Question 1

Functional and variations.

Consider the functional

(a) Show that if Δ = S[y + εg] - S[y] then to second order in ε,

(b) g(1) = g(2) show that by choosing

x2 +y' =c , y(1) = 0, y(2) = B,

where c is a positive constant, the term O(ε) in the expansion for Δ vanishes. Solve this equation for y(x) to show that a stationary path of S[y] is given by

                     y(x) = (B+7/3)x - 1/3x³ -B - 2y

(c) Show that provided that B> -7/3, the coefficient of ε² in the expansion of Δ is negative. What is the significance of Δ being negative?

Question 2

Euler-Lagrange equation.

(a) Write down the Euler-Lagrange equation for the functional

(b) Solve this Euler-Lagrange equation, and explain the significance of the solution.

Question 3

Changing variables in variational problems.

This question is about the functional

(a) Find the value of β such that when expressed in terms of the new independent variable u, where x = u^βx = uβ , the functional is equivalent to the functional

Where a = A^β and b = B^β

(b) Using the first-integral of the Euler-Lagrange equation associated with the functional obtained in part (a), show that its general solution is

y(u) = 1/d-cu'

whered and c are arbitrary constants.

(c)    Using the result derived in part (b), deduce that the general solution of

d²y/dx² - 2/y(dy/dx)² + 3/x(dy/dx) = 0

is

y = x²/dx² - c

Question 4

Lagrange's equations and Hamilton's principle. 

A particle of mass m moves on the surface of an inverted cone. In cylindrical polar coordinates (r, φ, z), the apex of the cone is at r = z = 0, and the height of the surface at a distance r from the axis is z = αr (with α > 0).

(a) Using r and φ as the generalised coordinates, show that the kinetic and potential energies of the particle are respectively

T = m/2[(1+α²)r² + r²ψ²]

and
V = mgar

(b) Write down the Lagrangian for this system, and hence derive the equations of motion. Show that the equation of motion for φ implies that r²φ, where K is a constant. Hence obtain an equation of motion for r that does not contain φ or its derivatives.     [12]   

(c) Show that there is a solution of the equations of motion where r and φ? take constant values, r₀ and ? respectively. Obtain a relation between ? and r₀.

(d) There also exists a solution in which r(t) makes small oscillations about r₀, with angular frequency ω.

By substituting r(t) = r₀ + ε sin(ωt) into the equation of motion for r and neglecting terms of order ε² and above, determine the frequency of these small oscillations, and show further that

ω/Ω = √3/1+α²