Central potential V = -k/r⁴

This simulation explores the non-Newtonian potential V(r) = -k/r⁴, in terms of the mechanical energy E, for k > 0, in dimensionless units:

The unit mass m is the particle mass (or the reduced mass for the 2-body system).
The unit length is the radius of the circular orbit: r₀ = 2(mk)^1/2/L.
The unit time is 4km²/L³.
In these units we have m = L = 4k = 1.

On the left appears in red the effective potential energy -1/4r⁴+1/2r²: its maximum is located at the point (r,E) = (1,1/4). You may use the mouse (or the controls below) to select the mechanical energy E and the initial polar distance r.
Particle's plane motion (or the relative motion in the 2-body problem) is displayed on the right. You may use the mouse to select the initial position: the program will automatically set r.
Put the mouse pointer over an element to get the corresponding tooltip.

Activities

With the default settings the particle moves along the circular trajectory.
To check that the latter is unstable, press Stop, set r = 3, and press Continue.Can you explain what you would expect and what really happens?
What should happen with E < 1/4? Use the simulation to check your guess. (To go near r = 0, you must set a small integration step: dt = 0.01, for instance.)

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.