Dynamics of a system of particles all influencing eachother

[jameswhite]

Take a closed system of particles. Each has it's own mass, but that's really about it for the sakes of this argument. Now suppose there is some standard force law that applies between particles, whatever it may be. (In general it will depend only on the separation between particles and thus be translation invariant but not always). Now, the force on some given particle is therefore dependent on the position of ALL the particles. Let's work in one dimension, x for simplicity...

So what we have here is an acceleration field which depends on x (all of them). Now suppose that we know the initial configuration of the system (positions and velocities of all the particles), how does one go about finding the configuration of the system at some later time, t?

I'm looking for equations here, not explanations. I understand the mechanics of the problem but I'm having trouble visualizing the mathematics. Mainly because my instinct is to integrate the acceleration function with respect to time to find the overall change in velocty... but the acceleration field depends on position, not time... This is throwing me off.

To summarise, I'm looking for the mathematical procedure one would follow to calculate the configuration of a system of this kind after a time, t has elapsed - given the initial configuration. (I want positions and velocities in case that isn't clear).

Purely Newtonian mechanics also, I'm not worried about SR just yet and let's not complicate things.

(I know my maths... I will understand whatever equations you give me, I just need a hand seeing the process here)

Thanks :)

EXTRA:
If anyone feels like it, my next question would be: What if the acceleration depends on both positions and velocities? Such as an electrically charged particle moving in a magnetic field. How would this affect the equations?

 

[CWatters]

This really isn't my field but if I remember correctly it gets very difficult for more than two particles..

http://en.wikipedia.org/wiki/Three-body_problem

 

[jtbell]

Suppose we have three particles moving in one dimension, and call the force exerted on particle i by particle j $F_{ij} = f(x_i, x_j)$ where the form of the function f is the same for all pairs of particles, changing only the subscripts on the x’s.

Write down Newton’s Second Law for each particle, using the force law between pairs of particles to find the net force on each particle. For particle 1:
$$m_1 a_1 = F_{1,net}\\
m_1 \frac{d^2 x_1}{dt^2} = f(x_1, x_2) + f(x_1, x_3)$$
and similarly for particles 2 and 3. This gives you three coupled second-order differential equations for $x_1(t)$, $x_2(t)$ and $x_3(t)$.

There is no general method for solving such a system analytically. However, there are numerical methods that calculate (to some degree of precision) the motions of the particles over a series discrete time steps. First, convert the system of three second-order equations into a system of six first-order equations by defining a new set of variables:
$$y_1 = x_1\\
y_2 = x_2\\
y_3 = x_3\\
y_4 = \frac{dx_1}{dt}\\
y_5 = \frac{dx_2}{dt}\\
y_6 = \frac{dx_3}{dt}$$
I leave it to you to verify for yourself that the three second-order equations described above become
$$\frac{dy_1}{dt} = y_4\\
\frac{dy_2}{dt} = y_5\\
\frac{dy_3}{dt} = y_6\\
\frac{dy_4}{dt} = \frac{1}{m_1}[f(y_1, y_2) + f(y_1, y_3)]\\
\frac{dy_5}{dt} = \frac{1}{m_2}[f(y_2, y_3) + f(y_2, y_1)]\\
\frac{dy_6}{dt} = \frac{1}{m_3}[f(y_3, y_1) + f(y_3, y_2)]$$
Considering the y’s together as a six-dimensional vector $\vec y$, and the six right-hand-sides as a vector-valued function of the y’s, we can write this concisely as
$$\frac{d \vec y}{dt} = \vec f (\vec y)$$
Many of the common methods for solving first-order differential equations numerically (e.g. Runge-Kutta) can be re-cast in vector form using the notation above. A Google search for something like “system of coupled first order differential equations” might help here.

I give Runge-Kutta only as an example of a method that you may already be acquainted with. I know from experience that it’s actually not a good method for this if you want accurate results over long periods of time. I once programmed a gravitational system simulator using a version of fourth-order R-K. When I used it to simulate a small solar system, I found that the orbits of the “planets” gradually spiraled in towards the “sun” (or maybe it was away from the “sun”) because the numerical errors in successive steps compounded in such a way that the total energy of the system was not conserved.

Regarding your extra question, I hope you can see that the system of six equations above can easily accommodate velocity-dependent forces.

Finally, we can move from one spatial dimension to three, by starting with Newton's Second Law in vector form for each particle, which multiplies the number of equations in the system by three.

 

[jameswhite]

Thank you jtbell, that's very helpful. I myself am actually in the process of programming a simulator for the evolution of particles under gravity. That is actually where I encountered this problem. I noticed that when particles started to fall towards each other they would pick up speed and then fly through each other (no collision detection just yet) and then fly apart. The issue being that due to the discrete time intervals, the particles would move too far apart before the next loop had a chance to calculate the force between them. The overall effect being that they would fly apart rather than settle into a harmonic oscillator.

I must admit, this is the first time I'm hearing about the three-body problem. I am quite confused by this, it seems to me than a simple mechanical, deterministic system like this should be infinitely predictable... Could you perhaps provide some insight into exactly what the problem is here. Is it just a short fall in our maths? Or is it something much more fundamental and if so, can you shed some light on how a seemingly deterministic system can become unpredictable in this way?

 

[pasmith]

I must admit, this is the first time I'm hearing about the three-body problem. I am quite confused by this, it seems to me than a simple mechanical, deterministic system like this should be infinitely predictable... Could you perhaps provide some insight into exactly what the problem is here. Is it just a short fall in our maths? Or is it something much more fundamental and if so, can you shed some light on how a seemingly deterministic system can become unpredictable in this way?

Chaos theory deals with this.

 

[UltrafastPED]

Thank you jtbell, that's very helpful. I myself am actually in the process of programming a simulator for the evolution of particles under gravity. That is actually where I encountered this problem. I noticed that when particles started to fall towards each other they would pick up speed and then fly through each other (no collision detection just yet) and then fly apart. The issue being that due to the discrete time intervals, the particles would move too far apart before the next loop had a chance to calculate the force between them. The overall effect being that they would fly apart rather than settle into a harmonic oscillator.

I must admit, this is the first time I'm hearing about the three-body problem. I am quite confused by this, it seems to me than a simple mechanical, deterministic system like this should be infinitely predictable... Could you perhaps provide some insight into exactly what the problem is here. Is it just a short fall in our maths? Or is it something much more fundamental and if so, can you shed some light on how a seemingly deterministic system can become unpredictable in this way?

(I) The problem starts with the lack of analytic solutions to the system of coupled differential equations.

(II) The second is with the computational methods available - we must always use a limited number of digits in a computation - and hence the calculations are done in some form of limited precision arithmetic, such as fixed point or floating point ... you can always do it better, but there are always error terms which come from the arithmetic.

(III) Finally, there is more than one way to approach every problem, and the first pass, obvious solutions are seldom optimal.

Problem (I) has been known studied for centuries; it is theoretical-mathematical in nature, and can be proven: https://en.wikipedia.org/wiki/Euler's_three-body_problem

Problem (II) is in the province of Numerical Methods; any course called numerical methods, or numerical analysis will go into great detail on the issues, and how to minimize them.
https://en.wikipedia.org/wiki/Numerical_analysis

Problem (III) has been discussed above, but there is always more to say. Recasting the problem into another "space" often helps, as does improved techniques for translating the differential equations into finite difference equations.

I took a seminar course in aeronautical engineering where Lie algebras were used to express the problems, and the numerical algorithms were generated from the corresponding Hamiltonian equations of motion via an application of the calculus of variations. When applied to various problems the stability was thousands to millions of times better than the standard approaches, and you could test it by a simple calculation on the operator matrix: if it remains unitary, the stability is good. So you simply run a tolerance test once per cycle.PS: And chaos theory has something additional to say when slight changes in boundary conditions move the trajectories more than you might guess from a mere linear perturbation scheme.

 

[Jano L.]

The issue being that due to the discrete time intervals, the particles would move too far apart before the next loop had a chance to calculate the force between them. The overall effect being that they would fly apart rather than settle into a harmonic oscillator.

...Could you perhaps provide some insight into exactly what the problem is here. Is it just a short fall in our maths?

The problem is that numerical calculation has to discretize continuous motion. Discretized scheme for solving the equations is almost never entirely equivalent to the original differential equations so errors are likely to occur and multiply.

One good way to fight the propagation of errors is to use variable time step. In purely gravitational system, there is no definite characteristic time or length scale and particles may move with characteristic times that differ by many orders of magnitude. If you want to calculate motion of large number of particles in reasonable time, individual time steps are a good idea.