Numerical integration using Weber force

In summary, the conversation revolved around the problem of computing the acceleration in an n-body system interacting according to the Weber force, which is defined as a function of the second derivative of position (r''). The participants discussed different methods for solving this problem, such as using finite difference schemes and energy conservation. It was also mentioned that the Weber force is widely rejected by contemporary physicists and not mentioned in mainstream textbooks on classical electromagnetism. Eventually, the participants arrived at the conclusion that for a circular motion, the Weber force is identical to the Coulomb force.
  • #1
parkner
15
0
I need to compute numericaly n-body sys. interacting acording to the Weber force:
http://en.wikipedia.org/wiki/Weber_electrodynamics

and I have a problem with the acceleration on rhs: r'', because the acceleration is unknown, due to the Newton law: F = ma, and we need just 'a' to do next step of integration.

For the standard Newton/Coulomb force there is no problem:
F = k/r^2 r^0; so, we can directly compute: a = F/m = ...

but if a force F depends on r'', this is impossible, because:
r'' = |a| + v_t^2/r, where: v_t is a tangential speed (velocity vector: r' = v = v_r + v_t).

Is it solvable the Weber force - usable in some way?
 
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  • #2
Maybe I would ask to move this issue to the section of physics... mathematicians probably don't know too much about numerical problems of solving equations of motion.

i correct a mistake in my recent post:
|r|'' = a_r + v_t^2/r
 
  • #3
I don't really understand the conceptual problem here. If you set up a finite difference scheme to solve the equations, you know an approximation to r'', in exactly the same way as you know approximations to r' and r. If you are doing the timestep from t to t+h, and you need an approximation for r'' that depends on the (unknown) solution at time t+h, use an iterative solution algorithm.

On the other hand if you are asking us to invent an efficient and accurate algorithm to solve that particular equation, that's a different question, and considering that your Wiki link says
The theory is widely rejected and ignored by contemporary physicists, and is not even mentioned in mainstream textbooks on classical electromagnetism.
you are unlikely to find the answer in a handbook of numerical methods.
 
  • #4
parkner said:
Maybe I would ask to move this issue to the section of physics... mathematicians probably don't know too much about numerical problems of solving equations of motion.

i correct a mistake in my recent post:
|r|'' = a_r + v_t^2/r

Especially those pesky chaos theorists. :rolleyes:
 
  • #5
No, i don't konow the r'', because the force is defined as a function of the r'' also.

With a standard electric force:
F = ke^2/r^2 r^0
and: F = ma, then we can solve: a = F/m = ke^2/mr^2 r^0, simply.

But with the Weber force we get:
a = F/m = ke^2/ma (1 - r'^2/2c^2 + r''/c^2) r^0;

and the r'' = |r|'', ie. a radial acceleration, so it depends obviously on tha 'a' in some way, which we just want to compute.

Numerical methods have nothing to the equation, which we want to integrate.
These methods work for any equation - computable of couse.
 
  • #6
parkner said:
Numerical methods have nothing to the equation, which we want to integrate.
These methods work for any equation - computable of couse.

That makes no sense at all to me.

Anyway, if you have already calculated r at times t-2h, t-h, and t, you can easily estimate the value of r'' at time t (fit a parabola through the 3 points) and use that to calculate F. (You may need to tweak that basic idea - I haven't done any error and stability analysis on it)

Or, you know the potential energy is a function of r and r', so you can set up a solution algorithm based on energy conservation (i.e. a weak formulation of the original differential equation).
 
  • #7
AlephZero said:
Anyway, if you have already calculated r at times t-2h, t-h, and t, you can easily estimate the value of r'' at time t (fit a parabola through the 3 points) and use that to calculate F.

You can't compute in this way, because the r'', which you suggest to compute just geometricaly, will be incorrect - some static version, ie. without any forces (you assume here: v = const, at a position r(t)).

Or, you know the potential energy is a function of r and r', so you can set up a solution algorithm based on energy conservation (i.e. a weak formulation of the original differential equation).
This is probably the same, bacause you need to konow acceleration to modify a velocity v, and a position r. So, we must compute the gradient of these potential, thus we get the Weber force.
 
  • #8
Maybe I underappreciated too much these 'estimation' method of r''.
This will be an extrapolation.
Very little precision... perhaps 2-nd order with three points, but locally, thus globally 1-st only.
I plan to compute with 8 order... at last. :)


This is probably just implicit ode problem, formally:
y'' = f(t,y,t',y''); or F(t,y, y,',y'') = 0

in every step we must to solve some system of equations, to get a force, ie. y''.

For the simplest case: two body only, we can compute this:

a = -k/r^2 (1 - r'^2/2c^2 + r.r''/c^2)

and in this case: r'' = a + v_t^2/r, thus the explicit form is:
[tex]a = -\frac{k}{r^2}\frac{1 - r'^2/2c^2 + v_t^2/c^2}{1 + k/r}[/tex]

and now we can directly compute the acceleration a.
Esspecialy for a cilcular motion we have: v_r = 0, v = v_t, so:

[tex]a = -\frac{k}{r^2} \frac{1 + v^2/c^2}{1 + k/c^2r}[/tex]; but now k/r is just v^2 thus:
[tex]a = -\frac{k}{r^2} \frac{1 + v^2/c^2}{1 + v^2/c^2} = -\frac{k}{r^2}[/tex]

This means for a circular motion the Weber force is identical with the Coulomb force. :)
 
Last edited:

1. What is numerical integration using Weber force?

Numerical integration using Weber force is a method used in computational physics to solve the equations of motion of a system by breaking it down into smaller segments and calculating their interactions using the Weber force.

2. How does numerical integration using Weber force differ from other integration methods?

Unlike traditional integration methods, which use a fixed time step, numerical integration using Weber force adapts the time step based on the forces present in the system. This allows for more accurate and efficient calculations.

3. What are the benefits of using numerical integration using Weber force?

Numerical integration using Weber force can accurately model complex systems with multiple interacting particles, making it a valuable tool in many areas of physics and engineering. It also allows for faster computation times compared to traditional integration methods.

4. What are some limitations of numerical integration using Weber force?

One limitation of this method is that it can be computationally expensive, as it requires calculating the Weber force for each individual segment. It also may not be suitable for systems with rapidly changing forces or those that involve discontinuous motion.

5. How is the accuracy of numerical integration using Weber force evaluated?

The accuracy of this method is often evaluated by comparing the results to analytical solutions or experimental data. Additionally, the size of the time step and the convergence of the solution can also be used as indicators of accuracy.

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