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ySEOtextz0. General Formulation, (3) Examples of Classical Lagrangian Formulation, (i) Newton's Particles and Euclidean Group, a. L(x1,x2;y1,y2)=(m/2)[(y1)2+(y2)2]-V((x1-x2)2), I(q1,q2)=dtL(q1(t),q2(t);0q1(t),0q2(t)), x1,x2,y1,y2R3;q1,q2R3(R), b. I(q1,q2)/qij(t)=0́L/qij(t)-d/dtL/[0qij(t)]=0{md2q1(t)/dt2+2[q1(t)-q2(t)]V'([q1(t)-q2(t)]2)=0, md2q2(t)/dt2+2[q2(t)-q1(t)]V'([q1(t)-q2(t)]2)=0 where: V'(x)=(d/dx)V(x), c. Symmetry 1. O(3) and SO(3) a. Definition, O(3)={R|RR3~3,RTR=1},RO(3)det R=}1, SO(3)={R|RO(3),det R=+1} b. Parametrization and Generators RSO(3),΃ƁR3;R=exp(-iL) where: LiR3~3,(Li)jk=-iijk {[Li,Lj]=iijkLk,Li=Li, R-1LiR=RijLj, [exp(-iL)]kl=kl cos2+(kl/2)(1-cos2)-klj(j/2)sin2 c. O(N) and SO(N), In the same way, O(N) and SO(N) are defined. For example, SO(2) is a set of matrices of

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