Greta wrote:Steve, what about those with, say, Marfan Syndrome, or those who generally have very long arms and thin fingers?
Good point! I guess we have to assume that God has/had thumb sizes and arm lengths that were broadly similar to the average genetically healthy adult human.
Jonathan A Bain wrote:An ellipse always returns to its starting point. Are you saying orbits are not ellipses?
Well, no, in my post back there I wasn't saying that. I was asking you a question. But since you ask, no, orbits generally are not ellipses. That's Kepler's idealisation for two body systems. If orbits were all perfect ellipses then Neptune would not have been discovered, because its discovery was due to calculations based on observations of orbital perturbations of already known planets using Newton's (not Kepler's) laws.
So a more general solution is the application of Newton's laws and a more general solution than that is Einstein's laws. As you know, the application of those laws to systems of three bodies or more has to be numerical, using finite time steps.
Clearly a situation in which a large body impacts with another large body and they both partially or wholly vapourize and/or break into an extremely large number of pieces is not a simple two body problem. So, although this doesn't in itself show your conclusion to be wrong, I don't think that the justification for that conclusion is that a body following an elliptical path always returns to the same point. We know, for example, that it is dynamically possible, in an orbital system described by Newton's laws, for a body to be ejected from that system such that it never returns.
Your recollection is likely not based on real gravity equations, rather there are vague diagrams originating from centuries past wrongly depicting a cannonball going into orbit.
I'm pretty sure they're based on modern computer simulations of Newton's laws applied to large numbers of mutually gravitating particles. I've written some simpler ones myself at various times. More complex ones appear to have been used to model collisions of large bodies to show such things as the formation of ring systems and satellites. I'll try to find some examples online, but there have been numerous gravitational simulations written over the years.
One of my gravity algorithms is currently ranked 1st out of over 3 million at google for the search "binary orbit software".
I searched for it and found this:
http://www.flight-light-and-spin.com/binary-orbits.htm
I presume that's it? Looks good. Very interesting. I'd be interested to see some of the code that does the calculations. In your explanation...
http://www.flight-light-and-spin.com/ma ... roblem.htm
...you talk about using a "quantum of time". I presume what you're talking about here is simply the use of a finite (small but non-zero) time step in the calculations? This is what is done in all numerical computer simulations of physical systems which can't be solved analytically. Not just planetary orbits. It's also used in physics engines in computer games.
To start writing a numerical simulation of many mutually gravitating bodies is essentially quite simple. It's a numerical solution to Newton's law of gravitation:
F = Gm
1m
2 / r
2
For each time step, and for each body, as an initial first order approximation, you simply work out the acceleration of each body using the above equation and F = ma, add the acceleration to the velocity and add the velocity to the position. This is the simplest possible form of numerical integration. Then you can start optimising by using higher order numerical integration methods like Runge–Kutta methods and so on.
As far as I can tell from your explanation on your website, this first approximation is what you did. It is a standard first approximation for solutions to this kind of system. I can see why you used the thing you refer to as a "holding matrix" in order to calculate all momentum changes before calculating all position changes, but I don't think this is a fundamental innovation. It doesn't alter the fact that the number of gravitational interactions to be calculated is proportional to the square of the number of bodies ((n
2 - n) / 2, I think), and it doesn't alter the fact that the accuracy of the model will always be dependent on the size of the finite time step chosen and the method of numerical integration chosen.
There are various other more complex algorithms, such as "particle-in-cell" algorithms, and various special case simplifications, that can reduce the required computing time for very large systems of particles such that it doesn't increase with the square of the number of particles. But there's always a trade-off between general applicability, accuracy, computing time and complexity of the algorithm.
Although not exactly the question of this topic, solving for n-body-gravity with numerous interacting gravity fields is actually a far more advanced algorithm than a single body orbit -which can only be an ellipse, or the object reaches escape velocity.
It's not necessarily a more advanced algorithm. The difference is in the way in which the problem can be solved: numerically or analytically. In an analytical solution to a problem, it is possible to write an equation that precisely defines the entire state of the system at any time t:
State = F(t)
In three-or-more-gravitating-body problems this cannot be done. Instead, the system has to be evolved from an initial time, t
0 by numerical integration.
But, at heart, the algorithm representing the numerical solution to the problem is pretty simple. It's the successive optimisations of the algorithm, and/or the use of simplifying assumptions for special cases, to yield more accurate models for a given amount of computing time that adds the algorithmic complications. As a general rule, if computing time is unlimited, algorithms can be relatively simple.
The interesting part, I think, is when you start to think about chaotic systems in which the inaccuracies of numerical solutions to problems blow up so large so quickly that no amount of computing power can keep up. But I guess that really is straying off topic.