Constant central force

• 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 = (L²/mk)^1/3.
• The unit time is mr₀²/L.
• In these units we have m = L = k = 1.

• On the left appears in red the effective potential energy r+1/2r²: its maximum is located at the point (r,E) = (1,3/2). 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

• Try different values for r and E to check that, although all orbits are bounded, most of them are not periodic, as predicted by Bertrand's theorem.
• However, the theorem does not preclude isolated periodic orbits. In fact, (apart from the circular orbit for E = 3/2) it is not that difficult to find some of them.Try near the following values: E = 1.7, 2.4, 3.05.