Mathematical Methods and Techniques in Engineering and Environmental Science

Functor and natural operators on symplectic manifolds

CONSTANTIN PĂTRĂCOIU Faculty of Engineering and Management of Technological Systems, Drobeta Turnu Severin University of Craiova Drobeta Turnu Severin, Str. Călugăreni, No.1 ROMANIA [email protected] http://www.imst.ro

Abstract. Symplectic manifolds arise naturally in abstract formulations of classical Mechanics because the phase–spaces in is the cotangent bundle of configuration manifolds equipped with a symplectic structure. The natural and operators described in this paper can be helpful both for a unified description of specific proprieties of symplectic manifolds and for finding links to various fields of geometric objects with applications in Hamiltonian mechanics.

Key-words: Functors, Natural operators, Symplectic manifolds, Complex structure, Hamiltonian.

1. Introduction. diffeomorphism f : M → N , a morphism The geometrics objects (like as vectors, covectors, T*f : T*M→ T*N , which covers f , where −1 tensors, metrics, e.t.c.) on a smooth manifold M, x = x :)*)(()*( x * → xf )( * NTMTfTfT are the elements of the total spaces of a vector So, the cotangent , )(:* → VBmManT and bundles with base M. The fields of such geometrics objects are the in corresponding vector more general ΛkT * : Man(m) → VB , are the bundles. bundle functors from Man(m) to VB, where Man(m) By example, the vectors on a smooth manifold M (subcategory of Man) is the category of m are the elements of total space of vector bundles dimensional smooth manifolds, the morphisms of this

(TM ,π M , M); the covectors on M are the elements category are local diffeomorphisms between of total space of vector bundles (T*M , pM , M). A manifolds. Recall that an operator is a rule field of vectors on the manifold M is a section of transforming the sections of a fiber bundles the vector bundles (TM ,πM , M); a field of M MpE ),,( into sections of another covectors on the manifold M is a section of the M ' MpE )',,'( . Regarding to the bundle functors vector bundles (T*M , pM , M). A k or an exterior form of degree k or a kform is a Λ T * , the d, transforms sections k k +1 section of the vector bundle ( ΛkT * M , p k ,M) of Λ T * M into sections of Λ T * M for every M manifold M and d commutes with local In fact, the vector bundle (TM ,π M , M) is the value diffeomorphisms. So, d is a natural operator from the of the functor, T: Man→VB, where Man is the functor k into functor k+1 and written: category of smooth manifolds(the morphisms of Λ T * Λ T * k k+1 this category is smooth maps between manifolds) d : Λ T *  Λ T * , k ∈N and k k +1 and VB is the category of vector bundles, the dM : Λ T * M→ Λ T * M for any ∈ mManM )( morphisms of this category are morphisms of vector bundles [2]. So, the tangent functor T , 2.Natural operator from tangent to associates to each manifold M , the cotangent functors on symplectic (TM ,π M , M) and to each smooth map f:M→N , a vector bundle morphism Tf: TM→TN , which manifolds. covers f. Recall that the couple (M,ω ) is an almost symplectic In the case of the cotangent functor T* we can not manifold if M is a smooth manifold and ω is an use the whole category Man, we use only almost symplectic form i.e. a nondegenerate 2form on manifolds of the same dimension the manifold M. The cotangent functor )(:* → VBmManT If an almost symplectic form ω∈ 2 (M) is closed, ω associates to each mdimensional manifold M , the is called symplectic form and the couple (M,ω ) is cotangent bundle (T*M ,π*M , M) and to each local called .

ISBN: 978-1-61804-046-6 162 Mathematical Methods and Techniques in Engineering and Environmental Science

If (M,ω ) is a symplectic manifold. Then, each So, Φ : T  T* , is an operator from T to T* . tangent space ( Tx M , ωx ) is symplectic vector Moreover, Φ is a regular operator because every space and the manifold M is necessarily of even smoothly parameterized family of vector fields is dimension. transformed into a smoothly family of covector fields. If 2n is the dimension of manifold M, the product Let F and G be two bundle functors over manifolds, n ω = ω ∧ ω ∧ ... ∧ ω (nfactors) never vanishes, M a smooth manifold, FM Mand G the fiber bundle thus M is orientable and any symplectic corresponding to M ; Γ(FM) and Γ(GM), the set of diffeomorphism preserve the volume. smooth section of this fiber bundle. By Darboux’s Theorem such a 2ndimensional 2 nn Recall that a natural operator A : F G is a system of manifold looks locally like ≅ CR with the regular operators n AM : Γ(FM) → Γ(GM) satisfying following standard symplectic form ω0 ∑ i ∧= dydx i , i=1 conditions: (i) For every section s∈Γ(FM) and every where 21 n 21 yyyxxx n ),...,,,,...,,( are coordinates 2 nn isomorphism f : M → N in category of manifold it in . ≅ CR . holds So symplectic manifolds, in contrast to A (Ff s f 1) = Gf A s f –1 Riemannian manifolds, have no local invariants. N o o o M o A between 2ndimensional (ii) AU (s  U ) = (AM s) U for every section s ∈ Γ(FM) symplectic manifolds M ω ),( and M ω ),( is and every open submanifold U of M. 11 22 a diffeomorphism : 1 → MMf 2 satisfying the Let be T* and, T the restriction of tangent and condition: f * = ωω 12 . We denote Simp(2n), the cotangent functors to category Symp(2n). category of 2ndimensional symplectic manifolds , Proposition. Φ : T T* is a natural operator between the morphisms in this category are the the two bundle functors T and T* . . Proof. Φ is a system of regular operators The Simp(2n) is a subcategory of Man(2n). 1 ΦM : X (M) → )(M , M ∈ Ob Symp(2n) We will consider the restriction of tangent and Let be M , N∈ Ob Symp(2n) , ω and ω the cotangent functors to category Symp(2n). M N Let (TM ,π , M) be the tangent bundle and corresponding sympletic forms. M The condition (i) from previously definition is (T*M ,π*M , M) the cotangent bundle of -1 –1 symplectic manifold M . The manifold M is ΦN( Tf o X o f ) = T*f oΦM X of endower with a symplectic structure i.e. a for every vector field X∈ X (M) and for every nondegenerate closed 2form ω∈ 2 (M). symplectomorphism f : M → N Then, each tangent space ( Tx M , ωx ) , x∈ M is Let be x∈M, f(x) = y∈N, Z∈ X (N) symplectic . –1 [ΦN( Tf o X o f ] (Z) y = ω N y(Txf Xx , Zy ) For each x∈ M we can define the map –1 [T*f oΦM X of ](Z)y ΦMx : Tx *M →T x M , = T*f(i ω M x)(Zy) . X x Xx → ΦMx(Xx) =i ω x =ω x(Xx , ) X x 1 – = ω M x(Xx , Tf Zy) Since ωx is nondegenerate this map is an 1– = f*(ω N y) (Xx , Tyf Zy) isomorphism between the tangent space Tx M and 1– T*x M. = ω N y(Txf Xx , Txf (Tyf )Zy )

Then, the map ΦM : T *M→T M , Φ/TxM =ΦMx , = ω N y(Txf Xx , Zy ) ∀ x∈M is an isomorphism of tangent fiber In the previous calculus we have used the equality bundles TM and cotangent fiber bundle T*M. ω M =*f ω N due the fact that f is a Let X (M) be the set of vector fields of M (the symplectomorphism. 1 sections of tangent bundle ) and (M) the set of The condition (ii) is satisfied because the geometric 1forms of M (the sections of cotangent bundle objects implied in definition of Φ do not depend on (T*M ,π*M , M)). the changes of coordinates. There is a onetoone correspondence between vector fields and 1forms of manifold M , A vector field X∈ X (M) is called symplectic if ω 1 iX given by the map ΦM :) X (M → ( M) , is closed.

ΦM (X) = Φ o X = iX ω.

ISBN: 978-1-61804-046-6 163 Mathematical Methods and Techniques in Engineering and Environmental Science

If is a vector field and the Lie and linear and X∈ X (M) L X →∈ JMx x :xx → x MTMTJ derivative along , the vector field is 2 X X∈X (M) x −= IdJ . symplectic if and only if Indeed, we L X ω =0. The almost complex structure J on the manifold M is know that Because is L X =iX o d + do iX . ω tamed by the symplectic form ω if ω(X,JX)>0, closed we have dω =0 . But, X is symplectic if ∀X∈ T(M) {0}; if moreover ω is Jinvariant, J is and only if i ω is closed i.e. if and only if said to be calibrated. We know that any symplectic X manifold have a lot a almost complex structure, the d( ω) = 0 iX space of almost complex structures on a given

L X ω = (iX o d) ω+ (do iX )ω symplectic manifold (M,ω) which are tamed (resp.

= iX (d ω)+d(iX ω) calibrated) by ω is nonempty and contractible(in = d(i ω) = 0 particular these spaces are connected). X Let J be an almost complex structure on the manifold if and only if X is symplectic. We denote the space of symplectic vector fields by M, tamed by the symplectic form ω. We define the map X (M ,ω ) g(X,Y) = ω(X,JY) - ω(JX,Y) , ∀X,Y∈ T(M). Proposition. Let be X∈ X (M) and L the Lie X Because bilinearity of ω and linearity of J follow that derivative The vector field X is symplectic vector g is a bilinear map. field ( X ∈X (M ,ω )) if and only if the Lie However g has the following properties: derivative commute with natural operator L X Φ : g(X,X) = ω(X,JX) ω(JX,X) T  T* i.e. if and only if the diagram = 2ω(X,JX)>0, ∀X∈ T(M){0}; Φ 1 2 2 X (M ) M → (M) g(JX,JY) = ω(JX,J Y ) ω(J X,JY) = ω(JX,-Y) - ω(X,JY) L X ↑ ↑ L X Φ 1 = ω(JX,Y) + ω(X,JY) X (M) → M (M) is a commutative one. = g(X,Y), ∀X,Y∈ T(M); 2 2 Proof. g(Y,X) = g(JY,JX) =ω(JY,J X) ω(J Y,JX ) = ω(JY,-X) - ω(Y,JX) (Φ M o L X )(Y) = Φ M (L X (Y)) =ω(X,JY) ω(JX,Y) = Φ M,Y ( [X ] ) =g(X,Y), ∀X,Y∈ T(M). = i ω YX ],[ Then, g is a Jinvariant Riemannian metric.

=( L X o iY - iY o L X )ω Let M be a symplectic 2ndimensional manifold and 2 n = L X (iY ω) - iY (L X ω). ω ∈ ( M) the symplectic form. Let ω be the If the vector field X is symplectic L ω = 0 then volume canonic form on M, J an almost complex X structure on the manifold M tamed by the symplectic (Φ M L )(Y) =L (i ω) o X X Y form ω and the Jinvariant Riemannian metric = L ( Φ ) (Y) X M g(X,Y) = ω(X,JY) ω(JX,Y). = (L Φ .)( Y) X o M Let F(M) be the set of real functions defined on M. So the equality holds. We can define the map Conversely, if the diagram commute k 2n-k . L X ω = 0 F: (M)× (M) → F(M) 0 ≤ k ≤ 2n. and X is symplectic vector field. (α,β)∈k(M)×2n-k (M) → F(α ,β) = s∈ o(M) such that α ∧β =sω n. 3. Codiferential operator and De The real function s is well defined because the space Rham laplacian. of 2nforms is 1dimensional. Like for Riemannian geometry we define the De So, F(α ∧β)(x) is a real number such that Rham laplacian (Hodge laplacian or Beltrami (α ∧β)(x) = F(α ∧β)(x)ω n(, xfor) any two kforms operator). Recall that J is an almost complex α , β and for any point x ∈ M. structure on a manifold M if J is a section of Proposition. For any nonnegative integer k≤ 2n, there End(TM) such that J2 d=, i.e.I J is a smooth field k 2n-k is an isomorphism ΨM : (→M) .( M) of complex structures on the tangent spaces, i.e. k 2n-k Proof. The map f1: (→M) ( (M))*

ISBN: 978-1-61804-046-6 164 Mathematical Methods and Techniques in Engineering and Environmental Science

k . 2n-k α ∈ ( M) → f1(α ) =F(α , )∈( ( M))* is If M is compact, (α ,β ) =def .= Ψ∧ M βα )( = an isomorphism. ∫M There is an isomorphism () n 2n-k 2n-k ), >><< ωβα is nondegenerate symmetric f2: (→M) ( (M))* ∫M k determined by the Riemannian metric g. bilinear form on (M) and −1 Then, Ψ =f 2 f : k( M) → 2n-k(M) is the M o 1 (ΨM α , ΨM β) = M α ΨΨ∧Ψ MM β ))(()( ∫M isomorphism. k(2n-k) Let ( U,u) be a local coordinate chart on M, =(1) M )( ∧Ψ βα ∫M u : x∈U→ u(x) = (x1,x2, … ,xn, y1,y2, … 2n k(2n-k)+k(2n-k) =(1) Ψ∧ M αβ )( ,yn)∈R such that the symplectic form ∫M n

ω  = ∧ dydx . Then, = Ψ∧ M βα )( =(α ,β ) . U ∑ i i ∫M i=1 k nn − )1( Thus, ( . , .) is a scalar product on (M) invariant n 2 ω = − )1( n! dx 1 ∧ dy 1 ∧ dx 2 ∧ to ΨM . k k −1 dy 2 ∧…dx n ∧ dy n . The operator δ: Λ T *  Λ T * is an adjoint k −12 2k operator for the exterior differential operator i.e. We denote e = dx , e = dy ; d , k k (α , d β ) = (δ α , β ) for any forms α ∈ k(M) ; k=1,2,…,n. For any positive integers M M β∈ k-1(M). Because ω is closed form, 1 21 sss k ≤<<<≤ 2... n , n-1 n-1 n-2 ΨM (ω)=ω , dω (= n1)dω ∧ω and we 1 21 ... 2 −kn ≤<<<≤ 2nttt such that have δMω = 0 ≠ ts ji ,∀(i,j)∈{1,2,…,k}×{1,2,…,2n-k} we The De Rham laplacian (Hodge laplacian) will be: 2 s1 s2 sk Θ =(d+δ) =dδ+δd. have ΨM(e ∧ e ∧…∧ e ) t The De Rham laplacian is self adjoin for ( . , .) i.e. = ± n! e t1 ∧ e t2 ∧…∧ e 2 −kn , ± is the sign of the permutation (α , Θβ) = (Θα ,β). Summarizing, if ΛkT * is the natural functor on the ( 21 k ,...,,,,...,, tttsss 221 −kn ). k category of symplectic manifold to the category of If α ∈ ,( M then,) ΨM (α) is the unique 2n-k vector bundles, then: form such that ∀ X1 , X2 ,…, X2n-kM∈ T . is natural operator from the functor k to n 1 2 d Λ T * ΨM (α)(X1 , X2 ,…, X2n-k ) ω = α ∧θ ∧θ the functor Λk+1T * and write d : ΛkT *  Λk+1T * ; ∧…∧θ 2n-k, where θss , s =1,2,…,2n-k is the 1 δ is natural operator from the functor Λk+1T * to forms corresponding to Xi , i =1,2,…,2n-k, via k k+1 k induced by musical isomorphism. the functor Λ T * and write δ : Λ T *  Λ T * ; k Evidently, if β = fω n we have Θ is natural operator from the functor Λ T * to n n k k k ΨM (β)=f ; Ψ (1) = ω , ΨM (ω )=1, the functor Λ T * and write Θ : Λ T *  Λ T * . M k k 2n-k We call the form α ∈ ( M) harmonic if ΘMα =0 ΨM (ω ) = ω . k (2n-k)k i.e. if and only if this form is closed (dMα =0) and For any α ∈ ( M) , ΨM (ΨM (α ))=(1) α (2 n-k)k coclosed (δMα =0). ⇔ΨM ΨM = (1) Id. o From precedent relations hold that the symplectic Then, ΨM is an invertible and 1 (2 n-k)k form ω is harmonic. ΨM =(1) ΨM. If the manifold M is compact, the Hodge theorem With the help of , we can define the operator: k ΨM hold: For any form α ∈ ( M) , k<2n there is three k k −1 δ: Λ T *  Λ T * , unique forms: k k-1 k-1 k k+1 δM : (→M) (M) β∈ (M) , γ∈ (M) , σ∈ (M) such that k -1 δM α =(1) ΨM ( d(ΨM (α))) α =d M β + γ +δMσ, with γ harmonic form. k k =ΨM ( d(ΨM (α))),∀α ∈ ,( M and) the Then for any form α ∈ (M) , k<2n there is the fiberwise scalar product forms β∈ k-1(M) and σ∈ k+1(M) such that << , >>: k(M)× k (→M) o (M) δ M α = δMd M β and d M α = d M δMσ. such that if α ,β∈ k (M), n We remark that the kform α=d M β + γ +δMσ is α ∧ ΨM (β)=β ∧ΨM (α)=<<α ,β >>ω .

ISBN: 978-1-61804-046-6 165 Mathematical Methods and Techniques in Engineering and Environmental Science

closed if and only if the form δ M σ is closed and pointed convex cone. So, in tangent bundle of a the k- form α is exact if and only if there is a k symplectic manifold we have natural fields of cones. By means of natural operator Φ : T  T* between form σk such that α = d M δMσ k. The vector field X on the manifold M is a two bundle functors T and T* , we can extend this classification of vector fields to 1 forms and allow to symplectic vector field if and only if iXω = d M f o define fields of cones on a corresponding manifold, to + γ1 +δMσ2, with f∈ ( M) , γ1 harmonic 1 associate fields of cones on the ktangent bundle, to 1 form, σ2∈ (M) and the form δMσ2 is closed. define natural fields of cones on the kcotangent The vector field X on the manifold M is a bundle, to define natural, positive and monotone if iXω is exact i.e. there operators between the ktangent and the kcotangent is a smooth function : MH → R such that bundle and . iXω = dH. The function H is called Hamiltonian function of X. 4. Conclusion. So, The vector field X on the manifold M is a Symplectic manifolds are special cases of a Poisson Hamiltonian vector field if and only if there is a manifold, arise naturally in abstract formulations of 1form σ such that: classical Mechanics. So, the study of symplectic 1 manifolds is motivating because the phase–spaces in i ω =d δ σ . Then, δ σ :M→R , is a X M M 1 M 1 Hamiltonian mechanics is cotangent bundle of Hamiltonian function of X. configuration manifolds (the set of all possible If 21 n 21 pppqqq n ),...,,,,...,,( are local coordinates configurations of a ), equipped with of manifold M and symplectic form in this local a symplectic structure. n The natural functors and operators described in this coordinates is ∑ ∧ pdqd iMiM and paper can be helpful both for a unified description of i=1 specific proprieties of symplectic manifolds and for H = M σδ 1 : M → R is the Hamiltonian function finding links to various fields of geometric objects of X, then with applications in Hamiltonian mechanics.

n  ∂ ∂ ∂ σδσδ ∂   M 1 M 1  References: X H = ∑ −  i=1  ∂ ∂ ii ∂ ∂pqqp ii  The curve (q(t), p(t)) is an integral curve for XH if 1. Arfken B.G., Weber J. H., Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, ∂ M σδ 1 ∂ M σδ 1 &i tq = & i tp )(,)( −= (Hamilton 2005). ∂pi ∂qi 2. Kolar I., Michor P. W. , Slovak J., Natural equations) operations in , Electronic edition. Remark. The natural functors and operators can be Originally published by SpringerVerlag, Berlin used to find new properties of geometric objects Heidelberg 1993, ISBN 3540562354. based on some already known. By example, using 3. McDuff D., Salamon D., Introduction to a semi–Riemannian metric g on manifold M , i.e. a Symplectic Topology, second ed., Oxford smooth symmetric tensor–field of type (0, 2) which Mathematical Monographs, Clarendon Press, Oxford assigns to each point ∈ Mx a nondegenerate University Press, New York, 1998. 4. Pătrăcoiu, C.; Titirez, A.M. Fields of cones on a inner product gx on the tangent space x MT of symplectic manifold, An. Univ. Timi»soara, Seria signature we can define the energy of the (k, r), matematica, vol. XLI, f2, 2003, 6978. vector field X∈ T(M): 5. Pătrăcoiu, C.; Field of cones on (k,r)-covelocities 1 vector bundle on a manifold. Balkan J. Geom. Appl. 3, : Mf → R, )( = XXgxf ),( . So, the 2 xxx No.1, 97102 (1998). MSC2000. vector field X is: 6. Papuc, D.I. ; Field of cones and positive operators time–like, if f < 0 ; on vector bundle, An. Univ.Timisoara, Seria matematica, vol. XXX, f1, 1992, 3958. nonspacelike or causal, if f ≤ 0 ; 7.P apuc, D.I. The geometry of a vector bundle null or lightlike, if f = 0 ; endowed with a cone field, Proceedings of the 3rd space–like, if f > 0 . International Workshop on Differential Geometry and The set of timelike vector fields and the set of its applications, Sibiu, vol. V, 1997, 314321. spacelike vector fields are the convex cone. The 8. M. Puta, Hamiltonian Mechanical Systems and set of nonspacelike or causal vector fields is a Geometric Quantization, Kluwer 1993

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