Particles trajectory in a bound orbit.

[sleventh]

Homework Statement

I am to find the particles trajectory to the first order of r/a knowing it to have the Yukawa potential

v(r)=V$$_{\circ}$$r$$_{\circ}$$$$/$$r * e$$^{-r/r_{\circ}}$$
= -k/r * e$$^{-r/a}$$

Homework Equations

$$\theta$$(r)= $$\int$$ (1/r$$^{2}$$)/$$\sqrt{2\mu (E-U-l^{2}/2\mu r^{2}}$$) dr

The Attempt at a Solution

My attempt goes as far as using the effective potential given to have a basic equation of motion. I infortunately do not know what is meant by "to first order of"

any help or recomended readings would be greatly apreciated. I am a bit lost.

 

[Jhero]

Hello,

To find the particle's trajectory to the first order of r/a, we can use the first order approximation of the Yukawa potential. This means that we will only consider the first term in the series expansion of the potential, which is -k/r.

To solve for the trajectory, we can use the equation of motion for a particle in a central potential:

m(d^2r/dt^2) = -k/r^2 * r

To simplify this equation, we can substitute r = 1/u, where u is the inverse of the distance from the center of the potential. This will give us:

m(d^2u/dt^2) = -k * u

This is a simple harmonic oscillator equation, and its solution is given by:

u = A * cos(ωt + φ)

Where A and φ are determined by the initial conditions of the particle, and ω is the angular frequency given by ω = √(k/m).

To find the trajectory, we can substitute back for u:

r = 1/A * cos(ωt + φ)

This is the equation for a circle with radius 1/A, centered at the origin. In other words, the particle will move in a circular orbit around the center of the potential, with a radius given by 1/A.

I hope this helps! For further reading, I would recommend looking into the equations of motion for central potentials and how to solve them. You may also want to look into perturbation theory, which deals with approximations and first order solutions.