# Special invariants with few constants of motion

1. Oct 28, 2009

### wkb13

Ordinarily, a system of N particles in d dimensions has 2Nd constants of motion, but there are certain invariants, like energy and angular momentum, that have a lot fewer. What's so special about these? Why do they have so few constants of motion?

2. Oct 29, 2009

### Gerenuk

Hope I understand this correctly. Not sure why you refer to everything as "constants".
The invariants are due to the special form of the interaction between particles which is $\propto\vec{r}/r^3$. If you have another fictional, crazy type of interaction, you wouldn't have energy and angular momentum conserved.

3. Oct 29, 2009

### wkb13

Yes, I guess "parameters of motion" would have been a more appropriate term. So, if I understand correctly, any inverse square interaction will have fewer than 2Nd parameters?

4. Oct 29, 2009

### Gerenuk

Yes, any inverse square law where action equals reaction (and I believe magnetism also doesn't harm), will conserve the sum of kinetic plus potential energy and also will conserve the the total angular momentum for the system and thus reduce the total number of parameters you need.

5. Oct 29, 2009

### Bob_for_short

The special feature of the total energy, momentum, and the angular momentum is that they are additive in particles. The inter-particle interaction potentials are not involved. It helps in certain simple cases (scattering, for example). The other integrals of motion are harder to find and they are not additive in particles.

6. Oct 29, 2009

### Gerenuk

What means additive? Every quantity can be added up. And we want to consider only those whose sum is constant over time.
And quantities like entropy are additive in the idealized case, but not conserved. So additivity doesn't play a role.

The interaction potential is the only determining feature. Just imagine a crazy unphysical force equation for the particles with index i:
$$F_i=\begin{cases} C & \text{if }i=j\\ 0 & \text{otherwise} \end{cases}$$
This physics would make particle j fly away to infinity and there is no conservation of energy of angular momentum.

7. Oct 29, 2009

### Bob_for_short

It means that the total momentum is a sum of particle momenta, for example.
I speak of inter-particle potentials, not of the external force. In presence of external force the additive conservation laws may not be valid.

8. Oct 29, 2009

### Gerenuk

That also applies for the x-component of particles. The total "x-component" of particle set A and B together is equal to the sum of their individual x-component sums. Yet, the quantity is not conserved during motion.

Correct. I should refer to the total force on each particle which should be of the form
$$F_i=\sum_{j\neq i} \frac{a_{ij}\hat{r}_{ij}}{r_{ij}^2}$$
$$a_{ij}=a_{ji}$$
for energy and momentum and angular momentum to work. That is necessary and sufficient I believe. Well, almost. I guess a conservative velocity dependent force like a magnetic field can also be added and yet the derivations for the conserved quantities would work.

9. Oct 29, 2009

### Bob_for_short

Component of what vector? If you speak of momentum, the total vector P is conserved:

dPx/dt = 0, dPy/dt = 0, dPz/dt = 0.

And Px = Σk(px)k, etc.

10. Oct 30, 2009

### Gerenuk

Oh come on. I forgot to say component of velocity, but it's really not hard to make up additive quantities that are not conserved. How about $x+v_y\cdot 1\mathrm{s}$ where x is x coordinate and v_y the y component of the velocity.

Last edited: Oct 30, 2009