Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism

0.25 MB·English

by  José F. Cariñena

Noncommutative Classical Dynamics on Velocity Phase
Space and Souriau Formalism
Jose´ F. Carin˜ena∗, He´ctor Figueroa† and Partha Guha‡§
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∗Departamento de F´ısica Teo´rica, Universidad de Zaragoza,
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n 50009 Zaragoza, Spain
a
J
4 †Departamento de Matema´ticas, Universidad de Costa Rica,
2
2060 San Pedro, Costa Rica

]
h
p ‡ Institut des Hautes E´tudes Scientifiques
-
h Le Bois-Marie 35, route de Chartres 91440 Bures-sur-Yvette France
t
a
m
§ S.N. Bose National Centre for Basic Sciences
[

2 JD Block, Sector-3, Salt Lake
v
Calcutta-700098, India
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9
January 27, 2015
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0 Abstract
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We consider Feynman-Dyson’s proof of Maxwell’s equations using the Jacobi identities
1
: on the velocity phase space. In this paper we generalize the Feynman-Dyson’s scheme by
v
i incorporating the non-commutativity between various spatial coordinates along with the ve-
X
locity coordinates. This allows us to study a generalized class of Hamiltonian systems. We
r
a explore various dynamical flowsassociated to theSouriau form associated tothis generalized
Feynman-Dyson’s scheme. Moreover, usingtheSouriauformweshowthatthesenewclasses
ofgeneralized systemsarevolumepreserving mechanical systems.
Keywords: Feynman problem, Souriau form, noncommutativity,generalized Hamiltonian dynam-
ics.
PACS numbers: 11.10.Nx,02.40.Yy,45.20.Jj.
1 Introduction
The study of exoticparticle models with non-commutativeposition coordinates was started in the
lastdecade. Thereareseveralphysicalphenomenaappearingincondensedmatterphysics,namely
1
semiclassical Bloch electron phenomena, fractional quantum Hall effect, double special relativity
models, etc., that exhibit such feature. All these models share the somewhat unusual feature that
thePoissonbracketsoftheplanarcoordinatesdonotvanish. Thisclassofdynamicalstructureshas
appeared in geometric mechanics and geometric control theory [1, 2, 3, 4]. In her thesis, Sa´nchez
deA´lvarez[5]indicatesacharacterization ofthePoissonstructureintermsofPoissonbrackets of
particularfunctionsonthetangentbundleTP ofaPoissonmanifoldP,anddiscussesitsfunctorial
properties.
A very noble derivation of a pair of Maxwell equations was originally proposed by Feynman,
but the exact details of his argument came to the scientific community from the work of Freeman
Dyson [6]. According to Dyson, Feynman showed him the construction and examples of the
Lorentz force law and the homogeneous Maxwell equations in 1948. A derivation of a pair of
MaxwellequationsandtheLorentzforceisbasedonthecommutationrelationsbetweenpositions
and velocities for a single non-relativistic particle. In general the locality property that different
coordinates commute is assumed. Due to increasing interest in non-commutative field theories,
it is worthwhile to consider the non-commutative analogue of Feynman approach. This destroys
the axiom of locality, which, according to Dyson, was the original aim of Feynman. Tanimura
[7] gave both a special relativistic and a general relativistic versions of Feynman’s derivation.
Land etal. [8] examined Tanimura’s derivation in the framework of the proper time method in
relativisticmechanics and showed that Tanimura’sresult then corresponds to thefive-dimensional
electromagnetictheorypreviouslyderivedfromaStueckelberg-typequantumtheoryinwhichone
gauges the invariant parameter in the proper time method. An extension of Tanimura’s method
hasbeenachieved[9]byusingtheHodgedualitytoderivethetwogroupsofMaxwell’sequations
with a magnetic monopole in flat and in curved spaces. A rigorous mathematical description of
Feynman’sderivationconnectedtotheinverseproblemforPoissongeometryhasbeenformulated
in [10] (see also [11]). Hughes [12] considered Feynman’s derivation in the framework of the
Helmholtzinverseproblemforthecalculus ofvariations(seealso [13] and[14]).
Infact, itwaspointedoutbyJackiwthatHeisenbergsuggestedinalettertoPeierls thatspatial
coordinates may not commute, Peierls communicated the same idea to Pauli, who informed it to
Openheimer;eventuallytheideaarrivedtoSnyder[15,16]whowrotethefirstpaperonthesubject.
Nowadays the physics in non-commutativeplanes is relevant not only in string theory but also in
condensed matters physics[17]. In the context of theFeynman’s derivationof electrodynamics, it
hasbeenshownthatnon-commutativityallowsotherparticledynamicsthanthestandardformalism
of electrodynamics [18]. Noncommutative quantum mechanics is recently the subject of a wide
rangeofworksfromparticlephysicstocondensedmatterphysics. Thishasalsobeenstudiedfrom
thepointofviewofFeynman’sformalismin [19].
The examples of exotic mechanics started to appear around 1995. Physicists obtained various
models such that the Poisson brackets of the planar coordinates do not vanish. Souriau’s orbit
method [20, 21, 22] was used to construct a classical mechanics associated with Le´vy-Leblond’s
exotic Galilean symmetry. In terms of the Souriau 2-form a wide set of Hamiltonian dynamical
systems have been described in [24, 25, 26, 27]. Le´vy-Leblond [28] has realized that due to the
commutativity of the rotation group O(2,R), the Lie algebra of the Galilei group in the plane
admitsasecondexoticextensiondefined by
[K ,K ] = iκI,
1 2
2
where κ is the new extension parameter. For a free particle the usual equations of motions are
unchanged and κ only contributes to the conserved quantities. It yields the non-commutativityof
thepositioncoordinates.
FeynmanproceduretoobtainMaxwell’sequationinelectrodynamicshasbeenreviewedunder
differentkindofsettings,andseveralnontrivialandinterestinggeneralizationsarepossible,seefor
instance [29, 30, 31, 32, 33, 34, 35, 36]. Recently, Duval and Horva´thy [29] successfully applied
thetechniquesofSouriau’s orbitmethod[37, 38, 39] tovariousmodels. Incidentally,oneofthese
modelscanbeviewedasthenon-relativisticcounterpartoftherelativisticanyonconsideredbefore
byJackiwandNair[32]. Mathematically,the‘exotic’modelarisesduetotheparticularproperties
of the plane. A wide set of dynamical systems can be derived from the Lagrange-Souriau 2-form
approachinthreedimensionsandthegeneralizationstohighernumberofdegreesoffreedomhave
been outlinedin [26].
Wong’s equations describe the interaction between the Yang-Mills field and an isotopic-spin
carrying particle in the classical limit. Feynman-Dyson’s proof offers a way to check the consis-
tencyoftheseequations[40]. See also[41]fora veryrecent paper.
In a slightly different context Kauffman [42] introduced discrete physics based on a non-
commutativecalculusoffinitedifferences. ThisgivesacontextfortheFeynman–Dysonderivation
ofnon-commutativeelectromagnetism. Morerecently,Kauffman[43]foundaninterestingwayto
describe mechanics in a curved background interacting with gauge fields in such a way that the
physical equations of motion emerge automatically from underlying algebraic relations in a non-
commutativegeometryandthisconstructiondependslargelyontheFeynman-Dysonconstruction.
Inaninterestingpaper,CorteseandGarc´ıa[44]studiedavariationalprinciplefornoncommutative
dynamicalsystemsintheconfigurationspace. Inparticulartheyshowedthatthenon-commutative
consistency conditions (NCCC), that come from the analysis of the dynamical compatibility, are
not the Helmholtz conditions of the generalized inverse problem of the calculus of variations.
It has been shown that the θ-deformed Helmholtz conditions are connected to a third-order time
derivativesystemofdifferentialequations. Noncommutativephasespaceshavebeenintroducedby
minimalcouplingsin[45]andthensomeofthemarerealizedascoadjointorbitsoftheanisotropic
Newton-Hookegroupsintwo-andthree-dimensionalspaces. Thishasbeen furthergeneralized to
realize noncommutativephase spaces as coadjoint orbit extensions of theAristotlegroup in a two
dimensionalspace[46].
InthisarticleweapplySouriau’sorbitmethodtostudyexoticmechanicsonthetangentbundle
orvelocityphasespace. SouriaufirstunifiedboththesymplecticstructureandtheHamiltonianinto
asingletwo-form. IthasanexoticsymplecticformandafreeHamiltonianandyieldsageneralized
Hamiltonianmechanics. Duval and Horva´thy used Souriau’s orbit method to construct a classical
planar system associated with Le´vy-Leblond’s two-fold extended Galilean symmetry. The four
dimensionalphasespaceisendowedwiththefollowingexoticform
θ
Ω = dp ∧dq + ǫ dp ∧dp ,
i i ij i j
2
where summationon repeated indices is understood. The exoticterm in the symplecticform only
exists in the plane. Following [47, 48] we also explore a volume-preserving flow on a symplectic
manifoldfromtheSouriau formassociated withthevelocityphasespace.
Many authors [29, 30, 49] have generalized this modification of the symplecticform by intro-
3
ducingtheso-calleddual magneticfield suchthat
1 1
Ω = dp ∧dq + g dp ∧dp + f dq ∧dq .
i i ij i j ij i j
2 2
The coefficients g and f are responsible of the noncommutativity of momenta and positions,
ij ij
respectively. TheclassicaldynamicsinnoncommutativespaceleadstononcommutativeNewton’s
secondlaw[50,51]. Thisgeneralizationcanbestudiedinvarioustypesofnoncommutativespace-
times;for instanceHarikumarand Kapur studiedin [52]the modificationto Newton’ssecond law
due to the kappa-deformation. In a very recent paper the modification of integrable models in the
kappa-deformed scheme is analyzed [53] and kappa-Minkowski space-time through exotic oscil-
lator is studied in [54]. Zhangetal. [55, 56] studied the 3D mechanics with non-commutativity,
where the potential may also depend on the momentum. They obtained the conserved quantities
by usingvan Holten’scovariantframework. Itis knownthattheSnydermodelhas theremarkable
property of leaving the Lorentz invariance intact. Recently, motivated from loop quantum gravity
an idea has been proposed to extend the Snyder model [15] to space-times of constant curvature,
byintroducinganewfundamentalconstantwhoseinverseisproportionaltotheinverseofthecos-
mological constant. More recently, classical dynamics on Snyder space-times has drawn a lot of
attentionto physicists[57, 58, 59, 60]. MoreoverSnyderdynamicsin curved space-time has been
extendedby Mignemietal.in [61,62]. See also[63].
Themainthemeofourpaperistoshowthatnon-commutativitybetweencoordinatesallowsus
toconstructvariousothergeneralizedclassesofdynamicalsystems. Toolsofnon-commutativege-
ometryoftenappearinquantumgravity. Usingadifferentialgeometrictheoryonnon-commutative
space-time Aschierietal.[64] defined θ-deformed Einstein-Hilbert action, and by means of their
technique of the deformation of the algebra of diffeomorphisms one can derive ⋆-deformed inte-
grable systems [65] and Newtonian mechanics [66]. Today we find non-commutativityin various
fields ofmodernphysicssuch as, graphene,Hydrogen atomspectrum,etc. [67, 68, 69].
Thispaperisorganizedasfollows: inordertothepaperbeself-contained,Section2isdevoted
to a review of multivectorfields, Poisson bivector, Schouten–Nijenhuis bracket and various other
geometricaltools. WegiveabriefgeometricaldescriptionofPoissonmanifoldsinSection3. Sec-
tion 4 is devoted to Souriau’s formalism of generalized symplectic forms. We illustrate Souriau’s
construction through examples. Section 5 is focused on the construction of Feynman-Dyson’s
scheme and its connection to Souriau’s method. Section 6 relates volume preserving mechanical
systemsand Souriau’sform. Wefinish ourpaperwithan outlookinSection 7.
2 Geometrical background
LetF(M)bethealgebraofC∞-classfunctions(thealgebraofclassicalobservables)onamanifold
M (the classical state space). We denote by Ωp(M) the space of C∞-class differentiable p-forms,
and by Ap(M) thespaceofC∞-class skew-symmetriccontravarianttensorfields oforderp, often
called p-vectors. By convention we set A0(M) = Ω0(M) = F(M) and Ap(M) = Ωp(M) = 0,
when p < 0. Then,
Ω(M) = Ωp(M) and A(M) = Ap(M),
p∈Z p∈Z
M M
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are Z-graded algebras under their exterior products; moreover both are anticommutative, so, for
instance,ifP ∈ Ap(M) andQ ∈ Aq(M)
P ∧Q = −(−1)pqQ∧P.
When α is a1-formand X isavectorfield theC∞(M)-class functionhα,Xigivenby
hα,Xi(x) := hα(x),X(x)i := α(x) X(x) , ∀x ∈ M,
defines apairingbetweenΩ1(M) andA1(M). Moregene(cid:0)rallyw(cid:1)henη inΩq(M) andP inAp(M)
are decomposable, so η = α ∧ ··· ∧ α and P = X ∧ ··· ∧ X , for α in Ω1(M) and X in
1 q 1 p i j
A1(M), weset
0 if p 6= q,
hη,Pi := hα ∧···∧α ,X ∧···∧X i =
1 q 1 p
(det hαi,Xji if p = q.
(cid:0) (cid:1)
Since the value of hη,Pi at a point only depends on the value of η and P at this point, we can
extend by bilinearity, in a unique way, this pairing to arbitrary elements η in Ω(M) and P in
A(M). Furthermore,it iseasy tocheck thatifη is inΩp(M), then
hη,X ∧···∧X i = η(X ,...,X ).
1 p 1 p
If X is a vector field on M, the inner product i(X) is a derivation of degree −1 on the graded
algebra Ω(M) and since the exterior derivative d is a derivation on Ω(M) of degree 1, it follows
thattheLiederivativewithrespect to X,givenby Cartan’s formula:
L := [i(X),d] = i(X)◦d+d◦i(X),
X
where[·,·]meansgradedcommutator,isagradedderivationofdegree0onΩ(M). L canalsobe
X
defined onA(M) astheuniquederivationofdegree0suchthat,forf inA0(M)and Y inA1(M),
L f = X(f) and L Y = [X,Y],
X X
where[X,Y]istheusualLiebracketonvectorfields. FurthermoretheSchouten–Nijenhuisbracket
isdefinedasanaturalextensionoftheLiederivativewithrespecttoavectorfieldonA(M). More
specifically, it is defined as the unique bilinear map [·,·] : A(M) ×A(M) → A(M) such that,
SN
forf andg inA0(M) = F(M),X ∈ A1(M), P ∈ Ap(M), Q ∈ Aq(M) and R ∈ Ar(M),
a-) [f,g] = 0
SN
b-) [X,Q] = L Q
SN X
c-) [P,Q] = −(−1)(p−1)(q−1)[Q,P]
SN SN
d-) [P,Q∧R] = [P,Q] ∧R+(−1)(p−1)qQ∧[P,R]
SN SN SN
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Fromthesepropertiesitreadilyfollowsthat[P,Q] belongstoAp+q−1(M),thereforethelast
SN
propertymeans thattheendomorphismd : A(M) → A(M) givenby
P
d Q := [P,Q] , (1)
P SN
is a derivation of A(M) of degree p− 1. A somewhat long, but otherwise easy, induction, based
on thedefining properties,gives
(−1)(p−1)(r−1) P,[Q,R] +(−1)(q−1)(p−1) Q,[R,P]
SN SN SN SN
+(−1)(r−1)(q−1) R,[P,Q] = 0, (2)
(cid:2) (cid:3) (cid:2) SN(cid:3)SN
(cid:2) (cid:3)
whichis called graded Jacobiidentity.
This,togetherwithbilinearity,c-),andthefactthat[P,Q] belongstoAp+q−1(M)meansthat
SN
A(M), equippedwiththeSchouten–Nijenhuisbracket, isagradedLiealgebrawhenthedegreeof
P in Ap(M) is declared to be p − 1, not p. So, for instance, vector fields would be the homoge-
neous elements of degree 0 under this new grading of A(M). To perform computations with the
Schouten–Nijenhuisbracket it is convenientto extend thedefinitionof theinteriorproduct. Ifη is
inΩ(M), f isafunctionand X ,...,X arevectorfieldsweset
1 p
i(f)η := fη and i(X ∧···∧X )η := i(X )◦···◦i(X )η.
1 p 1 p
In general, i(P)is defined insuch away thatthemapi(·): A(M) → E Ω(M) , whereE Ω(M)
isthespaceofendomorphismofΩ(M), isF(M)-linear. Inparticular, i(P ∧Q)η = i(P)(i(Q)η),
(cid:0) (cid:1) (cid:0) (cid:1)
forall P and QinA(M). Furthermore, whenη isap-form
(p−1)p
i(X1 ∧···∧Xp)η = i(X1)◦···◦i(Xp)η = η(Xp,...,X1) = (−1) 2 η(X1,...,Xp),
thereforefor anyP ∈ Ap(M)
(p−1)p
i(P)η = (−1) 2 hη,Pi. (3)
Unfortunatelyi(P),ingeneral,isnotaderivation,whichcomplicatescomputations. Nevertheless,
anothersimpleinductiongives
i [P,Q] = [i(P),d],i(Q) , (4)
SN
(cid:0) (cid:1) (cid:2) (cid:3)
where the brackets on the right are the usual brackets on thealgebra of endomorphismsof A(M).
Notice that when P = X is a vector field this reduces to the well-known relation among interior
productsand Liederivatives: i(L Q) = [L ,i(Q)].
X X
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If P is a p-vector and η is a (p−1)-form, then i(P)η = 0 and i(P)◦i(f)η = i(P)(fη) = 0.
These, togetherwith(3)and (4)entail
(p−2)(p−1)
hη,[P,f]SNi = (−1) 2 i([P,f]SN)η
(p−2)(p−1)
= (−1) 2 [i(P),d],i(Q) η
= (−1)(p−2)2(p−1)(cid:2)i(P)◦d◦i(f(cid:3))η −(−1)pd◦i(P)◦i(f)η
(cid:16)
−i(f)◦i(P)◦dη−(−1)pi(f)◦d◦i(P)η
(p−2)(p−1) (cid:17)
= (−1) 2 i(P)◦d◦i(f)η −i(f)◦i(P)◦dη
(p−2)(p−1)(cid:16) (cid:17)
= (−1) 2 i(P)(df ∧η)
= (−1)(p−1)2hdf ∧η,Pi
= (−1)(p−1)(p−2)hη ∧df,Pi
= hη ∧df,Pi.
Repited useofthisgives
hdf ∧···∧df ,Pi = hdf ∧···∧df ,[P,f ] i
1 p 1 p−1 p SN
= ··· = ··· [P,f ] ,f ,··· ,f . (5)
p SN p−1 SN 1
SN
h i
(cid:2) (cid:3)
3 Poisson Manifolds
A Poisson structure on M is a skew-symmetricR-bilinear map {·,·}: F(M)×F(M) → F(M)
satisfyingtheJacobiidentity:
{f,{g,h}}+{h,{f,g}}+{g,{h,f}} = 0, ∀f,g,h ∈ F(M),
and such that the map X = {·,f} is a derivation of the associative and commutative algebra
f
F(M), for each f ∈ F(M), or in other words, X is a vector field, usually called a Hamiltonian
f
vectorfield,andf issaidtobetheHamiltonianofX . Thispropertycharacterizingderivationsof
f
theassociativeandcommutativealgebraF(M),{g g ,f} = g {g ,f}+g {g ,f},calledLeibniz’
1 2 1 2 2 1
rule, is very important and gives a compatibility condition of the associative and commutative
algebrastructureinF(M) withtheLiealgebra giveninF(M) by thePoissonbracket.
To construct Poisson structures let Λ be an element of A2(M), if f ∈ A0(M) and g ∈ A0(M)
are twofunctions,using(5), wedefine athirdfunctionby
{f,g} := Λ(df,dg) = −Λ(dg,df) = −hdg ∧df,Λi = − [Λ,f] ,g . (6)
SN SN
By constructionX := [Λ,f] is avectorfield, and thedefiningpro(cid:2)perty b-)en(cid:3)tails
f SN
X (g) = L g = [X ,g] = −{f,g} = {g,f}. (7)
f Xf f SN
In particular
{g,{h,f}} = [Λ,g] , [Λ,h] ,f = X ,[X ,f] = L ◦L f.
SN SN SN g h SN SN Xg Xh
SN
h i
(cid:2) (cid:3) (cid:2) (cid:3)
7
By the same token {h,{f,g}} = −{h,{g,f}} = −L ◦ L f. On the other hand, the graded
Xh Xg
Jacobiidentity(2), thedefining propertyb-), and (7)entail
{f,{g,h}} = −{{g,h},f} = {[X ,h] ,f} = − Λ,[X ,h] ,f
g SN g SN SN
SN
h i
(cid:2) (cid:3)
= − X ,[h,Λ] ,f − h,[Λ,X ] ,f
g SN SN g SN SN
SN SN
h i h i
(cid:2) (cid:3) (cid:2) (cid:3)
= − [X ,X ] ,f − h,[Λ,X ] ,f
g h SN SN g SN SN
SN
h i
(cid:2) (cid:3) (cid:2) (cid:3)
= −L f − h,[Λ,X ] ,f .
[Xg,Xh] g SN SN
SN
h i
(cid:2) (cid:3)
Altogethergives
{f,{g,h}}+{g,{h,f}}+{h,{f,g}} = L ◦L −L ◦L −L f
Xg Xh Xh Xg [Xg,Xh]
(cid:16) (cid:17)
− h,[Λ,X ] ,f
g SN SN
SN
h i
(cid:2) (cid:3)
= − h,[Λ,X ] ,f .
g SN SN
SN
h i
(cid:2) (cid:3)
Furthermore, fromthegraded Jacobi identity
0 = Λ,[Λ,g] + Λ,[Λ,g] + g,[Λ,Λ]
SN SN SN SN SN SN
= 2 Λ,[Λ,g] + g,[Λ,Λ]
(cid:2) SN(cid:3) SN (cid:2) SN(cid:3) SN (cid:2) (cid:3)
= 2[Λ,X ] + g,[Λ,Λ] , (8)
(cid:2) g SN (cid:3) (cid:2) SN SN (cid:3)
therefore, by (5) (cid:2) (cid:3)
1
{f,{g,h}}+{g,{h,f}}+{h,{f,g}} = h, g,[Λ,Λ] ,f
2 SN SN SN
(cid:20) (cid:21)SN
h i
1 (cid:2) (cid:3)
= h, [Λ,Λ] ,g ,f
2 SN SN SN
(cid:20) (cid:21)SN
h i
1 (cid:2) (cid:3)
= − [Λ,Λ] ,g ,h ,f
2 SN SN SN
(cid:20) (cid:21)SN
h i
1 (cid:2) (cid:3)
= − hdf ∧dh∧dg,[Λ,Λ] i. (9)
SN
2
It follows that the bracket defined via a bivector field Λ is a Poisson structure exactly when
[Λ,Λ] = 0, and when this happens we say that Λ is a Poisson tensor. This elementary, but
SN
clever, computation was first performed by Lichnerowicz in [70], who also realized that, when Λ
isaPoissontensorand P isin Ap(M), thegraded Jacobiidentityimplies
0 = (−1)p−1 Λ,[Λ,P] − Λ,[P,Λ] +(−1)p−1 P,[Λ,Λ]
SN SN SN SN SN SN
= 2(−1)p−1 Λ,[Λ,P] .
(cid:2) SN(cid:3) SN (cid:2) (cid:3) (cid:2) (cid:3)
In other words, for Pois(cid:2)son tensors,(cid:3)the derivation d , defined as in (1) by d P = [Λ,P] ,
Λ Λ SN
satisfiesthecocyclecondition
d ◦d = 0.
Λ Λ
8
Ontheotherhand,from(8)weseethat,forPoissontensors,[X ,Λ] = 0,thistogetherwiththe
f SN
graded Jacobiidentityand (7)give
[X ,X ] = X ,[Λ,g] = − Λ,[g,X ] + g,[X ,Λ]
f g f SN SN f SN SN f SN SN
= Λ,[X ,g] = −[Λ,{f,g}]
(cid:2) (cid:3) (cid:2) f SN S(cid:3)N (cid:2) (cid:3)
= X . (10)
(cid:2) −{f,g} (cid:3)
The converse is also true: a Poisson structure determines a Poisson tensor. To see this let ξ
a
denoteasetoflocal coordinatesonM, then,usingthesummationindexconvention,
∂ ∂
X = X (ξ ) = {ξ ,f} ,
f f a a
∂ξ ∂ξ
a a
hence
∂f
{f,g} = X (f) = {ξ ,g} .
g a
∂ξ
a
Thus,
∂g ∂g
{ξ ,g} = −{g,ξ } = −{ξ ,ξ } = {ξ ,ξ } , (11)
a a b a a b
∂ξ ∂ξ
b b
and thelocal coordinateexpressionofthePoissonBracket becomes
∂g ∂f
{f,g} = {ξ ,ξ } . (12)
a b
∂ξ ∂ξ
b a
Therefore to compute the Poisson bracket of any pair of functions is enough to know the funda-
mentalPoissonbrackets
Λ = {ξ ,ξ }.
ab a b
Moreover, the value of {f,g} at a point m ∈ M does not depend neither on f nor on g but
only on df and dg, as explicitly shown in (12), hence from the Poisson structure we get a twice
contravariantskew-symmetrictensor
Λ(df,dg) := {f,g}.
Indeed, thelocalcoordinateexpressionofΛ is
∂ ∂
Λ = Λ ∧ ,
ab
∂ξ ∂ξ
a b
and ifξ¯= φ(ξ)isanotherset oflocalcoordinateson M, then,
∂φ ∂φ ∂φ ∂φ
Λ¯ = {ξ¯ ,ξ¯} = {φ ,φ } = {ξ ,ξ } a b = Λ a b ,
ab a b a b c d cd
∂ξ ∂ξ ∂ξ ∂ξ
c d c d
so the components of Λ do change like the local coordinates of a skew-symmetric twice con-
travariant tensorwhich, by (9), it is aPoisson tensor. Weare usingtheconventionthat inthelocal
expression of the wedge product only summands whose subindex on the left hand side term is
smallerthan thesubindexon therighthandsidetermdo appear.
9
For any function h ∈ F(M) theintegral curves of thedynamical vector field X are precisely
h
determinedby thesolutionsofthesystemofdifferentialequations
dξ
a
= {ξ ,h}, (13)
a
dt
and thedynamicalevolutionofafunctionf onM isgivenby
df
= {f,h},
dt
orinlocal coordinates
df ∂f ∂h
= Λ .
ab
dt ∂ξ ∂ξ
a b
Interesting examples of Poisson manifolds are symplecticmanifolds. A symplecticform ω on
M determines a bundle map ω♭: TM → T∗M over the identity, which gives rise to the corre-
spondinglinearmapbetween theirspaces ofsections,defined by
ω♭(X) (Y) := hω♭(X),Yi = ω(X,Y).
Sinceω isnon-degenerateω♭(cid:0)isactua(cid:1)llyanisomorphism;wedenoteω♯ theinversemapanddefine
abivectorΛby
Λ(α,β) := ω ω♯(α),ω♯(β) ,
if α and β are 1-forms. When α = df is ex(cid:0)act the corres(cid:1)ponding vector field is denoted by
X := ω♯(df), and we say X is the vector field associated to f with respect to ω. It is actually
f f
definedbytheequationi(X )ω = df. Furthermore,let{·,·}bethebracketassociatedtoΛvia(6).
f
Then the vector field associated to f with respect to ω is also the Hamiltonian vector field given
in (7), explaining why we use the same notation. If ξ , a = 1,...,m, denote local coordinates
a
and ∂/∂ξ ,...,∂/∂ξ , and dξ ,...,dξ are, respectively, the local basis of A1(M) and Ω1(M)
1 m 1 m
associated to ξ , let B = (B ) be the matrix of the linear map ω♭ relative to these bases, and
a ab
ω = ω dξ ∧dξ thelocalexpressionofthesymplecticform,then
ab a b
∂ ∂ ∂ ∂ ∂
ω = ω , = ω♭ = B dξ = B .
ab ac c ab
∂ξ ∂ξ ∂ξ ∂ξ ∂ξ
(cid:18) a b(cid:19) (cid:18) (cid:18) a(cid:19)(cid:19)(cid:18) b(cid:19) (cid:18) b(cid:19)
(cid:16) (cid:17)
Thus, B = (ω ), and the matrix of ω♯ associated to these bases is the inverse of B. On the other
ab
hand,
dω(X ,X ,X ) = X ω(X ,X ) +X ω(X ,X ) +X ω(X ,X )
f g h f g h g h f h f g
−ω([X ,X ],X )−ω([X ,X ],X )−ω([X ,X ],X ).
(cid:0) f g (cid:1)h (cid:0) g h (cid:1)f (cid:0) h f (cid:1) g
Nevertheless,ω(X ,X ) = Λ(dg,dh) = {g,h},so (7)entails
g h
X ω(X ,X ) = X ({g,h}) = −{f,{g,h}}.
f g h f
Also,by (10),[X ,X ] = [X(cid:0) ,X ] =(cid:1)X , hence
g h g h SN −{g,h}
ω([X ,X ],X ) = ω(X ,X ) = −{{g,h},f} = {f,{g,h}}.
g h f −{g,h} f
10