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1. (a) Suppose an Eulerian representation of a steady two-dimensional velocity field at a
point x = (x, y) is u = (y, −x). Find the Lagrangian position xL(ζx, ζy, t) of the fluid particle which is at position (ζx, ζy) at time t = 0. (b) Solve, by the method of characteristics, the partial differential equation for u(x, t) given by
t ∂u ∂t
- x ∂u ∂x = 0, with u = ex^ when t = 1.
(c) You are given that the 1 -dimensional motion of an isentropic gas is governed by the equations ( ∂ ∂t
∂x
u + 2 a γ − 1
∂t
∂x
u − 2 a γ − 1
where a^2 = γpρ , with p the pressure and ρ the density. A trivial solution of these equations, describing a stationary gas of density ρ 0 at pressure p 0 , is given by u = 0, ρ = ρ 0 , p = p 0 where ρ 0 and p 0 are constants. Suppose now that this stationary gas is perturbed slightly and let the small perturbation to the velocity field be |uˆ(x, t)| 1. Find the linearized equation governing the evolution of uˆ(x, t). Hence find the speed of small-amplitude linearized waves in this gas.
2. Consider the kinematic wave equation for u(x, t)
∂u ∂t
where k is a positive constant. (a) Verify that an implicit form of the solution to the kinematic wave equation is given by u = f (x − (u + k)t) where f is an arbitrary differentiable function. (b) Use part (a) to find an explicit solution to the problem when the initial condition is given as
u(x, 0) =
1 + 2x, − 1 / 2 ≤ x ≤ 0 , 1 − x, 0 ≤ x ≤ 1 , 0 , otherwise.
(c) A shock forms in the solution to this initial value problem at some time ts. Find ts and the position xs at which the shock first forms.
4. (a) Let A, B ∈ RN^ be symmetric, positive definite matrices and consider the
Lagrangian
L =
∑^ N
i,j=
aij q˙i q˙j −
∑^ N
i,j=
bij qiqj.
(i) Write down the Euler-Lagrange equations. (ii) Consider the case A = B = I, where I is the identity matrix. Show that this system is integrable. Solve the Euler-Lagrange equations. (b) A particle of mass m moves in R^3 under a central force
F (r) = − dV dr
(i) Write down the Lagrangian in spherical coordinates
(x, y, z) = (r cos(φ) sin(θ), r sin(φ) sin(θ), r cos(θ)).
(ii) Show that h, defined by h = mr^2 φ˙ sin^2 (θ) is an integral of motion.
