This week's exercises focus on solving two-body problems with central forces and the Lagrangian formalism (exercise 5). There are no numerical exercises. The first four exercises are examples taken from our discussions of two-body problems and the relevant theory is to be found in chapter 8 of Taylor. The last exercise involves chapters 6 and 7 of Taylor.
Consider a particle in an attractive potential
$$ V(r)=-\alpha/r. $$The quantity \( r \) is the absolute value of the relative position. We will use the reduced mass \( \mu \) and the angular momentum \( L \), as discussed during the lectures. With the transformation of a two-body problem to the center-of-mass frame, the actual equations look like an effective one-body problem. The energy of the system is \( E \) and the minimum of the effective potential is \( r_{\rm min} \).
The analytical solution to the radial equation of motion is
$$ r(\phi) = \frac{1}{\frac{\mu\alpha}{L^2}+A\cos{(\phi)}}. $$Find the value of \( A \). Hint: Use the fact that at \( r_{\rm min} \) there is no radial kinetic energy and \( E=-\alpha/r_{\rm min}+L^2/2mr_{\rm min}^2 \).
Consider again the same effective potential as in exercise 1. This leads to an attractive inverse-square-law force, \( F=-\alpha/r^2 \). Consider a particle of mass \( m \) with angular momentum \( L \). Taylor sections 8.4-8.7 are relevant background material. See also the harmonic oscillator potential from hw8. The equation of motion for the radial degrees of freedom is (see also hw8) in the center-of-mass frame in two dimensions with \( x=r\cos{(\phi)} \) and \( y=r\sin{(\phi)} \) and \( r\in [0,\infty) \), \( \phi\in [0,2\pi] \) and \( r=\sqrt{x^2+y^2} \) are given by
$$ \ddot{r}=-\frac{1}{m}\frac{dV(r)}{dr}+r\dot{\phi}^2, $$and
$$ \dot{\phi}=\frac{L}{m r^2}. $$Here \( V(r) \) is any central force which depends only on the relative coordinate.
Consider again a particle of mass \( m \) in the same attractive potential, \( V(r)=-\alpha/r \), with angular momentum \( L \) with just the right energy so that
$$ A=m\alpha/L^2 $$where \( A \) comes from the expression
$$ r=\frac{1}{(m\alpha/L^2)+A\cos{(\phi)}}. $$The trajectory can then be rewritten as
$$ r=\frac{2r_0}{1+\cos\theta},~~~r_0=\frac{L^2}{2m\alpha}. $$The solution to the radial function for an inverse-square-law force, see for example Taylor equation (8.59) or the equation above, is
$$ r(\phi) = \frac{c}{1+\epsilon\cos{(\phi)}}. $$For \( \epsilon=1 \) (or the energy \( E=0 \)) the orbit reduces to a parabola as we saw in the previous exercise, while for \( \epsilon > 1 \) (or energy positive) the orbit becomes a hyperbola. The equation for a hyperbola in Cartesian coordinates is
$$ \frac{(x-\delta)^2}{\alpha^2}-\frac{y^2}{\beta^2}=1. $$For a hyperbola, identify the constants \( \alpha \), \( \beta \) and \( \delta \) in terms of the constants \( c \) and \( \epsilon \) for \( r(\phi) \).
This exercise is a follow-up of homework 6. There we studied the so-called Lennard-Jones potential which is widely used in molecular dynamics calculations and in the simulations of quantum liquids. This potential is based on parametrizations from experiments. In molecular dynamics calculations the assumption is that atoms move according to the laws of Newton, given the correct model for interactions. We can say then that quantum-mechanical degrees of freedom stemming from complicated interactions between electrons and protons in an atom, are parametrized in terms of an effective potential.
We will limit ourselves to a two-body problem and the famous Lennard-Jones potential,
$$ \begin{equation} V(r) = 4\varepsilon\left((\frac{\sigma}{r})^{12} - (\frac{\sigma}{r})^6\right), \label{_auto1} \end{equation} $$where \( r \) is the distance between two atoms, \( r=\vert\boldsymbol{r}_i-\boldsymbol{r}_j\vert \), that is the norm of the relative distance vector \( \boldsymbol{r} \). The quantities \( \sigma \) and \( \varepsilon \) are parameters which determine which chemical compound is modelled. This potential is a good approximation for noble gases like helium, neon, argon and other.
The following gives you an opportunity to earn five extra credit points on each of the remaining homeworks and ten extra credit points on the midterms and finals. This assignment also covers an aspect of the scientific process that is not taught in most undergraduate programs: scientific writing. Writing scientific reports is how scientist communicate their results to the rest of the field. Knowing how to assemble a well written scientific report will greatly benefit you in you upper level classes, in graduate school, and in the work place.
The full information on extra credits is found at https://github.com/mhjensen/Physics321/blob/master/doc/Homeworks/ExtraCredits/. There you will also find examples on how to write a scientific article. Below you can also find a description on how to gain extra credits by attending scientific talks.
This assignment allows you to gain extra credit points by practicing your scientific writing. For each of the remaining homeworks you can submit the specified section of a scientific report (written about the numerical aspect of the homework) for five extra credit points on the assignment. For the two midterms and the final, submitting a full scientific report covering the numerical analysis problem will be worth ten extra points. For credit the grader must be able to tell that you put effort into the assignment (i.e. well written, well formatted, etc.). If you are unfamiliar with writing scientific reports, see the information here
The following table explains what aspect of a scientific report is due with which homework. You can submit the assignment in any format you like, in the same document as your homework, or in a different one. Remember to cite any external references you use and include a reference list. There are no length requirements, but make sure what you turn in is complete and through. If you have any questions, please contact us.
| HW/Project | Due Date | Extra Credit Assignment |
| HW 3 | 2-8 | Abstract |
| HW 4 | 2-15 | Introduction |
| HW 5 | 2-22 | Methods |
| HW 6 | 3-1 | Results and Discussion |
| Midterm 1 | 3-12 | Full Written Report |
| HW 7 | 3-22 | Abstract |
| HW 8 | 3-29 | Introduction |
| HW 9 | 4-5 | Results and Discussion |
| Midterm 2 | 4-16 | Full Written Report |
| HW 10 | 4-26 | Abstract |
| Final | 4-30 | Full Written Report |
You can also gain extra credits if you attend scientific talks. This is described here.
This opportunity will allow you to earn up to 5 extra credit points on a Homework per week. These points can push you above 100% or help make up for missed exercises. In order to earn all points you must:
Approved talks: Talks given by researchers through the following clubs: