Using Noether's Theorem to get conserved quantities

[Toby_phys]

Homework Statement

N point particles of mass mα, α = 1,...,N move in their mutual gravitational field. Write down the Lagrangian for this system. Use Noether’s theorem to derive six constants of motion for the system, none of which is the energy

Homework Equations

Noethers Theorem: If a change ($q_i \implies q_i+\delta q_i$) creates no change in the Lagrangian the conserved quantity is
$$\sum \dot{p_i}\delta q_i$$

The Attempt at a Solution

So my lagrangian is:
$$L=\frac{1}{2}\sum m_i \dot{r}^2_i-\sum_{i\neq j}V(|r_i-r_j|)$$

With this I can get 2 conserved quantities - momentum (from translational invariance) and angular momentum. How do I get the other 4?

 

[Orodruin]

With this I can get 2 conserved quantities - momentum (from translational invariance) and angular momentum. How do I get the other 4?

How many components of each?

 

[TSny]

So my lagrangian is:
$$L=\frac{1}{2}\sum m_i \dot{r}^2_i-\sum_{i\neq j}V(|r_i-r_j|)$$

Do any of the symbols here represent vector quantities? Are you meant to write the potential energy explicitly for gravitational interaction?

With this I can get 2 conserved quantities - momentum (from translational invariance) and angular momentum. How do I get the other 4?

By "momentum", are you referring to the total linear momentum of the system? Momentum is a vector quantity. If it is conserved, what can you say about each of its Cartesian components? {Edit: I see Orodruin already addressed this point.}