Here ∆t = tf − ti

Classical Mechanics 2, Spring 2014 CMI Problem set 4

Due by the beginning of lecture on Monday Jan 27, 2014 Harmonic oscillator and action principle

1. h5i Recall that the general solution of ¨x = −ω²x is x(t) = acosωt+bsinωt where a,b are constants of integration. Find the unique classical trajectory connecting x(ti) = xi and x(tf)= xf

assuming ω∆t , nπ for any integer n. Here ∆t = t_f − t_i. You may use the abbreviations ci =cosωti, sf =sinωtf etc.

2. h17iConsider a particle of mass m in the potential V(x) = ¹₂mω²x². Suppose x(t) is a trajectory between xi(ti) and xf(tf) and let x(t)+δx(t) be a neighboring path with δx(ti)=δx(tf)=0 .

(a) h5i Write the classical action of the path x+δx as a quadratic Taylor polynomial in δx.

Show that you get the following expression. What can you say aboutS₁? S[x+δx]=S₀+S₁+S₂=S[x]−

Z t_f

t_i

(mx¨+mω²x)δx dt+Z t_f t_i

"

1

2m(δx)˙ ²−1

2mω²(δx)²

# dt

(b) h2iFor what values ofκ is x(t)+δx(t) a legitimate neighboring path for the variation δx(t)= sinκ(t−t_i) ? (1) (c) h3iEvaluateS₂[δx] for all the allowed values of κ.

(d) h3iTake ∆t= t_f −t_i =10 s and ω=1 Hz. Find a path that can be made arbitrarily close to the trajectory x(t) , whose action islessthan that of x(t) .

(e) h3iTake ∆t= tf −ti =10 s and ω=1 Hz. Find a path that can be made arbitrarily close to the trajectory x(t) , whose action ismorethan that of x(t) .

(f) h1iWhat sort of an extremum of action is the classical trajectory?

1