# A Global Measure on a Gravitational System

1. Dec 3, 2013

### wildthing

Hi, I am a computer science engineer and I am trying to simulate the gravity on my computer; several objects with different masses randomly distributed in a Three-dimensional space. I would like to compute a global measure that describes the dynamic of that system at each instant T. My question is what kind of measure can I use?

2. Dec 3, 2013

### phyzguy

Your question is unclear. The system is uniquely characterized by the positions and velocities of each of the particles. You could plot this as a point in a 6N-dimensional phase space. What else are you looking for?

3. Dec 3, 2013

### wildthing

In my formalism, i see the system as a single entity, and i need to compute a measure that describes its state at each instant T. In other words i want to quantify the objects interactions, is there any measure that can do that kind of work or should i create mine?

4. Dec 3, 2013

### phyzguy

I'm repeating myself, but the state of the system at time T is described by the positions and velocities of each of the particles. In other words, if I have three particles, the state of the system at time T is given by:
$$\psi(T) = \{ x1(T),y1(T),z1(T),vx1(T),vy1(T),vz1(T),x2(T),y2(T),z2(T),vx2(T),vy2(T),vz2(T),x3(T),y3(T),z3(T),vx3(T),vy3(T),vz3(T) \}$$

You could look at derived quantities like the total energy, momentum, or angular momentum (all of which are constant), but it eventually all comes back to the description above.

5. Dec 3, 2013

### wildthing

I am repeating myself too, i am looking for a global measure (a scalar would be interesting) that can describe the evolution of the system. i don't want to use the raw position, velocity, acceleration, or something else of each object directly, but something that is computed from all these values and that produces a single value (if it is possible).

Last edited: Dec 3, 2013
6. Dec 3, 2013

### ShayanJ

I think the process that you guys just repeat yourself isn't so useful,therefore,I try to clarify instead of repeating.
Simple answer:There is no single variable which is able to give the configuration of the system you're interested in uniquely,So you have to use the positions and velocities of the objects or otherwise lose some information,which is a lot of information if you choose only one variable.

7. Dec 3, 2013

### wildthing

There are some information lost when using a single variable to describe the system, OK. Could you please tell me some measures that i can use (don't care about the loss)

8. Dec 3, 2013

### ShayanJ

The x component of of position of object 1
or the z component of velocity of object 4
or the kinetic energy of object 3
or the potential energy of the system
or the total energy of the system
or ...

But I should say,you haven't understood what I mean by information loss!
A state of a three-body system is uniquely determined by 18 variables.If you use less variables,you have less information so you're considering more than one state of the system which means you don't know which state is the actual state,so you can't have the evolution of the system!
The only way is having 17 constraint equations relating 18 variables which makes them depend on each other and only then one variable is enough,but you don't seem to have any constraints.

9. Dec 3, 2013

### gmax137

It is hard for the readers to guess what you have in mind. Maybe you're looking for the Hamiltonian (H=KE + PE) or the Lagrangian (L=KE - PE).

10. Dec 3, 2013

### voko

The Hamiltonian would not be useful here, because it is constant due to conservation of energy. The Lagrangian sounds more like it. But, due to conservation of energy again, one can just use the kinetic or potential energy alone to the same effect.

11. Dec 3, 2013

### ShayanJ

Lagrangian and Hamiltonian only help to determine the equations of motion but in specifying the configuration,they are just two other variables.

12. Dec 3, 2013

### phyzguy

Like gmax137 said, it is hard for us to guess what you have in mind, but here are some ideas:

(1) You could split the volume up into some number of discrete cells, and track the number of particles in each cell, or the density of particles in each cell. The cells could be rectangular, or they could be spherical shells.

(2) You could track the total kinetic energy of the particles, and/or the total potential energy (the sum of the two will of course be constant, so if you track one, you can calculate the other).

13. Dec 3, 2013

### wildthing

Suppose that you have two isolated systems composed each one of N1 and N2 number of objects (N1 is not necessary equal to N2), respectively.

i run the first system from Time T=0 to T=100.
i do the same with the second system.

My purpose is to compare the evolution of both. If the two systems have similar original configurations (spatial positions and masses), their evolutions are expected to be similar too. So, in my point of view (computer science engineer) i have to compute a measure M during the two runs. This process results in two histograms (for T=0 to T=100), one for the first system and another for the second. The problem will then become a simple histogram comparison.

The goal now is to determine the measure M? i can create an artificial one (it is not a problem) but I wanted to know if M could not be an already known measure (the total kinetic energy or the total potential energy seem to be an interesting choice)?

14. Dec 3, 2013

### Bobbywhy

"As the GRACE-twins fly in formation over the Earth the precise speed of each satellite and the distance between them is constantly communicated via a microwave K-band ranging instrument. As the gravitational field changes beneath the satellites - correlating to changes in mass (topography) of the surface beneath - the orbital motion of each satellite is changed. This change in orbital motion causes the distance between the satellites to expand or contract and can be measured using the K-band instrument. From this, the fluctuations in the Earth's gravitational field can be determined."
http://www.csr.utexas.edu/grace/science/gravity_measurement.html

15. Dec 3, 2013

### nasu

No, this is not true. If the initial speeds are not the same the "evolutions" may be wildly different.
Assuming that you are talking about a model of a real, mechanical system.
Of which I am not sure, so far.

The interactions between the components of the objects are essential too. So you need a lot more than spatial configuration and masses just to characterize the initial state.

16. Dec 3, 2013

### ShayanJ

I thought I made it clear enough,but it seems I should make it clearer.By choosing one variable,at every instant of time,you will be comparing a large number of possible states of the first system to a large number of possible states of the other system,so by this procedure,you won't be able to prove that the evolutions were the same,you can only say that the states of the systems at any instant belong to the same set of possible states. So the only thing you will prove is that the similarity of the evolutions of the systems is not impossible.
Also,a configuration for a system of particles with definite mass,is a set of positions plus velocities,so you can have two states of the same system in which masses and positions are equal but are two different configurations!

17. Dec 4, 2013

### wildthing

OK, let me express my self differently, Suppose that you have 3 isolated gravitational systems S1, S2, and S3, is there a means provided by physics, which allows to say something like 'S1 is similar to S2, S2 is not similar to S3 and so on'? (quantify the similarity (comparing two gravitational systems))

18. Dec 4, 2013

### gmax137

I think you are asking us to define what you mean by "similar."

But going past that, there are plenty of examples where seemingly insignificant differences in the initial conditions (positions and velocities) lead to drastic differences in the long-term behavior. (See "chaos" theory.) So, no, I do not think there is some "measure" of the initial conditions that can be used to predict the dynamic evolution (and thereby indicate "similarity" between systems with "similar" measures). But, maybe someone with a deeper background in classical mechanics can provide a better answer.

19. Dec 4, 2013

### wildthing

The similarity measure that i am talking about should be computed on a timescale e.g. T=0, T=100. At the end it would result in 3 histograms(3 systems) that have to be compared.

20. Dec 4, 2013

### nasu

It all depends on what you mean by similarity, as was already mentioned.
And it's not just a semantic problem. It is really about how do you want to use this and for what.

Of course you can pick up a parameter, for example speed of center of mass. And you can do histograms and so on. But these histograms will tell you no more than the fact that the too systems had "similar" values of the speed of the center of mass. Even this is problematic, because the histograms may look "similar" but the values of the speed may have completely different time evolution.
Imagine we want to see how "similar" are two persons. We can use the parameter "eye color". If they have the same eye color, they are similar but only from the point of view of eye color.
It does not say anything about other things. So would you consider them similar? Of course, if eye color is all that matter for your application, they are "similar".
For a shoemaker, two persons with the same foot size are "similar". But not completely. Even here, there is at least a second parameter needed: gender. So can you come up with a single parameter which will be used to uniquely distinguish shoes?

So I think you should think first what is your goal and then decide what similarity is relevant. So far you just have the method, using histograms. And you ask other people to find a meaningful goal for it.

Last edited: Dec 4, 2013