A potential V(r) = -k/r^4 A very superficial analysis

[yxgao]

What are the general features of this potential? Is the particle attracted or repelled? You don't need to perform a mathematical analysis, but qualitatively, what is the general shape of the orbits? Assume energy is positive. What happens if angular momentum is zero?

 

[arcnets]

Originally posted by yxgao
Is the particle attracted or repelled?

Attracted.

what is the general shape of the orbits?

Curved toward the center, but not closed (the're only closed for V \sim r^{-1}, Kepler motion). I guess they look like a pattern produced by a spirograph.

What happens if angular momentum is zero?

The body will fall into the center. From there, it will not be possible to calculate the further motion because of the singularity.

 

[M98Ranger]

The potential V(r) = -k/r^4 is a type of inverse fourth power potential, which is a common form of potential in classical mechanics. The general features of this potential include a strong repulsive force at short distances and a weak attractive force at large distances. This means that the particle will experience a strong repulsion when it is close to the source of the potential, but as it moves further away, the force becomes less and less attractive.

Assuming that the energy of the particle is positive, we can say that the particle will be repelled by this potential. This is because the potential is negative, and the force is in the direction opposite to the gradient of the potential. So, the particle will experience a force that is directed away from the source of the potential, causing it to move away from it.

Qualitatively, the orbits in this potential will have a hyperbolic shape. This is because the potential is a repulsive one, and the particle will have enough energy to overcome the repulsive force and escape to infinity. The exact shape of the orbit will depend on the initial conditions, but in general, it will be an open curve that approaches the source of the potential at one end and extends to infinity at the other end.

If the angular momentum of the particle is zero, it means that the particle is moving directly towards or away from the source of the potential. In this case, the particle will follow a radial trajectory, approaching or receding from the source of the potential without any change in direction. This is because the angular momentum is responsible for the change in direction of the particle's motion, and with zero angular momentum, there is no such change.