Central Force Problem; Need Help Solving Difficult Integral

In summary: You need to use the potential V to calculate the orbits for different values of the energy and angular momentum.
  • #1
logic smogic
56
0
Problem
A particle moves in a force field described by

[tex]F(r)=-\frac{k}{r^{2}}e^{-\frac{r}{a}}[/tex]

where k and a are positive.

Write the equations of motion and reduce them to the equivalent one-dimensional problem. Use the effective potential to discuss the qualitative nature of the orbits for different values of the energy and the angular momentum.

Attempt at Solution
I'm having trouble deriving the potential V. Given the force equation F(r), we should have

[tex]F(r)=-\nabla V[/tex]

Since the Lagrangian L is defined as [tex]L=T-V[/tex], we need to know the potential V. We can derive this by integrating the force equation. But I've searched through many integral tables, and come across nothing for this particular integrand!

Can anyone help with this simple step?
 
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  • #2
Why not just use Newton's second law directly? (You know that's what you're going to end up with anyway if you start with the Lagrangian).

[tex]m\ddot\vec{r}=F(r)\hat{r}[/tex]

To get the equivalent one-dimensional problem, you have to do some polar coordinate magic on the left-hand side.
 
  • #3
Fredrik said:
Why not just use Newton's second law directly? (You know that's what you're going to end up with anyway if you start with the Lagrangian).

[tex]m\ddot\vec{r}=F(r)\hat{r}[/tex]

To get the equivalent one-dimensional problem, you have to do some polar coordinate magic on the left-hand side.

The professor would like us to derive the equations of motion using the Lagrangian formulation.

Unless I'm mistaken, you can't arrive at an expression for the potential V by only using F=ma, right? Integrating the expression for force F(r) is the only way to arrive at an expression for the potential V(r).

The possibility of using Newton's laws aside - I can solve the integral via an exponential integral (and, therefore, possible a gamma function?), or by using an infinite sum or Taylor expansion. But I'd be surprised if they gave us a problem that was so difficult to solve analytically.

Thanks Fredrik, I'm still working on it.
 
  • #4
Have you remembered to use the Nabla operator in spherical coordinates when you set up your equations?
 
  • #5
Jezuz said:
Have you remembered to use the Nabla operator in spherical coordinates when you set up your equations?

[tex]V=-\int \int \int \frac{k}{r^{2}} Exp(-\frac{r}{a}) r^{2} sin(\theta) dr d\phi d\theta = -4 \pi k \int Exp(-\frac{r}{a}) dr = 4 \pi k a Exp(-\frac{r}{a})[/tex]

This works. Thanks for the idea.

I'm still confused about one thing, though. For your standard inverse-square law, calculated in spherical, you have

[tex]F(r)=- \frac{k}{r^{2}}[/tex]

giving (according to the text)...

[tex]V(r)=- \frac{k}{r}[/tex]

Wouldn't the r^2 cancel in the inverse square, and give a different potential (i.e. V ~ r, not ~1/r)? Just a side thought.
 
Last edited:
  • #6
Alright, going with it, I have used,

[tex]\frac{d}{dt} \frac{\partial L}{\partial \dot{r}} - \frac{\partial L}{\partial r} = 0[/tex]

for the equation of motion, I get

[tex]m \ddot{r} - 4 \pi k e^{-\frac{r}{a}} = 0[/tex]

Clearly the [tex]m \ddot{r}[/tex] term in the above equation is some additonal force (or is it the same as in the given force law?).

Either way, how do I reduce it to the equivalent one-dimensional problem?

Going on the hunch in parentheses above, would I substitute the force law in...

[tex]-\frac{k}{r^{2}} e^{-\frac{r}{a}}=4 \pi k e^{-\frac{r}{a}}[/tex]
[tex]\rightarrow r^{2} = -\frac{1}{4 \pi}[/tex] ?

Or do I convert from spherical to cartesian?

EDIT: Because of the concern mentioned in the post above, I've decided that this answer must be incorrect. My prof said he used a substitution to do the integral, but I've tried half a dozen substitutions, and none of them made the problem work. I've also tried integrating by parts multiple times, and searching for substitutions then, too. I've checked the errata, and it's not a typo. I'm assuming there's some clever substitution (other than the power, logarithmic, and exponential ones I've been trying) that makes it work.
 
Last edited:
  • #7
Problem solved!

There was a typo in my printing of the textbook, which did not appear in any of the errata available online. The *actual* expression given should be a potential V(r), not force law F(r), and is

[tex]V(r)=-\frac{k}{r}e^{-\frac{r}{a}}[/tex]

I'm still not sure what they mean by reducing the equation of motion I found,

[tex]m \ddot{r} + (\frac{k}{ar} + \frac{k}{r^{2}})e^{-\frac{r}{a}}=0[/tex]

to a one-dimensional expression. Any hints?
 

Related to Central Force Problem; Need Help Solving Difficult Integral

1. What is the central force problem?

The central force problem is a classical mechanics problem in which a particle is subjected to a force that is directed towards a fixed point, known as the center of force. This type of force is also known as a central force, and the goal is to determine the motion of the particle under this force.

2. What makes solving the central force problem difficult?

The main difficulty in solving the central force problem lies in finding the general solution to the equations of motion, which involves solving a difficult integral. This integral cannot be solved analytically in most cases, so numerical methods must be used to approximate the solution.

3. What are some common methods used to solve the central force problem?

Some common methods used to solve the central force problem include the Lagrangian method, the Hamiltonian method, and the Runge-Kutta method. These methods allow for the approximation of the solution to the equations of motion, which can be used to analyze the behavior of the particle under the central force.

4. How does the central force problem relate to real-world phenomena?

The central force problem has many applications in physics, including the study of planetary motion, the behavior of charged particles in an electric or magnetic field, and the motion of objects under the influence of gravity. It also has applications in engineering, such as in the design of satellites and spacecraft.

5. What are some tips for solving difficult integrals in the central force problem?

Some tips for solving difficult integrals in the central force problem include breaking the integral into smaller parts, using symmetry to simplify the integral, and using numerical methods such as Simpson's rule or the trapezoidal rule. It is also helpful to have a strong understanding of calculus and mathematical techniques for integration.

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