A particle of mass moves in the potential ( , )

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Oscillations and normal modes

Modes, Packets and Media, Ch The anharmonic oscillator: frequency, harmonics and drift

Problem

A particle of mass

m

moves in the potential

V(x) = c|x|^n

(

c > 0

n > 0

), with no damping or driving. ((a)) Starting from the period-as-an-energy-integral

T(E) = \sqrt{2m}\displaystyle\int_{x_-}^{x_+}\frac{\mathrm{d}x}{\sqrt{E-V(x)}}

, show by the substitution

x = (E/c)^{1/n}u

that

T \propto E^{1/n-1/2}

, with a proportionality constant independent of

E

. ((b)) For the pure quartic well (

n=4

), express this scaling in terms of the amplitude

A

instead of the energy

E

(using

E = cA^4

), and hence find the ratio

T_2/T_1

of the periods when the release amplitude is doubled (

A_2 = 2A_1