Symmetries of Dynamically Equivalent Theories

Transcription

1 13 Brzilin Journl of Physics, vol. 36, no. 1B, Mrch, 006 Symmetries of Dynmiclly Equivlent Theories D. M. Gitmn nd I. V. Tyutin Institute of Physics, University of São Pulo, Brzil nd Lebedev Physics Institute, Moscow, Russi Received on 31 Jnury, 005 A nturl nd very importnt development of constrined system theory is detil study of the reltion between the constrint structure in the Hmiltonin formultion with specific fetures of the theory in the Lgrngin formultion, especilly the reltion between the constrint structure with the symmetries of the Lgrngin ction. An importnt preliminry step in this direction is strict demonstrtion, nd this is the im of the present rticle, tht the symmetry structures of the Hmiltonin ction nd of the Lgrngin ction re the sme. This proved, it is sufficient to consider the symmetry structure of the Hmiltonin ction. The ltter problem is, in some sense, simpler becuse the Hmiltonin ction is first-order ction. At the sme time, the study of the symmetry of the Hmiltonin ction nturlly involves Hmiltonin constrints s bsic objects. One cn see tht the Lgrngin nd Hmiltonin ctions re dynmiclly equivlent. This is why, in the present rticle, we consider from the very beginning more generl problem: how the symmetry structures of dynmiclly equivlent ctions re relted. First, we present some necessry notions nd reltions concerning infinitesiml symmetries in generl, s well s strict definition of dynmiclly equivlent ctions. Finlly, we demonstrte tht there exists n isomorphism between clsses of equivlent symmetries of dynmiclly equivlent ctions. Keywords: Constrined systems; Symmetries; Lgrngin ctions I. INTRODUCTION The most of contemporry prticle-physics theories re formulted s guge theories. It is well known tht within the Hmiltonin formultion guge theories re theories with constrints. This is the min reson for long nd intensive study of the forml theory of constrined systems, see [1. It still ttrcts considerble ttention of reserchers. From the very beginning, it becme cler tht the presence of firstclss constrints mong the complete set of constrints in the Hmiltonin formultion is direct indiction tht the theory is guge one, i.e., its Lgrngin ction is invrint under guge trnsformtions. A next nturl, nd very importnt, step would be detil study of the reltion between the constrint structure nd constrint dynmics in the Hmiltonin formultion with specific fetures of the theory in the Lgrngin formultion, especilly the reltion between the constrint structure with the guge trnsformtion structure of the Lgrngin ction. An importnt problem to be solved in this direction would be strict demonstrtion, nd this is the im of the present rticle, tht the symmetry structures of the Hmiltonin ction nd of the Lgrngin ction re the sme. This proved, it is sufficient to consider the symmetry structure of the Hmiltonin ction. The ltter problem is, in some sense, simpler becuse the Hmiltonin ction is firstorder ction. At the sme time, the study of the symmetry of the Hmiltonin ction nturlly involves Hmiltonin constrints s bsic objects, see [, 3. It follows from the results of the rticle [4 tht the Lgrngin nd Hmiltonin ctions re dynmiclly equivlent. This is why in the present rticle we consider from the very beginning more generl problem: how the symmetry structures of dynmiclly equivlent ctions re relted. The rticle is orgnized s follows: In sec., we present some necessry notions nd reltions concerning infinitesiml symmetries in generl. A strict definition of dynmiclly equivlent ctions is given in sec. 3. Finlly, in sec. 4, we demonstrte tht there exists n isomorphism between clsses of equivlent symmetries of dynmiclly equivlent ctions. II. SYMMETRIES A. Bsic nottion nd reltions We consider finite-dimensionl systems which re described by the generlized coordintes q {q ; 1,,...,n}. The spce of the vribles q [l, q [l d t l q, l 0,1,...,N, q [0 q, d t d, 1 considered s independent vribles, with finite N, or with some infinite N, is clled the jet spce. The mjority of physicl quntities re described by so-clled locl functions LF which re defined on the jet spce. The LF depend on q [l up to some finite orders l N 0. The following nottion is often used[6: F q [0,q [1,q [,... F q [ for the LF. In wht follows, we lso del with so-clled locl opertors LO. LO Û A re mtrix opertors which ct on columns of LF f producing columns F A of LF, F A Û A f. LO hve the form K< Û A where u k A re LF. We cll the opertor u k A d t k, 3

2 D. M. Gitmn nd I. V. Tyutin 133 Û T K< A d t k u k A 4 the trnsposed opertor with respect to Û A. The following reltion holds true for ny LF F A nd f : F A Û A f [ Û T A FA f + d t Q, 5 where Q is n LF. The LO Û b is symmetric + or ntisymmetric respectively if Û T b ±Û b. Thus, for ny ntisymmetric LO Û b reltion 5 is reduced to the following: f Û b f b dq/, where Q is LF. Suppose the totl time derivtive of n LF vnishes. Then this LF is constnt. Nmely, df q [l t 0 F q [l const. 6 Indeed, let us suppose tht N re the orders of the coordintes q in the LF, i.e. F q [l F q [N. Then ccording to 6 the following reltion holds true [ F q [N q[n +1 F q[k+1. q[k t F + N 1 The right hnd side of the bove reltion does not depend on q [N+1. Thus, F/q [N 0, nd therefore F q [l must not depend on q [N. In the sme mnner we cn see tht F q [l must not depend on q [N 1 nd so on. If F q [l does not depend on ny q [l, then t F q [l 0 s well, nd we get F q [l const. We recll tht F A q [ 0 nd χ α q [ 0 re equivlent sets of equtions whenever they hve the sme sets of solutions. In wht follows, we denote this fct s F 0 χ 0. Vi OF we denote ny LF tht vnishes on the equtions F q [ 0. More exctly, we define OF ˆV b F b, where ˆV b is n LO. Besides, we denote vi Û ÔF ny LO tht vnish on the equtions F q [ 0. Tht mens tht the LF u tht enter into 3 vnish on these equtions, u OF, or equivlently Û f OF for ny LF f. We consider Lgrngin theories given by n ction S[q, S[q L, L L q [, 7 where Lgrnge function L is defined s n LF on the jet spce[7. The Euler Lgrnge equtions re δq d t l L 0. 8 q[l Any LF of the form O/δq is clled n extreml. For ny LF F q [ the opertion d EL F dq N d l F q [l 9 is clled the Euler Lgrnge derivtive with respect to the coordinte q. One cn see tht the functionl derivtive of the ction S coincides with the Euler Lgrnge derivtive of the Lgrnge function, δq d ELL dq. 10 The Euler Lgrnge derivtive hs the following property: To prove this, one my use the reltion d EL d dq q [k d q [k t + q b[l+1 q b[l 1 δ k0 q [k 1 + t + q b[l+1 q b[l q [k d q [k + 1 δ k0 q [k 1. Thus, one gets

3 134 Brzilin Journl of Physics, vol. 36, no. 1B, Mrch, 006 d EL d dq d k d q [k d k+1 q [k + d k k1 q [k 1 d d k q [k d d k 1 k1 q [k 1 d d EL dq d d EL dq 0. B. Noether symmetries Consider n infinitesiml inner[8 trjectory vrition δq inner vritions vnish together with ll their time derivtives t nd t. Nmely, q t q t q t + δq. 1 We suppose tht δq δq q [ is n LF. The corresponding first vrition of the ction cn be written s follows: ˆδL, 13 where the opertor ˆδ, which will be clled the trnsformtion opertor, cts on the corresponding LF s[9 ˆδ δq [k q [k ˆδ δq. 14 Two simple but useful reltions follow from 14: ˆδq δq, ˆδc i δ i q ci ˆδδi q. 15 The vrition 1 is symmetry trnsformtion of the ction S, or simply symmetry of the ction S, whenever the corresponding first vrition of the Lgrnge function is reduced to the totl time derivtive of LF. Nmely, δq is symmetry if ˆδL df, 16 where F is n LF. In this cse the first vrition 13 of the ction depends on the complete set of the vribles q [ t t nd t t only, ˆδL F t t1. Any liner combintion of symmetry trnsformtions is symmetry. Indeed, let δ i q be some symmetry trnsformtions, nd δq c i δ i q, where c i re some constnts. Then, tking into ccoun5, we obtin: ˆδ δi ql df i ˆδ δq L df, F ci F i. 17 Trnsformtion opertors tht correspond to symmetry trnsformtions re clled symmetry opertors. The bove-described symmetry trnsformtions re clled Noether symmetries. Below, we list some properties of the trnsformtion opertors nd of the symmetry trnsformtions: Any first vrition of the Lgrnge function cn be presented s ˆδL δq d ELL dq + dp δq δq + dp, 18 where P is n LF of the form P N m1 p m δq [m 1, p m N sl d s m L. 19 q[s One ought to remrk tht the sum 19 tht presents P is running only over those for which N > 0. However, it cn be extended over ll s since the moment p m tht correspond to the degenerte coordintes re zero. Thus, the prime over the sum bove cn be omitted. b Any trnsformtion opertor commutes with the totl time derivtive: [ ˆδ, d 0. 0 The ltter property is justified by the following reltions: [ d ˆδ δq [k+1 + δq[k q[k ˆδ d [ˆδq b[l+1 q [k t q b[l + ˆδ t + k, + q b[l+1 δq [k k, q [k q b[l, δq [k q b[l+1 q b[l q [k d ˆδ.

4 D. M. Gitmn nd I. V. Tyutin 135 c The commuttor of ny two trnsformtion opertors is trnsformtion opertor s well. Nmely, let ˆδ 1 q δq 1, nd ˆδ q δq. Then [ˆδ1, ˆδ ˆδ 3, ˆδ3 q ˆδ 1 δq ˆδ δq 1. 1 Indeed, one cn write: ˆδ 1 ˆδ ˆδ1 δq b[l q b[l + δq [k k, d l ˆδ ε1 δq b l q b[l + ˆδ ˆδ1 ˆδε δq [k 1 d k ˆδε δq b 1 k 1 δq b[l q b[l δq [k 1 δq b[l k, q b[l q [k + l, q [k + δq [k k, δq b[l δq[k 1 q [k 1 δq b[l q b[l q [k, q[k q b[l q [k. 3 Then subtrcting Eq. 3 from Eq., we obtin the reltion 1. In other words, the set of ll trnsformtion opertors form Lie lgebr. d The commuttor of the Euler Lgrnge derivtive nd trnsformtion opertor is proportionl to the Euler Lgrnge derivtive. Nmely, if ˆδq δq b, then [ del dq, ˆδ ˆQ b d EL dq b, ˆQ b d k q [k δqb. 4 To prove this property, one my consider sequence of equlities, ˆδF d EL dq ζ ˆδζ ˆδF ˆδˆδζ F + ˆδˆδζ δq F ζ d k F t ˆδ q [k + ˆδζ δq b d ELF dq b ˆδδ t ζ b + ˆQ b del F ˆδζ dq b, q ζ, where ζt is n rbitrry inner vrition, nd F is n LF. It is useful to keep in mind the following generliztion of reltion 4: [ d k d EL dq, ˆδ d k ˆQ b d EL dq b, 5 which follows immeditely from 0 nd 4. e The commuttor of two symmetry opertors is symmetry opertor s well. Indeed, let ˆδ 1 q δq 1, nd ˆδ q δq be symmetry trnsformtions, i.e., ˆδ 1 L df 1 /, nd ˆδ L df /. Then, tking into ccount 0 nd 1, we obtin [ˆδ1, ˆδ L ˆδ 3 L d F 3, F 3 ˆδ 1 F ˆδ F 1. 6 Thus, the set of symmetry opertors of the ction S forms Lie sublgebr of the Lie lgebr of ll trnsformtion opertors. f Symmetry trnsformtions trnsform extremls into extremls. The vlidity of this ssertion follows from the reltions proven below. Suppose ˆδ is symmetry opertor; then the following reltion tkes plce: ˆδ δq ˆQ b δq b. 7 Indeed, by virtue of 10, 11, nd 4, we cn write ˆδL ˆδ δq ˆδ d ELL dq d EL dq df d EL dq ˆQ b d EL L dq b ˆQ b δq b ˆQ b δq b. A generliztion of 7 bsed on the reltion 4 reds: ˆδ dk k δq dk ˆQ b k δq b. 8 g Symmetry trnsformtions trnsform genuine trjectories into genuine trjectories. Indeed, suppose tht q be genuine trjectory, tht is δq 0, 9 q nd δq be symmetry trnsformtion. Then the trnsformed trjectory q q + δq is lso genuine one. Indeed, by virtue of 7 nd 9, we get: δq q q+δq δq + ˆδ q δq q C. Trivil symmetries δ b ˆQ b δq b 0. q Below, we re going to describe so-clled trivil symmetries trnsformtions, which exist for ny ction. A symmetry trnsformtion is clled trivil symmetry trnsformtion whenever the corresponding trjectory vrition hs the form δq Û b δq b, 30

5 136 Brzilin Journl of Physics, vol. 36, no. 1B, Mrch, 006 where Û is n ntisymmetric LO, tht is Û T b Û b. Thus, trivil symmetry trnsformtions do not ffect genuine trjectories. One cn prove, see below, tht ny symmetry trnsformtion tht vnishes on the equtions of motion, δq O/δq, is trivil, nmely it hs the form 30. With the help of reltions 5 nd 18, we cn esily verify tht 30 is ctully symmetry trnsformtion. Indeed, ˆδL d ELL dq b Û b d ELL dq b + dp df + dp d F + P, where F nd P re some LF. Since trivil symmetry trnsformtions re proportionl to the equtions of motion, they do not chnge genuine trjectories, s ws lredy mentioned bove. The commuttor of symmetry opertor nd trivilsymmetry opertor is trivil-symmetry opertor. Nmely, if ˆδ 1 L df 1 /, ˆδ L df /, ˆδ q δ q ˆV b /δq b, then [ˆδ1, ˆδ L ˆδ 3 L, ˆδ 3 q δ 3 q b Û δq b, 31 where ˆV b nd Û b re some ntisymmetric LO. To verify 31, we remrk tht, ccording to 1, ˆδ 3 is symmetry opertor, with δ 3 q ˆδ 1 δ q ˆδ δ 1 q, where δ 1 q ˆδ 1 q. The term ˆδ 1 δ q cn be clculted with the help of 14, ˆδ 1 δ q δ q [ d k cb ˆV q c[k k δq b, nd the term ˆδ δ 1 q cn be clculted with the help of 7, ˆδ δ 1 q ˆδ ˆV b δq b + ˆV b ˆδ ˆδ δq b ˆV b δq b ˆV b ˆQ c b δq c. Thus, we obtin: ˆδ 3 q δ 3 q Û b /δq b, where Û b is n ntisymmetric LO of the form [ Û b δ q d k ˆV cb q c[k + ˆV c d k δ q b q c[k ˆδ ˆV b. We cll two symmetry trnsformtions δ 1 q nd δ q equivlent δ 1 q δ q whenever they differ by trivil symmetry trnsformtion: δ 1 q δ q δ 1 q δ q b Û δq b. 3 Here Û T b Û b. Let GS be the Lie lgebr of ll symmetries of the ction S. The trivil symmetries form the idel G tr S in the Lie lgebr GS. Then the clsses of equivlent symmetries form Lie lgebr G Ph S isomorphic to the quotient lgebr: III. G Ph S GS/G tr S. DYNAMICALLY EQUIVALENT ACTIONS Very often we encounter n ction S E [q,y L E q [,y [, 33 which contins two groups of coordintes q [ nd y [ such tht the Euler Lgrnge llow one to express ll y vi q [. It is convenient to cll S E [q,y the extended ction. One cn try to eliminte the vribles y from the extended ction to get some reduced ction, which depends now only on q, nd sk the question: Wht is the reltion between the extended nd the reduced ctions? There exist cse when this question hs definite nswer [, 5. Nmely, let us suppose tht the Euler Lgrnge E [q,y/δy 0 llow one to express uniquely the vribles y s LF of the vribles q, E [q,y δy Then we define the reduced ction S[q S[q S E [q,ȳ 0 y ȳ q [. 34 L E q [,ȳ [ L q [. 35 Let us compre the Euler Lgrnge tht correspond to both ctions. First consider the vrition of the reduced ction under rbitrry inner vritions δq,

6 D. M. Gitmn nd I. V. Tyutin 137 E [q,y [q δq i δq i + E [q,y [q yȳ δy α δȳ α yȳ δq i δq i. 36 In virtue of 34, the Euler Lgrnge of the reduced ction red [q δq E [q,y δq yȳ On the other hnd, the Euler Lgrnge of the extended ction S E [q,y re E [q,y δq 0, E [q,y δy 0 y ȳ q [. They re reduced to 37 in the q-sector. We cn see tht the extended ction nd the reduced ction led to the sme Euler Lgrnge for q. This is why the vribles y re clled the uxiliry vribles. The uxiliry vribles y cn be eliminted from the ction with the help of the Euler Lgrnge. Further, we cll the ctions S E [q,y nd S[q the dynmiclly equivlent ctions. One ought to stress tht the bove equivlence is consequence of the ssumption tht the vribles y re expressed vi q by mens of the equtions /δy 0 only. If, for this purpose, some of the equtions /δq 0 re used s well, then the bove equivlence cn be bsent. Of course, the solutions of the Euler Lgrnge for the reduced ction, together with the definition y ȳ, contin ll solutions of the Euler Lgrnge for the extended ction s it is esily seen from Eq. 36. However, the reduced ction cn imply dditionl solutions. Actions contining uxiliry vribles nd the corresponding reduced ctions hve similr properties, in prticulr, there exists direct reltion between their symmetry trnsformtions. As ws mentioned bove, we re going to relte the symmetry properties of the extended nd reduced ctions. To this end, it is convenient to mke n invertible coordinte replcement, q,y α q A q,z α, y z + ȳ q [l, in the extended ction. In fct, we re going to consider modified extended ction S[ q, which is obtined from the extended ction S E [q,y s follows: S[ q L q [ S E [q,z + ȳ L E q [,z [ + ȳ [. 38 The extended ction S E [q,y nd the modified extended ction S[ q re completely equivlent. They led to completely equivlent Euler Lgrnge. Thus, it is sufficient to study the reltion between the symmetry properties of the modified extended ction S[ q nd the reduced ction S[q. Note tht S[q S[ q, L q [ L q [. 39 Besides, the ction 38 cn be presented in the form S[ q S[q + S[ q, S[ q L, L E q [,z [ + ȳ [ L E q [,ȳ [. 40 The vribles z re uxiliry ones for the ction S[ q, nd, in prticulr, z 0 on the Euler Lgrnge. Indeed, δ S[ q δz 0 E[q,y δy The ltter implies: 0 y ȳ q [ z δ S δz α δ S δz α Û αβ z β 0. 4 Since eqution 41 hs the unique solution z 0, one cn esily verify tht Û is n invertible LO. The eqution 4 implies L z α ˆK αβ z β + d F, 43 where ˆK is symmetric LO, nd F is n LF. Besides, one cn write z α Û 1 αβ δ S δz β Û 1 αβ δ S δz β. 44 On the other hnd, due to the property 11, one cn write δ S δq d EL L dq d [ EL dq z α ˆK αβ z β. Then, tking into ccount 43, 44, nd the definition of the Euler Lgrnge derivtive, we get the following useful reltion: δ S δq ˆΛ α δ S δz α, ˆΛ α d l z ν ˆK νβ Û 1 βα, 45 q [l where ˆΛ α is n LO. IV. SYMMETRIES OF THE EXTENDED AND THE REDUCED ACTIONS There exists one-to-one correspondence isomorphism between the symmetry clsses of the extended ction S[ q nd the reduced ction S[q. Below, we prove set of ssertions, which justify, in fct, this correspondence.

7 138 Brzilin Journl of Physics, vol. 36, no. 1B, Mrch, 006 i If the trnsformtion δ q A δ q δz α, 46 is symmetry of the extended ction S, then the trnsformtion δq δ q 47 is symmetry of the reduced ction S. Indeed, let 46 be symmetry of the ction S. Then ˆδ δ q L d F, 48 where F is n LF. Considering 48 t z δz 0, we get ˆδ δq L d F, δq δ q, F F, where L is given by 39. Thus, ny symmetry of the ction S implies symmetry of the ction S. The symmetry δq obtined in such wy cn be clled the symmetry reduction of the extended ction. ii If the trnsformtion δq is symmetry of the reduced ction S, then the trnsformtion δ q A δq δz α, δz α ˆΛ T α δq, 49 where the LO ˆΛ defined by Eq. 45 is symmetry of the extended ction S. To prove this ssertion, let us consider the first vrition ˆδ δ q L of the Lgrnge function L. Since δq is symmetry of the reduced ction S, the reltion ˆδ δq L df/, where F is n LF, holds true. Thus, with the help of the property 15, one my write the vrition ˆδ δ q L in the form ˆδ δ q L ˆδδq + ˆδ δz L d ˆδδq F + + ˆδ δz L. 50 Now, we present the vritions ˆδ δq L nd ˆδ δz L with the help of reltion 18. Besides, tking into ccount the expression 49 for the vrition δz, we get ˆδ δ q L d F + P q + P z + δq δ S [ δq ˆΛ T α δq δ S δz α, 51 where P q nd P z re some LF. Using 45 nd 5, we my write δq δ S δq δq ˆΛ α δ S [ δz α ˆΛ T α δq δ S δz α + dg, 5 where G is n LF. Thus, the vrition ˆδ δ q L is reduced to the totl derivtive of n LF, ˆδ δ q L d F + P q + P z + G. Thus, δ q is symmetry of the extended ction S. iii Any symmetry of the form 0 δ q δz of the extended ction S is trivil. Since δ q is symmetry of the ction S, one cn write 53 ˆδ δ q L ˆδ δz L df, 54 where F is n LF. Tking into ccoun8, we my rewrite Eq. 54 s δz α δ S δz α df, 55 where F is n LF. The left-hnd side of eqution 55 cn be trnsformed, with the help of 4 nd 5, to the form δz α δ S δz α δzα Û αβ z β [ Û T βα δz z β + df, where F is n LF. Thus, the eqution 55 my be reduced to z β f β dφ, f β Û T βα δz, 56 where f Q [ nd Φ Q [ re some LF. Let us present the LF Φ s Φ Q [ Φ 0 q [ + Φ 1 Q [, Φ 0 Φ, Φ 1 N Φ αk Q [ z α[k, N <. 57 It follows from eqution 56 tht dφ 0 / 0. According to 6, the ltter implies Φ 0 const. From 56, we get the eqution where N+1 ϕ αk z α[k 0, 58 ϕ α0 f α Φ α0, ϕ αn+1 Φ αn, ϕ αk [ Φ αk 1 + Φ αk, k 1,...,N. 59 The generl solution of Eq. 58 is N+1 ϕ αk m αk βsl z β[s, m αk βsl m βs lαk, 60 where m αk βsl Q [ re some LF. Then the LF Φ αk nd f α cn be found from Eq. 59: Φ αk f α N+1 m, N k m0 N+1 d m [ m αk+m+1 βl z β[l, d m [ m αm βl z β[l ˆm αβ z β, 61

8 D. M. Gitmn nd I. V. Tyutin 139 where ˆm αβ is n ntisymmetric LO. Thus, we get from 56 δz α ˆM αβ δ S [ Û δz β, ˆM αβ T 1 αγ Û 1 δβ ˆmγδ, 6 where ˆM αβ is n ntisymmetric LO. Therefore, the symmetry 53 is trivil. iv Suppose both trnsformtions δ q 1 nd δ q to be symmetries of the extended ction S such tht their reductions coincide, tht is δ q 1 δ q δq. 63 Then these symmetries re equivlent, δ q 1 δ q, 64 which mens tht δ q 1 nd δ q differ by trivil symmetry. Thus, we hve to prove tht the trnsformtion q q δ q 1 δ q δ q 1 δ q, q z δz 1 δz 0, is trivil symmetry of the extended ction S. In virtue of Eq. 63, the LF q my be presented s q ˆm αz α, 65 where ˆm is n LO. With the help of 44, we get for q the following expression: q ˆM β δ S δz β, 66 where ˆM ˆmÛ 1 is n LO. Let us present the trnsformtion q in the form q 1 q + q, where nd 1 q ˆM AB δ S δ q B, ˆM AB 0 ˆM β ˆM T αb 0, 67 0 q σ. 68 The trnsformtions 1 q is trivil symmetry since the LO ˆM AB is ntisymmetric, tht is ˆM T AB ˆM AB. Thus, q is symmetry of the extended ction S. Besides, the ltter symmetry hs specil form 68. It ws proven in item c tht ny symmetry of such form is trivil. Therefore, the symmetry q is trivil s well. v Let trnsformtion δ q be trivil symmetry of the extended ction S. Then its reduction δq is trivil symmetry of the reduced ction S. According to this ssumption, we my write δ q A δ q ˆM b δ S δq b + ˆM β δ S δz β δz α ˆM T bα δ S δq b + ˆM αβ δ S δz β, 69 where the locl opertors ˆM b nd ˆM αβ re ntisymmetric. Then the reduction δq δ q of the trnsformtion 69 reds δq b ˆm δq b, ˆmb ˆM b. 70 The LO ˆm b is ntisymmetric. Thus, 70 is trivil symmetry of the reduced ction S. vi Let symmetry δq of reduced ction S be trivil. Then ny extension of this symmetry to the symmetry δ q of the extended ction S is trivil s well. Since δq is trivil symmetry, one cn write δq b ˆm δq b, where ˆm b is n ntisymmetric LO. Consider the following extension of the symmetry δq : δ δ q 1 q, δ 0 q ˆm b δ S δq b, 71 which is trivil symmetry of the extended ction S. Any other extension of δq differs from δ q 1 by trivil symmetry, ccording to item iv. Therefore, ny extension of the trivil symmetry is trivil symmetry s well. Concluding, we cn see tht there exists n isomorphism between clsses of equivlent symmetries of dynmiclly equivlent ctions. Since the Lgrngin nd Hmiltonin ctions re dynmiclly equivlent, one cn study the symmetry structure of ny singulr theory considering the first-order Hmiltonin ction. Acknowledgement Gitmn is grteful to the founions FAPESP, CNPq for permnent support nd to the Lebedev Physics Institute Moscow for hospitlity; Tyutin thnks RFBR nd LSS for prtil support. [1 P. G. Bergmnn, Phys. Rev. 75, ; P.M. Dirc, Cn. J. Mth., ; Lectures on Quntum Mechnics Yeshiv University, New York 1964; E. C. G. Sudrshn, N. Mukund, Clssicl Dynmics: A Modern Perspective Wiley, New York 1974; D. M. Gitmn nd I.V. Tyutin, Quntiztion of Fields with Constrints Springer-Verlg, Berlin 1990; M. Henneux

9 140 Brzilin Journl of Physics, vol. 36, no. 1B, Mrch, 006 nd C. Teitelboim, Quntiztion of Guge Systems Princeton University Press, Princeton 199 [ V. A. Borochov nd I. V. Tyutin, Physics of Atomic Nuclei, 61, ; ibid 6, [3 D. M. Gitmn nd I. V. Tyutin, Resenhs IME-USP, v.6, U116/3, ; J. Phys. A 38, [4 D. M. Gitmn nd I. V. Tyutin, Nucl. Phys. B 630, [5 M. Henneux, Phys. Lett. B38, ; I. A. Btlin nd I. V. Tyutin, Int. J. Mod. Phys. A11, ; D. M. Gitmn, nd I. V. Tyutin, Nucl. Phys. B 630, [6 The functions F my depend on time explicitly, however, we do not include t in the rguments of the functions. [7 The functions L my depend on time explicitly, however, we do not include t in the rguments of the functions. [8 Inner vritions vnish together with ll their time derivtives t nd t. [9 Sometimes, we mrk the trnsformtion opertor below by the corresponding vrition.