Next Page
Previous Page
Home
【SEOtext】0. 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,y2∈R3;q1,q2∈R3(R), b. δI(q1,q2)/δqij(t)=0⇔∂L/∂qij(t)-d/dt∂L/∂[∂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|R∈R3×3,RTR=1},R∈O(3)⇒det R=±1, SO(3)={R|R∈O(3),det R=+1} b. Parametrization and Generators ∀R∈SO(3),∃θ∈R3;R=exp(-iθL) where: Li∈R3×3,(Li)jk=-iεijk {[Li,Lj]=iεijkLk,Li†=Li, R-1LiR=RijLj, [exp(-iθL)]kl=δkl cos√θ2+(θkθl/θ2)(1-cos√θ2)-εklj(θj/√θ2)sin√θ2 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