Note 214(4): Solution for r as a function of t

This is really excellent, and exactly what is needed! The solution r(t) can be animated and there may be a way of finding theta(t).

To: EMyrone@aol.com
Sent: 29/04/2013 14:23:00 GMT Daylight Time
Subj: Re: 241(4) : Calculation of x from the Minkowski Force and Animation

I just had a bit time to check the note. The term for x^2 is correct. gamma can a bit simplified, see %o5. Surprisingly the integral for t can be solved analytically, giving expression %o17. For the inverse function r(t) a polynome of 4th order has to be solved. The only real and non-trivial solution is shown at the end, %o22.

Horst

Verschickt: Mo, 29 Apr 2013 12:48 pm
Betreff: Fwd: 241(4) : Calculation of x from the Minkowski Force and Animation

It could be that the latest animation software can deal with this. If not, a fitting program can be used to interpolate and then animate. I will have a look at the relativistic momentum.

To: EMyrone
Sent: 29/04/2013 10:38:41 GMT Daylight Time
Subj: Re: 241(4) : Calculation of x from the Minkowski Force and Animation

A problem for animation may be that time steps are not uniform but come out from the calculation. For a precise animation the pairs (t_n, r_n) would have to be interpolated for a uniform time grid. Perhaps the r grid can be defined in a way so that t comes out nearly equidistant (for example by appying Kepler’s law).

An alternative would be to write the equation in 3 D for the relativistic momentum with cartesian coordinates. This would describe the most general case and could certainly be solved easier numerically than for the polar coordinates. Of course a projection to 2D would be sufficient in the cases that we are considering currently.

Horst

Verschickt: Mo, 29 Apr 2013 10:39 am
Betreff: 241(4) : Calculation of x from the Minkowski Force and Animation

This is a method of calculating x from the Minkowski force equation for nearly circular orbits, and it is given by eqs. (11) and (12). The orbit may be animated from eq. (26) as shown. The integral has to be evaluated numerically. I suggest something like Simpson’s rule from the Numerical Algorithms Group (NAG) Library or a contemporary package such as Mathematica, Maple or MOTECC or ESSL. I can also evaluate x for all orbits, from eq. (1), but that will be more complicated. Another method would be to integrate eq. (1) directly, and an even simpler method would integrate the relativistic momentum directly. I will explore these ideas in further notes.

241(4).pdf


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