Dirac Brackets in Geometric Dynamics Annales De L’I

ANNALES DE L’I. H. P., SECTION A ´JEDRZEJ SNIATYCKI Dirac brackets in geometric dynamics Annales de l’I. H. P., section A, tome 20, no 4 (1974), p. 365-372 <<a href="/goto?url=http://www.numdam.org/item?id=AIHPA_1974__20_4_365_0" target="_blank">http://www.numdam.org/item?id=AIH</a><a href="/goto?url=http://www.numdam.org/item?id=AIHPA_1974__20_4_365_0" target="_blank">P</a><a href="/goto?url=http://www.numdam.org/item?id=AIHPA_1974__20_4_365_0" target="_blank">A</a><a href="/goto?url=http://www.numdam.org/item?id=AIHPA_1974__20_4_365_0" target="_blank">_1974__20_4_365_0</a>> © Gauthier-Villars, 1974, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (<a href="/goto?url=http://www.numdam.org/conditions" target="_blank">http://www.numdam. </a><a href="/goto?url=http://www.numdam.org/conditions" target="_blank">org/conditions</a>). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce ﬁchier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques <a href="/goto?url=http://www.numdam.org/" target="_blank">http://www.numdam.org/ </a>Ann. Inst. Henri Section A : Poincaré, Vol. n° 365 XX, 4, 1974, Physique théorique. in Dirac brackets geometric dynamics 015ANIATYCKI Jedrzej of The Canada. Alberta, University Calgary, of Mathematics Statistics and Science Department Computing is formulated in the of of constraints in ABSTRACT. framework of - Theory symplectic dynamics Geometric significance secondary of Dirac Global existence geometry. is constraints and of Dirac brackets given. brackets is proved. 1. INTRODUCTION The successes of the canonical of with quantization the dynamical systems afinite number of of of freedom, degrees necessity quan- gravi- theory experimental of of the field tization and the that electrodynamics, hopes quantization could resolve in quantum tational field difficulties encountered of the have to rise canonical structure of given thorough investigation field theories. It has been found that the formu- cases standard Hamiltonian most lation of of is in the dynamics inadequate and physically interesting dueto existence of constraints. Methods electrodynamics gravitation with constraints have been of with several by developed dealing dynamics authors and it has been realized that the standard Hamiltonian dynamics can be formulated in terms of constraints (1). has been amathematical The aim of of constraints Hamiltonian dynamics given very elegant formulation in the framework of a(2). theory symplectic geometry this in is to formulation of the paper give symplectic As aresult the of the classification of geometric and of Dirac brackets is dynamics. significance Further, the of constraints given. globalization See refs. the references there. and [4], [5], [6], quoted C) [2], [3], [8] See refs. and [1] [9]. (~) eg. Henri Poincaré - de l’Institut Section A - n° 4 - 1974. XX, Annales vol. 366 JEDRZEJ SNIATYCKI theresults discussedin theliteraturein termsof local coordinatesis obtained. the fundamental with notion constraints is that As in the case of the Hamiltonian dynamics, of in the dynamical systems geometric analysis of amanifold. Basic of manifolds are symplectic properties symplectic reviewed in Section 2. awith constraints can be constraint Adynamical system represented by asubmanifold of is described and manifold. constraint Therefore, symplectic dynamics manifold aawhere is M, by triplet (P, P, of canonical (P, cv) symplectic Mis asubmanifold of which is called acanonical Ele- system [12]. their relations to and the mentary properties with systems, Lagrangian arelation between cano- systems nical homogeneous Lagrangians, and its reduced phase space are in Section 3. discussed system 4Section contains aof constraints. The discussion secondary generalization of Dirac classification of constraints is in Section 5. Sec- given tion 6is devoted to an ofthe of Dirac brac- analysis of their geometric significance akets and existence. proof global 2. SYMPLECTIC MANIFOLDS Pis amanifold and OJ manifold is awhere Aa(3) pair (P, OJ) symplectic symplectic form on P. The most of ais important example symplectic of is furnished in this case the structure of the manifold in by space dynamics system, bracket. phase Pthe and cv is the arepresents phase space dynamical Lagrange be amanifold afunction on P. There exists Let (P, OJ) symplectic and fsuch form avector field called the Hamiltonian vector field unique that of f, J~ = 2014 d f, where and Jdenotes the left interior are Hamiltonian vector fields of aproduct v f avector field. It by correspond- v f vg and then their Lie brackets is to functions ing fg, respectively, [v f’ ~] the Hamiltonian vector field to the Poisson bracket and the Jacobi corresponding ( f, g) for Further, ( f, g) an identity of f and g. ~(~) = 2014 vg( f ), the Poisson brackets is immediate of the Jacobi that 6D is consequence identity of fields and the closed. for the Lie bracket vector assumption 3. CANONICAL SYSTEMS - Acanonical is awhere DEFINITION symplectic 3 . l. M, (P, OJ) (P, triplet cv) system asubmanifold of P. Mis amanifold and is one from the if Canonical Lagrangian dynamics and passes systems appear All manifolds considered in this are finite dimensional, (3) paper paracompact of class C ". Annales de l’Institut Henri Poincaré - Section ADIRAC BRACKETS IN DYNAMICS 367 GEOMETRIC with to the canonical of systems dynamics homogeneous Lagrangians. Let Lbe aone aon aconical X. function of defined homogeneous degree domain We denote Dof the TX of bundle tangent space configuration space transformation FL : D - T*X the by Legendre given by for the fibre derivative of Land 0the Liouville form on T*X defined, is the by Ewhere n : T*X -~ Xaeach v bundle by p(Tn(v)) cotangent canonical TpT*X, The is FL, (T*X, de) system projection. triplet range submanifold ofT*X. canonical There are two subsets FL be is arange provided aLet (P, TPM Kand NM, co) system. of associated to as follows: M, (P, co) and N=KnTM. Nis called the characteristic set for canonical of M. The The set wdynamical significance of Naobtained from aFL, (T*X, range homogeneous Lagrangian d0) system with on XaLis given by Lagrangian system the following. the 3 . 2. - A curve in Xsatisfies if PROPOSITION Lagrange-Euler equa- where y denotes only FL. , yLand tions the to the corresponding prolongation Lagrangian is an this curve of N. Proof of of to TX, integral proposition is in the reference and it will be omitted. given [11] system for aof Mwhich can be canonical Thus, connected M, (P, points curves of Nare related in aby integral be physically meaningfull related time evolution as ain 3.2 or manner; they may by Proposition the maximal can be atransformation. Eby gauge Therefore, M, given point pNmanifold of it exists, integral as the p, through provided interpreted of history p. if Nis aDEFINITION 3 . 3. Acanonical is M, (P, w) regular system of TM. be and since cv is closed subbundle asubbundle Frobenius acanonical Then Nis Let M, (P, cv) regular system. is also involutive. of NTM, theorem Hence, unique by each EMis contained in amaximal (4), point pintegral manifold of N. Let P~ denote the set of Mthe quotient by equivalence relation defined maximal manifolds of that is each element N, described dynamical system by aintegral of the in P~ our cano- represents history by M ~ Pr denote the canonical nical and let M, (P, system p : projec- submersion MP~ admits aation. If then there exists differentiable structure such that is paformwr on P~ such that unique symplectic wThe manifold called the to of is reduced (Pr, symplectic wr) phase space on Pr constants of Functions M, (P, CD). correspond (gauge invariant) in the manner described and cvr rise to their Poisson motion, algebra gives in 2Section (5). All theorems on differentiable manifolds used in this can be found in ref. (~) (~) paper [7]. This Poisson was first studied in ref. [2]. algebra Vol. n° 4 - 1974. XX, JEDRZEJ SNIATYCKI 368 4. CONSTRAINTS SECONDARY on In If is not points then dim on which dim e M. open (P, M, regular ppcv) depends particular Np the set S° of EM0is an submanifold inadmissible Np S° as of if it is not this set M, Discarding physically constraints here. Consider the class ofall manifolds empty. [3] have lead Dirac to the notion of secondary generalization Mofwhich is given n Ydim contained in and Y’ such that TY Nis asubbundle of TY allow here (we order in this class defined as follows Y0), Ylet denote the partial if and if Yis aof Y’. follows from Zorn’s lemma that submanifold It only there to the exist maximal manifolds in class. we are lead this Thus, following. DEFINITION 4.1. Aconstraint manifold of acanonical secondary maximal manifold is a Scontained in Msuch that NnTS M, (P, cv) system ais subbundle of TS (6). Sbe submanifold cannot use Frobeniustheoremto define the reduced aconstraint manifold of If Sis not an Let M, secondary of (P, be involutive. open one case Mthen nTS need not NIn this If NnTS phase space. relation, equivalence maximal manifolds define an is involutive its and let integral set of denote the Srelation. that the that there this by such Nquotient differentiable structure on Ps Suppose aexists tion canonical Ps projec- S - is then asubmersion. Since ofOJ M is the characteristic set TS is contained in the characteristic set ps : Ps nand NTS c TM, Ns <ul style="display: flex;"><li style="flex:1">of </li><li style="flex:1">defined </li></ul>there exists non- Therefore The form the reduced OJ unique Sby 03C9rS only constraint and a2-form on <ul style="display: flex;"><li style="flex:1">such that </li><li style="flex:1">S</li></ul>is Ps N03C1*S03C9rS. 03C9rS if and if nTS manifold Thus, adegenerate Ns. Sphase space Ps ofthe has natural structure secondary of a symplec- if if TS nNtic manifold If this condition and if <ul style="display: flex;"><li style="flex:1">S</li><li style="flex:1">holds </li></ul>only then the structure Ns. is asubmanifold of Saof as all as constraint canonical P, OJ) secondary manifold of is the same that of cases aM, (P, cv). exactly regular This situation we in of in interest S, system (P, appears physics, regular therefore in the shall limit our considerations to following canonical systems. 5. ACLASS OF CANONICAL SYSTEM on afirst Mbe acanonical Afunction Pis called Let M, (P, OJ) system. fon function constant if its Poisson bracket with zero on M. This condition class function every can be reformulated as follows: is identically This definition is aof the in ref. definition (~) globalization [10]. given Annales de I’Institut Henri APoincaré - Section 369 DIRAC BRACKETS IN for each v GEOMETRIC DYNAMICS which Pif and Eon 0. Functions first class if, K, v(f) only f is are constraint not first class are called second class functions. Since the aMcan P - dim be described of system submanifold 0, apply locally by equations the notion ofa class can be extended f (p) to iM, 1, ..., dim acanonical system. regular acanonical is the 5.1. - Class of DEFINITION of M, by (P, regular system dim K - dim N, (dim N) locally M (’). integers pair If ais ofclass then ..., can be described system where all (P, M,cv) (n, k) 1, of in and 0, 1, ..., k, equationsf (p) 0, j are first class andall are second and class, it cannot functions £. functions gj be described tions. If k of with more than n first class func- by any system equations 0we call Mafirst class submanifold of as it can be (P, OJ) aof iwhere all second class 0, 1, ..., n, locally by system equations f (p) given if n Mis called aaare first class. 0, Similarly, functions f of In this case is submanifold manifold. (P, (M, OJM) symplectic two theorems which we hold shall later. need There For PROPOSITION 5.2. canonical M, any regular (P, co), system Nis an even number. dim K - dim M -dim N. even P~ dim Kdim P -dim Mand dim dim forms dim Proof. dim P~ are and PSince both Pand P~ admit symplectic is even. P - dim P~ numbers. K - dim Ndim dim Hence, a5 . 3. Let be canonical system j’ and g of PROPOSITION M, regular (P, class with k> 0. there exist two functions such that Then, (n, lc) For each E M there exists on aof in Pand 0M’ Proof - pneighbourhood pUp two functions and such that ngpMn Mfp fp of Up gp Up Up and nwe 1. associate to the Further, Mcomplement ( fp, gp) Up of the Min Pand to closure functionsf ’ covering zero. This there g’ equal identically we and since of Phave obtained aof is paracompact, P, way athis exists finite and For each aof locally covering }. partition refinement { to the such we have two subordinated on Ua Ua unity { functions covering { that and g03B1 U (1 Letf gfollows, and be Pdefined as functions on for each Then, is to W. M. due This definition (7) Tulczyjew (unpublished). Vol. n° 4 - 1974. XX, JEDRZEJ SNIATYCKI

Dirac Brackets in Geometric Dynamics Annales De L’I

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