∈ A(v H ). In fact dα, being exact, is closed. Furthermore L v H (dα) = d(L v H α) = 0 since L v H α is zero. (i) By Proposition 2.21, f k (p) is a globally conserved quantity. As ι(v p )ω = ±df k (p) due to Equation (1), it is an element of A(v H ) because of the preceding observation. (ii) By Proposition 1.15, l k (f 1 (x 1 ), . . . , f 1 (x k-1 ), H) is conserved. We compute d(ι(v 1 ∧ • • • ∧ v k-1 )H) = (-1) k-1 (ι(v 1 ∧ • • • ∧ v k-1 )dH) = -(ι(v 1 ∧ • • • ∧ v k-1 ∧ v H )ω) = -ς(k)l k (f 1 (x 1 ), . . . , f 1 (x k-1 ), H), so l k (f 1 (x 1 ), . . . , f 1 (x k-1
), H) is exact, and in particular closed.

The homological point of view

In this subsection we rephrase the "generation of conserved quantities" via a co-momentum in a homological fashion. Let g be a Lie algebra acting on a pre-n-plectic manifold (M, ω), let H be a Hamiltonian (n-1)-form, and v H a Hamiltonian vector field of H. Assume that the action is locally H-preserving, i.e. L vx H is closed for all x ∈ g. The map g → H n-1 dR (M ), x → [L vx H] measures how far the action is from being globally H-preserving. This map is 0 on [g, g] and can thus be defined on H 1 (g) = g/[g, g]. Furthermore it can be extended to a map on the whole Lie algebra homology.

Proposition 2.27. For every k = 1, . . . , dim(g) the map

A : H k (g) → H n-k dR (M ), [p] → [L vp H] is well-defined. Proof. Let p ∈ Z k (g). We first check that L vp H is closed: Putting v p = l v l 1 ∧ ... ∧ v l k one has dL vp H = (-1) k+1 L vp dH = -ι(v ∂p )dH + l k i=1 (-1) i ι(v l 1 ∧ • • • ∧ vl i ∧ • • • ∧ v l m )L v l i dH = 0,
where the first equality follows from Definition 2.19, the second from Remark 2.20 and the last one from ∂p = 0 and the closeness of L v l i H. Let q ∈ Λ k+1 g. Similarly to above, we write v q = l v l 1 ∧ ... ∧ v l k+1 . We check that L v ∂q H is exact. By the definition of Lie derivative, this follows since

ι(v ∂q )dH = (-1) k+1 L vq dH - k+1 i=1 l (-1) i ι(v l 1 ∧ • • • ∧ vl i ∧ • • • ∧ v l k+1 )L v l i dH = -dL vq H
is exact. Again, here in the first equality we used Remark 2.20 and in the second that L v l i H is closed since the action is locally H-preserving.

Remark 2.28.

(1) If the action is globally H-preserving, the map g → H n-1 dR (M ) is zero, but the higher components of A do not necessarily vanish. This is exhibited by Example 2.15:

ι(v a ∧ v b )dH = -ι( ∂ ∂x ∧ ∂ ∂y )(dx ∧ dy) = -1 is closed but not exact. (2)
If the action is strictly H-preserving, then the map A is identically zero. Indeed, for every p ∈ Z k (g) we have L vp H = 0, as can be seen applying Lemma 2.18 to H.

When a co-momentum map exists, we can be more explicit:

Lemma 2.29. If (f ) is a co-momentum map (f ) for the g-action, then the map A can be written as follows: for all p ∈ Z k (g),

A([p]) = -ς(k)[L v H f k (p)].
The last equality holds because ϑ * g (d H) = d H for all g ∈ G, as a consequence of dL vx H = L vx d H = 0 for all x ∈ g. Hence the Hamiltonian vector fields of H are exactly the Hamiltonian vector fields of H. Proposition 3.2. Consider a connected compact Lie group G acting on the pre-n-plectic manifold (M, ω). Let v be a G-invariant vector field on M with L v ω = 0. Suppose that H n dR (M ) = 0. Then v is the Hamiltonian vector field of some Hamiltonian form H which is G-invariant, i.e. strictly preserved by the action.

Proof. The condition L v ω = 0 implies that ι v ω is closed, and by our cohomological assumption it is exact:

ι v ω = -d H for some H ∈ Ω n-1
Ham (M ). Now we average both sides of the above equation. Notice that ι v ω is G-invariant, since v and ω are, hence the averaged form H := G ϑ * g ( H)µ(dg) (where µ is the normalized Haar measure on G) satisfies the same equation:

ι v ω = -dH.
An interesting special case is the one of volume forms: Corollary 3.3. Consider a connected compact Lie group G acting on (M, ω), where ω is a volume form. Let v be a G-invariant vector field on M which is divergence-free (i.e. The following statement is a variation of Proposition 3.2, in which the compactness assumption on G is replaced with the condition H n-1 dR (M ) = 0 and which leads to a globally preserved Hamiltonian. Proposition 3.5. Consider a connected Lie group G acting on the pre-n-plectic manifold (M, ω). Let v be a G-invariant vector field on M with L v ω = 0. Suppose that H n dR (M ) = 0 = H n-1 dR (M ). Then v is the Hamiltonian vector field of some Hamiltonian form H which is globally preserved by the action.

L v ω = 0). Suppose that H dim(M )-1 dR (M ) = 0. Then v is
Proof. Since L v ω = 0 and H n dR (M ) = 0 we have ι v ω = -dH for some (usually not

G- invariant) H ∈ Ω n-1
Ham (M ). The form ι v ω is G-invariant, since v and ω are. Hence for all x ∈ g we have 0 = L vx dH = d(L vx H). The condition H n-1 dR (M ) = 0 implies that L vx H is exact.

Induced actions of isotropy subgroups

Let G act on a pre-n-plectic manifold (M, ω). In this whole subsection we fix p ∈ Z k (g) ⊂ Λ k g, for some k ≥ 1 (see Definition 2.10). We denote by G p the corresponding isotropy group for the adjoint action of G on Λ k g, and by g p its Lie algebra. Explicitly, g p = {x ∈ g : [x, p] = 0}. Remark 3.6. Let p ∈ Z k (g) and x ∈ g. From Lemma 3.12 it follows that x ∧ p ∈ Z k+1 (g) iff x ∈ g p . Lemma 3.7. The form ι(v p )ω ∈ Ω n+1-k (M ) is closed, and invariant under the action of G 0 p , the connected component of the identity in G p . Proof. The equality d(ι(v p )ω) = 0 follows upon applying Lemma 2.18. The invariance holds since for every y ∈ g p we have L vy ι(v p )ω = ι([v y , v p ])ω + ι(v p )L vy ω = 0, where the bracket is defined analogously to Lemma 3.12 later on. Now assume there is a co-momentum map (f ) : g → L ∞ (M, ω). Proposition 3.8. A co-momentum map for the action of G 0 p on (M, ι(v p )ω) is given by (f p ) : g p → L ∞ (M, ι(v p )ω) with components (j = 1, . . . , n -k):

f p j : Λ j g p → Ω n-k-j (M ), q → -ς(k)f j+k (q ∧ p).
Furthermore, if the co-momentum map (f ) is G-equivariant, then (f p ) is G 0 p -equivariant. Proof. We first show that (f p ) is a co-momentum map. Let q ∈ Λ j g p . We have

ς(k + j)ι vq (ι(v p )ω) = ς(k + j)ι(v p∧q )ω = -f k+j-1 (∂(p ∧ q)) -d(f k+j (p ∧ q))
using in the second equality that (f ) is a co-momentum map (see eq. ( 1) ). From Lemma 3.12 we obtain ∂(p ∧ q) = (-1) k p ∧ ∂(q) = (-1) kj ∂(q) ∧ p since p ∈ Z k (g) and q ∈ ∧ j g p . Using ς(k + j) = ς(k)ς(j)(-1) kj+1 we hence obtain ς(j)ι vq (ι(v p )ω) = -f p j-1 (∂(q)) -df p j (q). For the equivariance statement, notice that for all y ∈ g p :

L vy (f p j (q)) = L vy (f k+j (p ∧ q)) = f k+j ([y, p ∧ q]) = f k+j (p ∧ [y, q]) = f p j ([y, q]
), where we used the equivariance of (f ) in the second equality.

Remark 3.9. The existence of a co-momentum map (f ) implies that ι(v p )ω is exact with primitive -ς(k)f k (p), by Equation [START_REF] Audin | Torus actions on symplectic manifolds[END_REF]. Assume further that f k is G-equivariant. Then this primitive is G 0 p -invariant, for L vy f k (p) = f k ([y, p]) = 0 for all y ∈ g p . Hence, by [2, Lemma 8.1], an equivariant co-momentum map for the action of G 0 p on (M, ι(v p )ω) is given by (j = 1, . . . , n -k):

Λ j g p → Ω n-k-j (M ), q → (-1) k ι(v q )(f k (p)).
Notice that this co-momentum map may differ from the one given in Proposition 3.8.

Finally, we consider Hamiltonian forms.

Proposition 3.10. Let H ∈ Ω n-1
Ham (M ) be G-invariant, then ι(v p )H is G 0 p -invariant and it is a Hamiltonian form with respect to ι(v p )ω with Hamiltonian vector field v H .

Proof. The G 0 p -invariance of ι(v p )H is shown exactly as in Lemma 3.7. For the second statement, using Lemma 2.18 we compute d(ι(v p )H) = (-1) k ι(v p )dH = ι X H (ι(v p )ω).

Remark 3.11. Consider the case k = n -1. Then ι(v p )ω is a 2-form, and from Proposition 3.10 we recover the fact that ι(v p )H is a conserved quantity (a special case of Proposition 3.15 later on).

Example 3.19. Let N × G → N be a right action and assume that, in the set-up of the preceding proposition, the vector field w and the forms a and b be G-invariant. Then H (see eq. ( 4)) is invariant under the induced G-action on M = Λ k T * N . Assuming furthermore db = 0, we can choose a := ι w b. Specialize to k = 2 and N = G with the action by N × G → N, (n, g) → g -1 •n. Thus we have:

• for x ∈ g = T e G and g ∈ G is v x (g) = -(r g ) * (x), where r g (h) = h•g for h ∈ G. In particular, the generators v x of the action are right-invariant vector fields on G.

• w is a left-invariant vector field, i.e. it exists w ∈ g such that for g ∈ G, w(g) = (l g ) * ( w), where l g Denote by (f ) the co-momentum map recalled in Remark 3.18. For any x ∈ g we compute

(h) = g•h for h ∈ G, • b is a closed left-invariant 2
L w h f 1 (x) = ι v h x π * (L w b) = -π * (d(ι vx a)), so f 1 (x) being a strictly conserved quantity is equivalent to ι vx a being a constant function on N . Evaluating the function ι vx a = b(w, v x ) at g ∈ N = G one obtains -b( w, Ad g -1 (x)). (5)
It is clear that the function ( 5) is not constant in general. For instance, take G = SL 2 (R).

A basis for g := sl 2 (R) is

h = 1 0 0 -1 , e = 0 1 0 0 , f = 0 0 1 0 , and [h, e] = 2e, [h, f ] = -2f, [e, f ] = h.
So notably all elements of g are boundaries. The form b := e * ∧ f * ∈ Λ 2 g * is closed (actually exact) with respect to the Chevalley-Eilenberg differential. Taking w := f and x := h ∈ g = B 1 (g) one computes that the function [START_REF] Frégier | A cohomological framework for homotopy moment maps[END_REF] attains the value 2βδ at g = α β γ δ ∈ G, hence it is not a constant function on G. We conclude that for this choice of x ∈ B 1 (g), the form f 1 (x) is not strictly conserved.

Applications of conserved quantities

In §2 we saw that many conserved quantities exist on pre-n-plectic manifolds endowed with a co-momentum map. In this section we show some geometric consequences of the existence of conserved quantities on a manifold M , by looking at maps from a compact oriented manifold Σ into M . In most of our statements M does not need any additional geometric structure (but we specialize to the pre-n-plectic case e.g. in Proposition 4.9). In §4.1 we consider conserved quantities whose degree, as differential forms on M , equals dim(Σ). In §4.2 we extend some of the results to arbitrary degrees.

A general Kelvin's circulation theorem

Let M be a manifold and v ∈ X(M ) a vector field. Let Σ be a compact, oriented ddimensional manifold, and σ 0 : Σ → M a smooth map. We view Σ as a "membrane" in M , which evolves under the flow of the vector field, and want to find quantities which are unchanged under the evolution. The following proposition can be viewed as a general version of Kelvin's circulation theorem, as we explain in Remark 4.6 below.

Proposition 4.1. Let Σ be a compact, oriented d-dimensional manifold (possibly with boundary), v a vector field on M with flow φ t , and σ 0 : Σ → M a smooth map. Consider

σ t := φ t • σ 0 : Σ → M . If α ∈ Ω d (M ) is a differential form,
Hence by compactness of Σ,

d dt Σ σ * t α = Σ σ * t (L v α) = Σ d(σ * t γ) = 0,
where in the first equality we used Equation ( 6), in the second that L v α = dγ for some form γ, and in the last one Stokes theorem.

Remark 4.3. The sufficiency of Condition (ii) in Proposition 4.1 is not surprising. By assumption v preserves α up to an exact form, and by Stokes' theorem the contribution given by exact forms vanishes upon integration over Σ.

The following statement addresses a variation of condition (iii) in Proposition 4.1. We conclude that the above expression equals t • c([σ 0 ]).

We present an example for Proposition 4.1 and Proposition 4.4.

Example 4.5. Let M be a manifold, v a vector field, and α ∈ Ω d (M ). Take a map σ 0 : S d → M defined on the d-dimensional sphere, denote by σ t the composition of σ 0 with the time t flow φ t of v. The number S d (σ t ) * α is independent of the time parameter t if the following occurs: either i) L v α is exact, or ii) L v α is closed and σ 0 is homotopy equivalent to a constant map. This follows from Proposition 4.1 (ii) and (iii). Further, assuming that L v α is closed, one obtains a well-defined group homomorphism Remark 4.6 (Kelvin circulation theorem). A variant of Proposition 4.1 (ii) for a timedependent vector field v t and time-dependent differential form α t ∈ Ω d (M ) is the following: if L v t α t + d dt α t is exact and Σ has no boundary, then the number

π d (M, x) → R, [σ 0 ] → S d
Σ (σ t ) * (α t )
is independent of the time parameter t.

We mention this because the Kelvin circulation theorem in fluid mechanics can be understood as a special case of the above. Let v t = i v t i ∂ x i be a time-dependent vector field on R 3 , and

  the Hamiltonian vector field of some G-invariant Hamiltonian form H. Remark 3.4. If M is compact and simply connected, then H dim(M )-1 dR (M ) vanishes by Poincaré duality and the above result is applicable.

  -form, i.e. it exists a b ∈ Λ 2 g * which is closed under the Chevalley-Eilenberg differential (the dual of the Lie algebra homology differential) and b(g) = (l g -1 ) * ( b) = ((l g -1 ) * ) * ( b) for all g ∈ G.

Proposition 4 . 4 .

 44 Let Σ be a compact, oriented manifold without boundary of dimension d, v a vector field on M with flow φ t . If α ∈ Ω d (M ) is locally conserved, then for every fixed time t, one obtains a well-defined mapF t : [Σ, M ] → R, [σ 0 ] → Σ (σ t ) * α -Σ (σ 0 ) * α.Here [Σ, M ] denotes the set of smooth homotopy classes of maps from Σ to M , σ 0 : Σ → M denotes a smooth map and σ t := φ t • σ 0 : Σ → M . Further, the dependence on t is linear:F t [σ 0 ] = t • c([σ 0 ]) where c([σ 0 ]) := Σ (σ 0 ) * (L v α).Proof. We haveΣ (σ t ) * αv α) dswhere the last equality is obtained as in the proof of Proposition 4.1. Now recall that L v α is a closed form on M . Hence by Stokes' theorem the term in the square bracket depends only on the homotopy class of σ s , which agrees with the homotopy class of σ 0 since σ s = φ s • σ 0 .

(

  σ t ) * α -S d(σ 0 ) * α defined on the d-th homotopy group of M based at some point x, and where the dependence on t is linear. This follows from Proposition 4.4 and the following argument to show that the group homomorphism property. We denote the group multiplication of π d (M, x) by * . It is given by the following composition, where p denotes a distinguished point on the sphere:f * g : (S d , p) → (S d /S d-1 , p) = (S d ∨ S d , p) (f ∨g) -→ (M, x)Choosing appropriate representatives of the respective homotopy classes we may assume that f, g and f * g are smooth. Then for α ∈ Ω d (M ) we calculate:S d (f * g) * α = S d \{p} S d \{p} (f ∨ g) * α = S d f * α + S d g * α.

  Constructing preserved Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 3.2 Induced actions of isotropy subgroups . . . . . . . . . . . . . . . . . . . . . 3.3 Co-momentum maps for g ⊕ R . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Multisymplectic analogue of magnetic terms . . . . . . . . . . . . . . . . . .

	3 Examples and constructions
	3.1

Introduction 1 Conserved quantities in multisymplectic geometry 1.1 Multisymplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Conserved quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The algebraic structure of conserved quantities . . . . . . . . . . . . . . . . 2 Conserved quantities from homotopy co-momentum maps 2.1 Actions on multisymplectic manifolds . . . . . . . . . . . . . . . . . . . . . . 2.2 Conserved quantities from locally H-preserving actions . . . . . . . . . . . . 2.3 Conserved quantities from globally H-preserving actions . . . . . . . . . . . 2.4 Conserved quantities from strictly H-preserving actions . . . . . . . . . . . 2.5 The homological point of view . . . . . . . . . . . . . . . . . . . . . . . . . .

  [START_REF] Cantrijn | On the geometry of multisymplectic manifolds[END_REF] an example, announced in §2.4, exhibiting a globally conserved quantity (with respect to a strictly H-preserving action) that is not strictly conserved. Finally, §4 is devoted to applications: we present a general version of Kelvin's circulation theorem, and more generally we show that conserved quantities on a manifold M induce conserved quantities on spaces of maps into M .

	For a discussion of the relation between conserved quantities in multisymplectic geometry
	on one side and classical field theory on the other side, we refer to Schreiber [10, §1.2.11].
	Acknowledgements: M.Z. thanks Chris Rogers for constructive comments, especially
	during the first phase of the project, and Daniel Peralta Salas for making him aware of
	the Kelvin Circulation Theorem. M.Z. was partially supported by grants MTM2011-22612
	and ICMAT Severo Ochoa SEV-2011-0087 (Spain), Pesquisador Visitante Especial grant
	88881.030367/2013-01 (CAPES/Brazil), by IAP Dygest (Belgium), by the long term struc-
	tural funding Methusalem grant of the Flemish Government. We thank the IECL Metz-
	Nancy (UMR 7502, France), KU Leuven and Ruhr-University Bochum (via its Research
	School Plus, funded by Germany's Excellence Initiative [DFG GSC 98/3]) for making pos-
	sible mutual visits of the authors.

  Definition 2.7. Let g → X(M, ω), x → v x be an infinitesimal action. It is called (a) locally H-preserving if L vx H is closed for all x ∈ g. (b) globally H-preserving if L vx H is exact for all x ∈ g. (c) strictly H-preserving if L vx H = 0 for all x ∈ g.

	Remark 2.8. Usually a differential form would be called "preserved by an infinitesimal
	action" if condition (c) is fulfilled.

  is globally conserved by any Hamiltonian vector field v H of H. Proposition 2.14. If p ∈ B k (g) ⊂ Z k (g), then f k (p) is globally conserved by any Hamiltonian vector field v H of H.

  then the number

Σ

(σ t ) * α is independent of the time parameter t if one of the following conditions holds:

(i) α is strictly conserved by v, (ii) α is globally conserved by v and Σ has no boundary, (iii) α is locally conserved by v and there exists a compact, oriented manifold with boundary N such that Σ = ∂N and a map σ0 :

N → M with σ0 | ∂N = σ 0 . Remark 4.2. Since Σ is compact, there exists an ε = ε(σ 0 ) > 0 such that φ t is defined at least on (-ε, ε) × σ 0 (Σ) ⊂ R × M .

Obviously in (i) and (ii) we can consider |t| < ε = ε(σ 0 ). Mutatis mutandis we consider only |t| < ε(σ 0 ) in case (iii). Notice that if dim(M ) = d, or more generally if α is closed, then α is globally conserved. Proof. We only prove that condition (ii) suffices, for the other implications follow analogously. The diffeomorphisms φ t satisfy d dt (φ * t α) = φ * t (L v α). Precomposing with the pullback (σ 0 ) * we obtain d dt (σ * t α) = σ * t (L v α).

Proof. Let p ∈ Z k (g). We have A([p]) = [L vp H] = (-1) k [ι(v p )ι v H ω] using the definition of Lie derivative for multivector fields (see Remark 2.20). We can express this in terms of the co-momentum map using

Passing to the cohomology class finishes the proof. Lemma 2.29 has several consequences: Remark 2.30.

(1) The form f k (p) is locally conserved if p ∈ Z k (g) and globally conserved if p ∈ B k (g).

Hence we recover Proposition 2.12 and Proposition 2.14. (2) There is a canonical injective map J :

as follows immediately from the definitions. The map A factors as

for every k, where the first map is induced by f k multiplied by -ς(k). In particular, the map A takes values in the subspace J( C loc (H) C(H) ) of H dR (M ).

(3) If the action is strictly H-preserving, by Remark 2.28 [START_REF] Callies | Homotopy moment maps[END_REF], f k (p) is globally conserved for all p ∈ Z k (g). Hence we recover Proposition 2.21.

Examples and constructions

In the previous section we proved the existence of conserved quantities in the following set-up: G is a Lie group acting on a pre-n-plectic manifold (M, ω), H ∈ Ω n-1 Ham (M ) is locally, globally or strictly preserved, and (f ) is a co-momentum map . Here we give several constructions that assure the existence of a preserved Hamiltonian H and of a co-momentum map, in §3.1 and §3.2. Further, in §3.3 and §3.4 we make additional remarks on conserved quantities.

Constructing preserved Hamiltonians

In this subsection we consider a natural geometric situation in which the above machinery can be applied. We do not assume the existence of an invariant Hamiltonian form here, but we always assume a smooth action ϑ : M × G → M of a connected Lie group G on a pre-n-plectic manifold (M, ω) such that ϑ * g (ω) = ω for all g ∈ G. Lemma 3.1. Consider a connected compact Lie group G acting on the pre-n-plectic manifold (M, ω), and let H ∈ Ω n-1

Ham (M ) which is locally preserved by the action. Then there exists

Ham (M ) which is strictly preserved by the action and has the following property: any Hamiltonian vector field of H is also a Hamiltonian vector field of H.

Proof. Define H := G ϑ * g ( H)µ(dg), the average of H using the normalized Haar measure µ(dg) on G. Then H is strictly preserved by the action. Furthermore we have

Co-momentum maps for g ⊕ R

We extend the results of §2.4 under a non-degeneracy assumption for ω. We assume that (M, ω) is an n-plectic manifold, H ∈ Ω n-1 Ham (M ) and that (f ) : g → L ∞ (M, ω) is a comomentum for a strictly H-preserving infinitesimal action g → X(M, ω), x → v x . By Lemma 1.7 (i) the generators of the action commute with the Hamiltonian vector field v H of H, so the infinitesimal g-action on M extends to an action of the direct sum Lie algebra g :

In the sequel we will make use of the following Lemma several times. Recall that the differential ∂ was defined in Def. 2.2. Lemma 3.12. Let p ∈ Λ k g and q ∈ Λ l g. Then

where

Proof. It is sufficient to prove the assertion for monomials p = x 1 ∧ . . . ∧ x k and q = x k+1 ∧. . .∧x k+l . In that case ∂(p∧q) is given by a sum over indices i, j with 1 ≤ i < j ≤ k + l. Splitting it into sums over i < j ≤ k, k < i < j and i ≤ k < j proves the assertion.

Lemma 3.14. There is a canonical extension of (f ) to a co-momentum map ( f ) for the g-action, determined by

for all k ≥ 1 and x 1 , . . . , x k-1 ∈ g.

Proof.

We have to check that Equation (1) is satisfied. Without loss of generality assume

g, and notice that [x i , c] = 0 for all i implies that ∂(p⊗c) = (∂p)⊗c, by Lemma 3.12. Using the definition of fk , Equation (1) applied to p ⊗ c reads

Using ι v H ω = -dH we see that this equation is satisfied by Lemma 2.18, since H is ginvariant and using the identity ς(k)ς(k -1) = (-1) k .

Notice that, even when (f ) is equivariant, ( f ) is not equivariant in general. For instance,

Further, it can usually not be made equivariant by an averaging procedure since the group

If we assume that L v H H = 0, then the g-action strictly preserves the Hamiltonian H, hence we can apply Proposition 2.21 to the g-action and obtain globally conserved quantities for v H for all elements of Z k (g). As the latter is isomorphic to Z k (g) ⊕ (Z k-1 (g) ⊗ c R ), these globally conserved quantities are those we already know from Proposition 2.21, plus those arising from Z k-1 (g) ⊗ c R . Somewhat surprisingly, it turns out that the latter are globally conserved quantities even without the assumption L v H H = 0. This fact is not predicted by Prop. 2.12, which only ensures the existence of locally conserved quantities.

Proposition 3.15. Assume that (M, ω) is an n-plectic manifold, H ∈ Ω n-1 Ham (M ) and that (f ) : g → L ∞ (M, ω) is the co-momentum of a strictly H-preserving Lie algebra action. The fk (p ⊗ c) as in eq. ( 3) is a globally conserved quantity, for all p ∈ Z k-1 (g).

Proof. Because of Cartan's formula it suffices to show that ι v H d fk (p ⊗ c) is exact. We will show that it actually vanishes. By Lemma 3.14 and using eq. ( 1) we get

Applying Lemma 3.12 to the Lie algebra g we see that ∂(p ⊗ c) = 0. Further v c = v H , so by the skew-symmetry of ω we get ι v H ι(v p⊗c )ω = 0, which finishes the proof.

Multisymplectic analogue of magnetic terms

In this subsection we explain how to generalize the well-known magnetic term from symplectic geometry to the multisymplectic situation (compare [4, §7] for this construction) and provide the example announced in §2.4. Proposition 3.17. Let k ≥ 1 and N be a manifold, b ∈ Ω k (N ) and w a vector field on N , such that L w b = da for some a ∈ Ω k-1 (N ) (i.e. b is globally conserved by w). Denote the canonical lift of w to M = Λ k T * N by w h . Then w h is a Hamiltonian vector field on (M, ω = dθ + π * db), with the following Hamiltonian (k-1)-form:

Proof. Upon observing ι w h (π * b) = π * (ι w b) and consequently L w h (π * b) = π * (L w b), we have:

where in the third equality we used L w h θ = 0, as in the symplectic case.

In §2.4 we announced an example of a strictly H-preserving action on a multisymplectic manifold admitting a co-momentum (f ) such that, for some boundary x ∈ B 1 (g), f 1 (x) is a globally conserved quantity that is not strictly conserved. We now provide this example. use the standard metric on R 3 to obtain from v t the 1-form α t = i v t i dx i . One computes

is exact. By the above, it then follows that Σ (σ t ) * α t is independent of t.

Upon rewriting the exactness of ( 7) as (v t • ∇)v t + ∂u ∂t = -∇w, with ∇ the usual gradient in R 3 , we recognise the first of the isentropic Euler equations (see e.g. [3, page 15]). It is well-known that this equation implies the classical Kelvin circulation theorem ([3, page 21]) which is exactly the time-independence of Σ (σ * t )(α t ) in this case.

Under symmetry assumptions, our methods lead to other conserved quantities, as we now explain. In fluid mechanics, the time-dependent vector field v t is divergence free, i.e. L v t ω = 0 for all t, where ω = dx 1 ∧dx 2 ∧dx 3 is the standard volume form on R 3 . In the presence of a compact group of symmetries -i.e. of an action of a compact Lie group G on R 3 preserving ω and v t -it turns out by Corollary 3.3 that, for every fixed value of t, the vector field v t is the Hamiltonian vector field of a time dependent G-invariant Hamiltonian 1-form of (R 3 , ω). Further, ω is exact with G-invariant primitive, so that the action of G on (R 3 , ω) admits a co-momentum map [2, §8.1]. The latter, by virtue of Proposition 2.21, delivers further (time-independent) globally conserved quantities for v t , for each value of t. One can then apply the above variant of Proposition 4.1 (ii) to the time-dependent vector field v t and to the newly obtained globally conserved quantities.

Transgression of conserved quantities

Proposition 4.1 (ii) fits in the following framework. Let Σ be a compact, oriented manifold (without boundary) and M a manifold. Given a vector field v on M , there is a naturally associated vector field v on M Σ = C ∞ (Σ, M ), the space of smooth maps from Σ to M . It is given as follows:

for all σ ∈ M Σ . Notice that, denoting by φ t the flow of v on M , the flow of v maps σ ∈ M Σ to φ t • σ ∈ M Σ . Similarly, associated to a differential form on M there is a differential form on M Σ of lower degree. It is defined by the transgression map

where ev : Σ × M Σ → M is the evaluation map and Σ denotes the integration along the fiber (cf. eg. Proof. The transgression map commutes with de Rham differentials, and furthermore we have (ι v α) = ι v α . Therefore it commutes with Lie derivatives in the following sense:

The other cases follow similarly.

where in the second equality we used that ev| Σ×{σ} = σ.

Now we specialize to a pre-n-plectic manifold (M, ω) together with a vector field v H which is Hamiltonian for some H ∈ Ω n-1 Ham (M ). Notice that (v H ) is a Hamiltonian vector field of H on (M Σ , ω ), as follows from

A special case of Proposition 4.7 reads: Proposition 4.9. Consider a pre-n-plectic manifold (M, ω) together with a vector field v H which is Hamiltonian for some H ∈ Ω n-1

Ham (M ). Let Σ be a compact, oriented manifold (without boundary) of dimension d. If α ∈ Ω k (M ) is a globally conserved quantity for v H , then α ∈ Ω k-d (M Σ ) is a globally conserved quantity for (v H ) , i.e. L (v H ) α is an exact form.

Remark 4.10. Consider a G-action on (M, ω) for which H is strictly preserved. The Gaction on M gives rise to a G-action on M Σ = C ∞ (Σ, M ), simply given by (g•σ)(p) := g•σ(p) for all σ ∈ M Σ and p ∈ Σ. It can be checked that for a infinitesimal generator v x of the action on M (x ∈ g), the corresponding infinitesimal generator of the action on Σ is (v x ) .

Hence the lifted G-action preserves ω and H . In [2, §11] and [5, §6] it is shown that a co-momentum map for the action of G on (M, ω) transgresses to a co-momentum map for the action on (M Σ , ω ). This is consistent with the fact, that firstly, certain components of co-momentum maps are globally conserved quantities (Proposition 2.21) and, secondly, globally conserved quantities transgress to give globally conserved quantities (Proposition 4.9).


Anthony Zubiaur

Tristan Asset

Fabien Deschamps

Laetitia Dubau

Frédéric Maillard

Pt and hollow PtCo nanoparticles supported on carbon xerogels for PEM fuel cell catalyst layers

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Pt and hollow PtCo nanoparticles supported on carbon xerogels for PEM fuel cell catalyst layers

Anthony Zubiaur 1 , Tristan Asset 1,2 , Fabien Deschamps 1 , Laetitia Dubau 2 , Frédéric Maillard 2 , Nathalie Job 1 1 University of Liège -Department of Chemical Engineering, Liège, Belgium 2 Université Grenoble Alpes, LEPMI, Grenoble, France Carbon black remains the most widespread support to prepare catalytic layers of Proton Exchange Membrane Fuel Cells (PEMFCs). The composition and the pore texture of these carbons are key variables to the device performance, through (i) the ability of the support to disperse metals, (ii) the electrolyte/carbon contact and (iii) the reactant/product mass transport within the porosity [1]. However, in the case of carbon black, the pore texture of the final electrode is not controllable since it largely depends on its processing, and not on the material itself.

Replacing these conventional carbons by nanostructured materials, such as carbon xerogels (CXs), allows for improving their performance and understanding the impact of the pore texture on the catalytic layer performance [2]. CXs are very pure carbons with tunable meso-macropore sizes ranging from a few nm to a few µm, with pore volumes as high as 3 cm³/g and large specific surface area (~600 m²/g). Those texture-tailored supports were used to synthesize Pt catalysts with excellent dispersion (particle size ~3-4 nm, Fig. 1a) and high mass fractions (min. 40% wt.) [3], which is mandatory to manufacture thin electrodes.

Although the final electrodes display excellent intrinsic catalytic activities with interesting mass transport properties compared to those elaborated from supported catalysts on carbon black, Pt nanoparticles are not stable under operating conditions. This is why the same type of nanostructured carbon were used to synthesize supported hollow PtCo nanoparticles (Fig. 1b) [4]. The addition of Co leads to the Pt lattice contraction, and thus to much higher catalytic activity while the hollow structure allows for relatively good metal utilization ratio despite the large particle size. PtCo/CX catalysts with 10 wt.% Pt were prepared; residual Co content ranged from 0.3 to 4.6 wt.%. Comparison between commercial Pt/carbon black and hollow PtCo/CX catalysts in MEA (H 2 /O 2 ); the Pt content is constant (0.033 mg Pt /cm² electrode ), and cells were cycled up to 30,000 times between 0.6 and 1.0 V to accelerate ageing [4].

These catalysts were thoroughly characterized using physico-chemical and electrochemical techniques, including Accelerated Stress Tests (ASTs) in Membrane-Electrode Assembly configuration, i.e. in complete fuel cell using a classical Pt/carbon black at the anode, and PtCo/CX at the cathode. ASTs consists in sharp voltage increase and decrease between 0.6 and 1.0 V to accelerate the catalyst ageing. Results show that the hollow PtCo/CX catalysts display much better stability than pure Pt/CX and Pt/carbon black samples. Indeed, while the performances of Pt nanoparticles continually decrease with ageing time, the current-voltage curves obtained with PtCo/CX hollow particles fully stabilize: no difference is observed between 10,000 and 30,000 ageing cycles. The reasons for enhanced stability probably lies in the large size and regular shape of the hollow nanoparticles. Current research goes towards the increase of the PtCo/CX metal loading so as to reach higher current densities.

Figure 1 .

 1 Figure 1. (a) TEM picture of a Pt/CX catalyst (40% wt.) prepared by impregnationreduction process [3]. (b) TEM picture of hollow PtCo/C nanoparticles [4]. (c)

Acknowledgements

NJ thanks the Walloon Region for funding (project HYLIFE, grant n° 1410135).


Mohamed Elbaa
email: mohamed.el-baa@univ-lorraine.fr

Mohamed Halit
email: m.halit@mail.lagh-univ.dz

Kévin Berger
email: kevin.berger@univ-lorraine.fr

El Hadj

Bruno Douine
email: bruno.douine@univ-lorraine.fr

Analytical modeling of a coil in a ferromagnetic circuit including a superconductor pellet

Keywords: Analytical Model, Electromagnets, Inductance, Superconducting pellet

In the design phase of the inductor required for the magnetization of High Temperature Superconductor (HTS) bulks, modeling and simulation are necessary for the design of this inductor. In this paper we have developed an analytical model to calculate the inductance for a coil of a ferromagnetic circuit with gap including a superconducting pellet. This model is based on the determination of the magnetic vector potential from solving the Laplace's and Poisson's equations in different regions of interest, using the separation of variables method in which the Cartesian coordinates are used. The boundary and continuity conditions between regions are considered to determine the global solution. This analytical resolution is carried out using a computation code developed under MATLAB. The results obtained are compared with those obtained by a numerical simulation based on the finite element method implemented under COMSOL. A remarkable concordance is observed between both approaches.

I. INTRODUCTION

Superconductivity has been the subject of several theories and research works to unravel the mystery of this phenomenon. The magnetization of superconducting pellets is an interesting way to investigate the trapped field in superconducting machines. Several researches have been devoted to studying the possibility of taking advantage of the significant magnetic induction produced by magnetized pellets for applications in Electrical Engineering [START_REF] Berger | Magnetization and Demagnetization Studies of an HTS Bulk in an Iron Core[END_REF], [START_REF] Berger | 3-D Modeling of Coils for Pulsed Field Magnetization of HTS Bulk Pellets in an Electrical Machine[END_REF]. The most widely used method for magnetizing superconductors bulk's is to apply a pulsed magnetic field (PFM method). Since a long time, analytical models have been proposed to study the behavior of the magnetic field created by ironless circular coils [START_REF] Urankar | Vector potential and magnetic field of current-carrying fi-nite arc segment in analytical form, Part III: Exact computation for rec-tangular cross section[END_REF], [START_REF] Garrett | Calculation of Fields, Forces, and Mutual Inductances of Current Systems by Elliptic Integrals[END_REF] and more recently for an axisymmetric coil in a ferromagnetic circuit [START_REF] Elbaa | Analytical Modeling of an Inductor in a Magnetic Circuit for Pulsed Field Magnetization of HTS Bulks[END_REF].

In this paper, we extend a previous work [START_REF] Elbaa | Analytical Modeling of an Inductor in a Magnetic Circuit for Pulsed Field Magnetization of HTS Bulks[END_REF] by adding a superconducting pellet in the gap area where a Cartesian coordinates are used to make the problem close to the real. The analytical solutions obtained using MATLAB are compared and validated with those obtained numerically through a finite element method implemented under COMSOL.

II. PROBLEM DESCRIPTION AND ASSUMPTIONS

The geometry of the studied problem is shown on Fig. 1. The regions I, III, IV, V, and VI correspond to the air and the region II corresponds to a circular coil of N turns with rectangular cross-section. The whole system was limited so that     57 0, and 0,

y y x x 
. The superconducting pellet is modeled as a perfect diamagnetic, which corresponds to a zero magnetic vector potential in the pellet and on its edges. However, the ferromagnetic circuit is modeled by an infinite permeability, this implies that the magnetic field is not calculated inside the ferromagnetic circuit.

i J i II A i I III IV V VI        (1)
Then, the expressions of the magnetic vector potential in the different regions can be expressed as: , cos

       
n n y IV n IV n y IV n n A y a e A x y x x A be                    (5)         2 2 2 1 , 2 sinh cos V V n n n n A x y a y x x      (6)         2 3 4 1 , 2 sinh sin VI VI n n n n A x y a y x x      (7)
where n is a positive integer, and a are the integration constants. The relations between those integration constants are determined by applying the interface conditions between the different regions. Finally, we have to solve here a system of nine linear equations with nine unknowns by rewriting this system in matrix and vectors format.

        2 1 6 2 / cos cos , n n n J J n x x           7 5 2 12 32 21 / , / , , 2 n n n n n x n x x xx               

III. RESULTS AND DISCUSSION

Fig. 3 shows the radial and the axial components of the magnetic flux density along the x-axis in Region III obtained with analytical and numerical models where a true concordance is observed between these two approaches.

IV. CONCLUSION

In this work and by using a Cartesian coordinates, we have demonstrate that results provide by the analytical model show a very good match with numerical ones.

Fig. 1 .

 1 Fig. 1. Geometry of the studied problem: Open magnetic circuit with an air gap of a length e, where a superconducting pellet is implemented. The interface conditions are then used to obtain the tangential component of magnetic field to be null. The homogeneous Dirichlet boundary conditions are considered in regions I, II, III, V and VI. However, homogeneous Neumann boundary conditions are used in regions IV, V and VI. A magnetic vector potential formulation in Cartesian coordinate is used in this problem, where only the zcomponent exists and it depends only on the x and y coordinates.

  A01 et A02 are integration constants due to the Neumann boundary conditions in region IV. It can be determined by applying the condition of continuity of the magnetic vector potential at the interfaces between regions as

Fig. 2 .

 2 Fig.2. Comparison of the distributions of the magnetic vector potential obtained from the analytical calculation (on the left) and the numerical calculation (on the right).

Fig. 3 .

 3 Fig.3. Bx and By components of the magnetic flux density along the x-axis in the Region III at y = (y1+y2)/2.