Question

Assigment 1.2. [10 points] A particle of mass m moves along x axis under the action of the force F--kx2n1 where n is an integer number. Show that this motion is periodic [2 points]. Denote by A the amplitude of the oscillations, so that | xISA. Find how the period T depends orn the amplitude A (1) for n21 [2 points]; (2) n 0 [2 points]; (3) n--1 [2 points]; (4) n <-1 [2 points).

I can do the first problem which is show the motion is periodic. The rest questions are hard for me. I found a similar question on the p27 of ‘mechanics’ by landau which shows on the second picture

But I can’t understand the math. Please help.

Answer 1

It is easy to understand beta function and you go through maturities formula which directly put in eqn so, it y^n = u y = (u)^1/n dy/du = 1/n (u)^1/n -1 so T = 2 squareroot 2m/E (E/A)^1/n integral^1_0 1/n (u)^1 - n/n du/squareroot 1 - u T = 2/n squareroot 2m/E (E/A)^1/n integral^1_0 (u^1/n -1) du/squareroot 1 - u T = 2/n squareroot 2m/E (E/A)^1/n integral^1_0 u^1/n/u squareroot 1 - u du so, T = 2/n squareroot 2m/E (E/A)^1/n squareroot (E/A)^1/m squareroot pi gamma (1/n)/gamma (1/2 + 1/n) T = 2/n squareroot 2 pi m/E (E/A)^1/n gamma (1/n)/gamma (1/2 + 1/n) please more derail phi of beta function go math but not need g thick. \r\n Now derive eqn (2) y = (A/E)^1/n x [y^n = A/E x^n y = (A/E)^1/n (x^n)^1/n y = (A/E)^1/n x] dy/dx = (A/E)^1/n times L dy = (A/E)^1/n middot dx (E/A)^1/n middot dy = dx equation (3) Now, from eqn (1) T = 2 squareroot 2m/E integral