Why isn't energy conservation equation sufficient to determine the motion?

Question

The star of large mass $M$ is orbited by planets of masses $m_1$ and $m_2$. The energy conservation equation for this system is$$\frac{1}{2}m_{1}v_{1}^{2}+\frac{1}{2}m_{2}v_{2}^{2}-MG\left(\frac{m_1}{r_1}+ \frac{m_2}{r_2}\right)-\frac{m_1m_2G}{r_{12}}=E,$$where $v_1$, $v_2$ are the velocities of the two planets, and $E$ is the constant total energy.

Since the mass of the star is supposed to be very much larger than the planetry masses, its motion has been neglected.

This is an example from the book Classical Mechanics by R. Douglas Gregory. In the book, he says

Since this system has six degrees of freedom (four if the motions are confined to a plane through $O$), the energy conservation equation is by no means sufficient to determine the motion!

While I understand the fact that this system has 6 degrees off freedom, I can't quite understand why the energy conservation equation would not be sufficient to determine the motion.

Answer 1

In order to solve a problem with a number of variables we need to get to a point where we have the same number of equations. For example, if we want to solve the motion of a single particle, we need 3 equations to describe $x,y,z$. Say we find some sort of geometric constrain, such as showing the problem is on a plane, we reduce the problem. Energy is a scalar, and therefor gives us only 1 equation, momentum can give us 3 equations, one for each axis. from your equation, we can find $v_2$ in terms of $v_1$, but how can you continue? This isn't enough information to find the position of the planets as a function of $t,\theta,r$ and so on. In problems that only involve a central force: $$\tau=(r\hat{r})\times(F\hat{r})\rightarrow \frac{dL}{dt}=0\rightarrow L=const$$ So we have conservation of angular momentum in addition to conservation of energy. You may want to read about the solution to the 2-body problem, in which the solution is to reduce the problem from 6 dimensions to 2.