8.8** Two masses m₁ and m₂ move in a plane and interact by a potential energy U (r) = kr². Writedown their Lagrangian in terms of the CM and relative positions R and r, and find the equations ofmotion for the coordinates X, Y and x, y. Describe the motion and find the frequency of the relativemotion.

Given, Two masses m1 and m2 move in a plane and interact by a potential energy Ur=12kr2 The kinetic energy is determined by T=12MR.2+12m1m2m1+m2r.2Put R.2=X.2+Y.2r.2=x.2+y.2T=12M(X.2+Y.2)+12(m1m2m1+m2)(x.2+y.2) The Lagrangian can be written as L=T-UL=12MX.2+Y.2+12m1m2m1+m2x.2+y.2-12kr2L=12M(X.2+Y.2)+12(m1m2m1+m2)(x.2+y.2)-12k(x2+y2)The equation of motion can be determined as For variables x and y ddt∂L∂x.-∂L∂x=0ddtm1m2m1+m2x.+kx=0m1m2m1+m2x..+kx=0.........(1)ddt∂L∂y.-∂L∂y=0ddtm1m2m1+m2y.+ky=0m1m2m1+m2y..+ky=0.........(2) From the above equation, the frequency can be obtained ω=(m1+m2)km1+m2For variables X and Y, Equation of motion can be determined as ddt∂L∂X.-∂L∂X=0ddtMX.=0MX..=0.........(3)ddt∂L∂Y.-∂L∂Y=0ddtMY.=0MY..=0.........(4) From motion equations (3) and (4) the motion of the object is linear.

Answered: 1 8.8** Two masses m₁ and m2 move in a…