# Proof that, for any force law in Newtonian Mechanics, if a planet gets closer and then further from its star repeatedly, the orbit is bounded

I have run simulations of planetary motion using various force laws that are functions of distance $$f(r)$$, including but not limited to

$$f(r)=\frac{1}{r^2}$$ $$f(r)=\sin(r)$$ $$f(r)=r!$$ $$f(r)=\frac{1}{r}$$ $$f(r)=\frac{1}{r^3}$$ $$f(r)=e^r$$ $$f(r)=\ln(r)$$ $$f(r)=r^r$$ $$f(r)=r$$

I have noticed that the simulations only seem to produce elliptical orbits for $$f(r)=r^{-2}$$ and $$f(r)=r$$ with the star being at the foci of the elliptical orbit in the case of $$f(r)=r^{-2}$$ and at the center of the ellipse in the case of $$f(r)=r$$. Some force laws don't seem to allow non circular bounded orbits, such as $$f(r)=r^{-3}$$, however other force laws such as $$f(r)=r^{-1}$$, $$f(r)=\sin(r)$$, $$f(r)=r!$$, $$f(r)=r^r$$, and many others seem to allow for stable orbits based on simulations that I ran.

$$f(r)=r^{-1}$$ produces orbits that what one might refer to as flower petal shaped, and these orbits seem to be stable based on simulations that I have run. Force laws involving the force being a trigonometric function of distance tend to produce wavy orbits with some of these orbits being bounded. $$f(r)=e^r$$ as well as $$f(r)=r!$$ seems to produce orbits that have a shape that is often referred to as star shaped.

Based on the simulations I have run it seems for any force law $$f(r)$$ that allows non circular bounded orbits, that if a planet goes all the way around it's star, and if a planet gets closer, and then further from its star repeatedly that the orbit is bounded/stable.

Is there a mathematical proof that this is the case for any force law $$f(r)$$ using Newtonian Mechanics?

• Bertrand's theorem states that only two types of central potential allows bound and closed orbits. en.wikipedia.org/wiki/Bertrand%27s_theorem – K_inverse Sep 24 '18 at 7:51
• Actually if you read what it says about Bertrand's Theorem carefully, it says that there are only two central potentials, in which all bounded orbits are also closed orbits, not necessarily that there are no bounded orbits outside those potentials. – Anders Gustafson Sep 24 '18 at 8:18
• The title statement (v1) is not true for $n\geq 3$ bodies. – Qmechanic Sep 24 '18 at 8:18