A classical particle with spin realized

Transcription

1 A classical particle wit spin realized by reduction of a nonlinear nonolonomic constraint R. Cusman D. Kemppainen y and J. Sniatycki y Abstract 1 In tis paper we describe te motion of a nonlinear nonolonomically constrained system wic after reduction realizes a nonrelativistic classical particle wit spin. 1 Introduction Te usual pysical interpretation of spin is tat of internal angular momentum. Te main dierence between classical spin and angular momentum of a rigid body is tat te lengt of te spin vector is xed a priori, wile te lengt of te angular momentum vector J is a dynamical variable. Tis suggests tat te dynamics of a particle wit spin sould be related to tat of a rigid body by restricting te pase space to tose points were te lengt of te angular momentum vector is xed and ten reducing te rigid body degrees of freedom. In te present paper we carry out tis program and obtain Souriau's formulation of te dynamics of a particle wit spin, see Souriau [5]. Since te constraint given by xing te lengt of te angular momentum vector is nonlinear in velocities we are lead into te controversial eld of dynamics of systems wit nonlinear nonolonomic constraints, see Arnol'd [1] or Naimark and Fufaev [4]. Te main problem of deciding wat te Matematics Institute, Budapestlaan 6, University of Utrect, 3508TA Utrect, te Neterlands ydepartment of Matematics and Statistics, 2500 University Dr. N.W., Calgary, Alberta, Canada, T2N 1N February

2 dynamics of suc a system sould be is avoided ere, because te postulated constraint is preserved and weknow te dynamics of te rigid body. Tus we are left wit te problem of decribing te reduction of a system wit nonlinear nonolonomic constraints. To solve tis problem we use te procedure of Bates and Sniatycki [2]. We avetwocecks tat tis approac is correct: (i) our procedure gives te well establised Souriau model and (ii) it is equivalent to Marsden-Weinstein reduction. 2 Te unconstrained system Consider a uniformly carged sperically symmetric rigid body wit all its principal moments of inertia equal to I aving a magnetic moment qj. Impose a constant magnetic eld b =(b 1 ;b 2 ;b 3 ). Matematically tis unconstrained system is described as follows. Its conguration space is te tree dimensional rotation group SO(3) and its pase space (after trivialization by left translation) is SO(3) so(3), were so(3) is te Lie algebra of 3 3skew symmetric matrices. On pase space we ave te symplectic form!(a; X) (AY 1 ;Z 1 ); (AY 2 ;Z 2 ) = Ik(Y 1 ;Z 2 ), Ik(Y 2 ;Z 1 )+Ik(X; [Y 1 ;Y 2 ]); (1) were (A; X) 2 SO(3)so(3), (AY i ;Z i ) 2 T (A;X) SO(3)so(3) for i =1; 2, and k :so(3)so(3)! R :(X; Y )!, 1 tr XY is te Killing metric. For 2 more details about tis Lie group model for te rigid body, see Cusman and Bates [3]. Te Hamiltonian of te unconstrained system is were : SO(3) so(3)! R :(A; X)! 1 2 Ik(X; X)+qI k(ad AX; B); (2) B = 0,b 3 b 2 b 3 0,b 1,b 2 b 1 0 Te second term in (2) represents te interaction of te carged rigid body wit te magnetic eld. A straigtforward calculation sows tat te Hamiltonian vector eld X associated wit te Hamiltonian as integral curves wic satisfy 8 < A _ = A(X + q Ad A,1B) (3) : _X =0 1 A. 2

3 Tere are two Hamiltonian actions of SO(3) on our unconstrained system. First, te \body" action, corresponding to te cange of frame xed in te body, wic is given by te lift of rigt multiplication to te tangent bundle T SO(3) of SO(3). After left trivialization te body action becomes : SO(3) (SO(3) so(3))! SO(3) so(3) : C; (A; X)! (AC,1 ; Ad C X): (4) Second, te \space" action, corresponding to rotation in pysical space, given by left multiplication on T SO(3). After left trivialization te space action becomes SO(3) (SO(3) so(3))! SO(3) so(3) : C; (A; X)! (CA;X): (5) Te rigt action (4) is a symmetry of te unconstrained system wereas te left action (5) is not, due to te presence of te magnetic eld. Te momentum of te left action J :SO(3) so(3)! so(3) : (A; X)! IAd A X; (wic is pysically te angular momentum of te body), is not conserved by te unconstrained system. 3 Te constrained system Now constrain te Hamiltonian system (; SO(3) so(3);!) to te submanifold M = n (A; X) 2 SO(3) so(3) k(x; X) =1 o : (6) Tis corresponds to requiring tat te magnitude of te angular momentum of te rigid body is I. (In a similar way we can look at any nonzero magnitude of te angular momentum.) Because te magnitude of te angular momentum is conserved, te unconstrained vector eld X is tangent tom. By its very denition M is a nonlinear nonolonomic constraint. M determines a constraint 1-form ' on SO(3) so(3) given by '(A; X)(AY; Z) =k(x; Y ) (7) 3

4 Because M is not a subbundle of TM, te constrained system (M;X jm;') is not linear nonolonomic and terefore falls outside te teory of Bates and Sniatycki. Te 1-form ' and te tangent bundletm of M determine a constraint distribution H on SO(3) so(3) dened by H (A;X) = ker '(A; X) \ T (A;X) M = n (AY; Z) 2 T (A;X) SO(3) so(3) k(x; Y )=k(x; Z) =0 o : (8) Observe tat te constrained vector eld X jm does not lie in te constraint distribution H. To remedy tis, we note tat H (A;X) is a symplectic subspace of T (A;X) (SO(3) so(3));!(a; X). Terefore we may write T (A;X) (SO(3) so(3)) = H (A;X) H? (A;X) for every (A; X) 2 (SO(3) so(3)). Here H? is te symplectic perpendicular of H (A;X). (A;X) We can decompose X jm into its components on H (A;X) and H? : (A;X) X (A; X) =X H (A; X)+X H? (A; X): A calculation sows tat for every (A; X) 2 M, and X H (A; X) = A(q Ad A,1B, qk(x; Ad A,1B)X); 0 (9) X H? (A; X) = A(X + qk(x; Ad A,1B)X); 0 : (10) Looking forward to te next section, (see te proof of (14)), we observe tat is killed by te reduction process. X H? 4 Symmetry and its reduction We nowsow tat te nonolonomic system (M;X H jm;') as a symmetry. Consider te action given by (4). Since te constraint manifold M is invariant under, te induced action e = j(so(3)m) is dened. Because te 1-form ' is e -invariant, it follows tat te constraint distribution H is 4

5 also e -invariant. Consequently, e isasymmetry of te nonolonomic system (M;X H jm;'), tat is, T (A;X) C X H (A; X) = X H ( C(A; X)) for every C 2 SO(3) and every (A; X) 2 M. To gainasomeinsigtinto te dynamical meaning of (M;X H jm;') we remove its SO(3) symmetry following te procedure of Bates and Sniatycki [2]. (We also use teir notation.) Te innitesimal generator of e inte direction Y 2 so(3) is given by tevector eld X Y (A; X) =(,AY;,[X; Y ]). Tus te SO(3) symmetry distribution V on M is V (A;X) = n X Y (A; X) 2 T (A;X) M Y 2 so(3) o : A calculation sows tat te distribution V \ H on M is given by (V \ H) (A;X) = n (,AY;,[X; Y ]) 2 T (A;X) M k(x; Y )=0 o : Consequently, te distribution U = fu 2 Hj! H (u; v) =0 for every v 2 V \ Hg on M is U (A;X) = n (AY; 0) 2 T (A;X) M k(x; Y )=0 o : (11) Note tat X H (A; X) 2 U (A;X) for every (A; X) 2 M, since Now consider te map k(q Ad A,1B, qk(x; Ad A,1B)X; X) =0: : M! so(3) :(A; X)!,Ik ] (Ad A X)=: (12) Because is constant on e orbits, it induces a map e : M=SO(3)! so(3) on te orbit space M=SO(3). Since te adjoint action of SO(3) is transitive on S = fx 2 so(3) j k(x; X) =1g, te range of (and ence of e) iste SO(3) coadjoint orbit O = n Ad t A,1 A 2 so(3) o ; 5

6 were =,Ik ] (X 0 ) for some X 0 2 S. From te fact tat ber,1 () is a single e orbit, it follows tat te map e is a dieomorpism of M=SO(3) onto O. Terefore is te reduction map of te SO(3) symmetry of (M;X H jm;'). We know tat te distribution U puses down under to te reduced distribution H on O. A calculation sows tat H = n, ad t AdAY k(x; Y )=0; Y2 so(3) o = n, ad t AdAY Y 2 so(3) o = T O : We also know tat te 2-form! H on H puses down under to a symplectic form! H on O. Again a calculation sows tat! H (), ad t AdAY 1 ;,ad t AdAY 2 =,Ad t A [Y 1 ;Y 2 ] : Tus te reduced vector eld is X H () = T (A;X) X H (A; X) =,q ad t B ; were B = B, k(b;ad AX)Ad A X and is given by (12) =,q ad t B: (13) We now sow tat we obtain te same result using Marsden-Weinstein reduction. Below we sow tat X H? (A; X) 2 ker T (A;X) (14) for every (A; X) 2 M. For te moment assume (14). Applying Marsden- Weinstein reduction on M to remove te spatial SO(3) symmetry from te vector eld X jm, we nd tat te reduced vector eld is T (A;X) X (A; X) = T (A;X) X H (A; X) = X H ((A; X)): Tis is te same as (13). We nowprove (14). First, note tat T (A;X) : T (A;X) M! T O :(AY; Z)!,ad t AdAY, Ad t A,1(Ik ] (Z)): (15) 6

7 Using (15) and formula (10) for X H?,we obtain T (A;X) X H? (A; X) = (1 + qk(x; Ad A,1B))T (A;X) (AX; 0) But for every Z 2 so(3) we see tat = (1 + qk(x; Ad A,1B)) ad t AdAX Ik] (Ad A X) : ad t AdAX Ik] (Ad A X) Z = Ik(Ad A X; [Ad A X; Z]) = Ik([Ad A X; Ad A X] ;Z)=0: Tus (14) follows. Te reduced vector eld X H (13) is Hamiltonian on (O ;! H ) wit Hamiltonian : O so(3)! R :! 1 2 (k[ ()), q(b): (16) Pysically, te reduced system (; O ;! H ) is a classical nonrelativistic particle wit spin as dened by Souriau. Tus intrinsic spin in classical mecanics is realized by reducing a nonlinear nonolonomically constrained system. References [1] Arnol'd, V. \Dynamical Systems" vol. III, Encyclopedia of Matematical Science, Springer Verlag, New York, [2] Bates, L. and Sniatycki, J., Nonolonomic reduction, Reports on Matematical Pysics, 32 (1993), 99{115. [3] Cusman, R. and Bates, L., \Global aspects of classical integrable systems", Birkauser, Basel, [4] Naimark, J.I., and Fufaev, N.A., \Dynamics of noolonomic systems", Transl. of AMS, vol. 33, Providence, [5] Souriau, J.-M., \Structure of Dynamical Systems: a symplectic pysics point of view", Birkauser, Boston, 1997 = \Structure des Systemes Dynamique", Dunod, Paris,