Do orbits with positive energy tend to infinity?

Question

Consider any potential field $$V = V(x)$$ (not limited to gravitational potential field, but we only consider time-independent ones) in 3-d space that satisfies the following conditions:

1. The potential field is everywhere negative, i.e. $V(x) < 0, \forall x \in \mathbb{R}^3.$
2. The potential field tends to zero at infinity, i.e. $\lim_{x \to \infty} V(x) = 0.$

Now consider a partical moving in this potential field. Suppose it has positive total energy $E$ (= kinetic energy $T$ + potential energy $V$). Does its orbit tend to infinity? Or is there a possibility that its orbit is bounded?

I only know that in the case of Kepler orbits in gravitational field, every orbits with positive energy is hyperbolic. I have complete no idea about the general case.

Answer 1

Consider an attractive central force field with a magnitude that varies as the inverse fourth power of radial distance i.e. $F(r) = \frac {km} {r^4}$. Then we have $V(r) = - \frac {km} {3r^3}$. For a circular orbit with constant speed $v$ and radius $r$ we have

$\displaystyle \frac {mv^2} r = \frac {km} {r^4} \\ \displaystyle \Rightarrow \frac 1 2 mv^2 = \frac {km} {2r^3} \\ \displaystyle \Rightarrow \frac 1 2 mv^2 + V(r) = \frac {km} {2r^3} - \frac {km} {3r^3} = \frac {km} {6r^3}$

So we have found a family of bounded orbits with positive total energy.

(Note that Bertrand's Theorem tells us that these orbits are not stable these circular orbits are not necessarily stable).