Quiz 1#

Physics 141/241: Computational Physics I

Instructor: Javier Duarte

Spring 2023

Due date: Friday, April 14, 2023 5pm

Total points: 10

Submission Instructions#

Problem 1 (141/241) [10 points]#

In this problem, you will prove that leapfrog integration has exact time-reversal symmetry. Recall the kick-drift-kick leapfrog update equations, which tell you how to go from \((x_n, v_n) \to (x_{n+1}, v_{n+1})\):

\[\begin{split} \begin{align} v_{n+1/2} & = v_n + a(x_n) \Delta t / 2\\ x_{n+1} &= x_n + v_{n+1/2} \Delta t \\ v_{n+1} &= v_{n+1/2} + a(x_{n+1}) \Delta t / 2 \end{align} \end{split}\]

Consider the time-reversed problem, with variables we’ll denote with a prime: \(t', x', v', a'\). By flipping the “arrow of time,” some variables are negated with respect to the forward problem. That is, \(t' = -t\) and \(v' = -v\), but \(x' = x\) and \(a' = a\) are unchanged.

Prove that if you start from the end point in the time-reversed problem \(x'_{n} = x_{n+1}\) and \(v_{n}' = -v_{n+1}\), then apply the leapfrog update rule, you arrive back at the starting point: \(x'_{n+1} = x_{n}\) and \(v'_{n+1} = -v_{n}\).

Hint: recall that \(\Delta t' = - \Delta t\).

Problem 2 (141/241) [10 points]#

For a uniform sphere of mass \(M\) with radius \(R\), prove that the gravitational potential \(\Phi\) inside the sphere is related to the mass density \(\rho\) by the following equation (from lecture 5):

\[ \begin{align} \Phi(r) & = -2\pi G\rho(R^2 - \frac{r^2}{3}) & r \leq R \end{align} \]

where \(\rho = M/(4\pi R^3 /3)\).

Hint: Start from the equations derived from the infinitesimal spherical shell:

\[\begin{split} \begin{align} \Phi(r) &= G\int_{r_0}^r dr' ~\frac{M(r')}{r'^2}\\ M(r) & = 4\pi\int_{r_0}^r dr'~ r'^2 \rho(r') \end{align} \end{split}\]