Conservation of mechanical energy in a system

[ImAnEngineer]

Suppose we have a universe consisting of three planets. Their velocities and positions at t=0, and the gravitational constant are known. One can calculate, using differential equations, the positions of the planets at any time.
Assuming that the only force acting on the planets is gravity and that whenever the planets collide it is in an completely elastic manner, will there be conservation of mechanical energy?

Also, if one wants to calculate the potential energy at any time t, how can that be done? What do we take as "reference point"? The center of mass of all masses in the universe ?

 

[Gerenuk]

Yes, total energy is conserved if the only forces acting are inverse square laws.

The potential energy has to be taken over all distinct pairs AB, BC, AC so that the total potential energy for pointlike particles is
$$V=-\frac{Gm_Am_B}{|\vec{r}_A-\vec{r}_B|}-\frac{Gm_Bm_C}{|\vec{r}_B-\vec{r}_C|}-\frac{Gm_Cm_A}{|\vec{r}_C-\vec{r}_A|}$$
Note that to be exact for extended planets there should be a detailed calculation for all constituents since some parts of the planet are closer than other parts.
Potential energy is like an indefinite integral, so that it is only defined up to an added constant. Which constant you add doesn't matter (here I added no constant), as potential energy is only used for the conservation law.

 

[ImAnEngineer]

Yes, total energy is conserved if the only forces acting are inverse square laws.

The potential energy has to be taken over all distinct pairs AB, BC, AC so that the total potential energy for pointlike particles is
$$V=-\frac{Gm_Am_B}{|\vec{r}_A-\vec{r}_B|}-\frac{Gm_Bm_C}{|\vec{r}_B-\vec{r}_C|}-\frac{Gm_Cm_A}{|\vec{r}_C-\vec{r}_A|}$$
Note that to be exact for extended planets there should be a detailed calculation for all constituents since some parts of the planet are closer than other parts.
Potential energy is like an indefinite integral, so that it is only defined up to an added constant. Which constant you add doesn't matter (here I added no constant), as potential energy is only used for the conservation law.

Thank you, that formula makes a lot of sense :) . I was allowed to treat the planets as point particles.

I used the formule in my mathematica file. However I get a smaller amount of mechanical energy at t=50 than at t=0 (decrease of 10.6%). I assume I did nothing wrong, because in the assignment it is asked why there is a change in mechanical energy.

Knowing that you don't know the details, it's probably hard to say with certainty why that is. But do you have any suggestions as to what it could possibly be?

 

[Gerenuk]

You can first check if your program is OK. For example take two planets and see if you get perfect ellipses.

Apart from that I can only imagine that numeric instabilities in the calculations pile up as you do more of them. Like rounding errors or so. You could try introducing artifical very small errors in the calculation (add a small number at some steps) and then see if adding this number makes a lot of difference in the end result. If it does, then probably the rounding error eventually also prevail. You could try using a high precision math library and see if the result changes.

 

[Bob S]

Also check to verify that the total angular momentum is conserved.

Bob S

 

[ImAnEngineer]

You can first check if your program is OK. For example take two planets and see if you get perfect ellipses.

Apart from that I can only imagine that numeric instabilities in the calculations pile up as you do more of them. Like rounding errors or so. You could try introducing artifical very small errors in the calculation (add a small number at some steps) and then see if adding this number makes a lot of difference in the end result. If it does, then probably the rounding error eventually also prevail. You could try using a high precision math library and see if the result changes.

Indeed, I get ellipses or hyperbolae, or parbolae depending on the chosen initial conditions.
I'll ask my instructor what this is about... kinda strange.