rdp@teddy.UUCP (09/23/85)
[]
This is quest for some general information.
One often hears that the two-body problem (two bodies interacting
gravitationally) is completely solvable (I guess that means that
one can completely describe the motions and ineteractions of these
two bodies in an isolated system), but when the problem involves any
more tha two bodies (3 or "many"), then there does not exist a known
solution for describing the system completely. About this I have several
questions:
1. Why is the three- or many-bodied problem unsolvable? (Note I
realize that, given the asccuracy with which we can navigate
about the solar system, then the problem, while unsolved, is
approachable with some spectacularily good approximations).
2. Do the three-body problems apply for systems where the mass of
one of the bodies is vanishingly small compared to the others
(such as in a Voyager/Jupiter/Sun system)?
3. Since general relativity seems to approach gravitation not as
a force acting over a distance, but more as a deformation in the
geometry of space-time (a wild simplification, I agree), can the
three- (or many-) bodied problem be solved as a geometry problem?
In other words, is the difficulty associated with a Newtonian
view of gravity and the attendant mechanisms, or does general
relativity suffer the same way?
4. Is the solution to all this merely one of computational
fortitude? (Has JPL solved the problem simply by brute
force, or has the brute force merely made their approximations
less approximate?)
AN ensuing discussion might be of value, unless the answer is really
very simple and obvious, which it does not seem to be.
Dick Pierceethan@utastro.UUCP (Ethan Vishniac) (09/26/85)
> [] > One often hears that the two-body problem (two bodies interacting > gravitationally) is completely solvable (I guess that means that > one can completely describe the motions and ineteractions of these > two bodies in an isolated system), but when the problem involves any > more tha two bodies (3 or "many"), then there does not exist a known > solution for describing the system completely. About this I have several > questions: > > 1. Why is the three- or many-bodied problem unsolvable? (Note I > realize that, given the asccuracy with which we can navigate > about the solar system, then the problem, while unsolved, is > approachable with some spectacularily good approximations). > It is not unsolvable in the sense that when one is given initial conditions it is not particularly difficult to calculate the future evolution of the system indefinitely far into the future. It is unsolvable in the formal sense that the number of constants of motion is less than the number of constants of integration in the problem. Therefore the general features of the evolution of the system can (apparrently) only be solved for by brute force. The constants of motion are: angular momentum (three components), momentum of the center of mass (three components), total energy of the system, and total mass (however since the mass of each object is separately conserved this last is trivial). > 2. Do the three-body problems apply for systems where the mass of > one of the bodies is vanishingly small compared to the others > (such as in a Voyager/Jupiter/Sun system)? > In such a case it can be solved as a perturbation problem. Difficulties arise only after absurdly long times. > 3. Since general relativity seems to approach gravitation not as > a force acting over a distance, but more as a deformation in the > geometry of space-time (a wild simplification, I agree), can the > three- (or many-) bodied problem be solved as a geometry problem? > In other words, is the difficulty associated with a Newtonian > view of gravity and the attendant mechanisms, or does general > relativity suffer the same way? > GR has the same problems here. > 4. Is the solution to all this merely one of computational > fortitude? (Has JPL solved the problem simply by brute > force, or has the brute force merely made their approximations > less approximate?) See above. > Dick Pierce People have had *some* success finding approximate constants of motion that help give some insight in the way the system moves around phase space. -- "Support the revolution Ethan Vishniac in Latin America... {charm,ut-sally,ut-ngp,noao}!utastro!ethan Buy Cocaine" ethan@astro.UTEXAS.EDU Department of Astronomy University of Texas
rpw3@redwood.UUCP (Rob Warnock) (09/27/85)
+--------------- | One often hears that the two-body problem (two bodies interacting | gravitationally) is completely solvable... but when the problem involves any | more tha two bodies (3 or "many"), then there does not exist a known | solution for describing the system completely. About this I have several | questions: 1. Why is the three- or many-bodied problem unsolvable? +--------------- It has to do with what kinds of differential equations have "closed form" solutions. Certain kinds of simultaneous differential equations are not expressible in "closed form", and this happens to be one of them, which means: You cannot write a set of equations (a "solution") for which the left-hand sides are the positions/velocities of the bodies and the only independent variable on the right-hand side is "time". (A.k.a.: "Some formulas don't have integrals." Purists: Please excuse me for the over-simplification.) The "open form" can, of course, be expressed. I.e., the right-hand side is a function of time and all of the positions and velocities of all the bodies, and the left-hand side of each equation is the derivative of exactly one of those quantities (except time). By choosing a "small enough" increment of time ("delta_T"), one can predict "as closely as you choose" what the positions/velocities are at "T + delta_T", given the positions/velocities at "T". Do this enough times, and you can predict "as closely as you choose" the resulting positions/velocities at "T + K*delta_T", for any "K". The trick comes in knowing: a. What "delta_T" do you need to preserve accuracy across "K" iterations? b. How precise does your arithmetic need to be to preserve accuracy across "K" iterations? c. How many bodies do you need to consider (i.e., how small a body do you have to include) to get enough accuracy? +--------------- | ...realize that, given the asccuracy with which we can navigate | about the solar system, then the problem, while unsolved, is | approachable with some spectacularily good approximations). +--------------- Right. Unsolvable IN CLOSED FORM has nothing to do with solvable "as closely as you like" by iterative approximation. Obviously, NASA and friends have answered these questions "closely enough". +--------------- | 2. Do the three-body problems apply for systems where the mass of | one of the bodies is vanishingly small compared to the others | (such as in a Voyager/Jupiter/Sun system)? +--------------- Good question. No, there is a special solution for the "2-big/1-small" problem. Basically you solve the two-body problem, then assume that the third body does not perturb the "body" comprised of the two-body system (the "2-big" bodies are considered a single body in solving a two-body problem with the "1-small". Works fine for (say) spaceship/Earth/Moon. Unfortunately, there are so many massive bodies in the Solar System that ship/Earth/Moon isn't (practically) good enough. You need ship/Earth/Moon/Sun/Jupiter/etc. +--------------- | 3. ...In other words, is the difficulty associated with a Newtonian | view of gravity and the attendant mechanisms, or does general | relativity suffer the same way? +--------------- Yes. Yes, only worse. +--------------- | 4. Is the solution to all this merely one of computational | fortitude? (Has JPL solved the problem simply by brute | force, or has the brute force merely made their approximations | less approximate?) +--------------- See above. It is NOT SOLVABLE "in closed form", but we do very well with iterative solutions of the "open form" equations (unknowns on both sides), thank you. +--------------- | AN ensuing discussion might be of value, unless the answer is really | very simple and obvious, which it does not seem to be. | Dick Pierce +--------------- Open to constructive flames, I remain... Rob Warnock Systems Architecture Consultant UUCP: {ihnp4,ucbvax!dual}!fortune!redwood!rpw3 DDD: (415)572-2607 USPS: 510 Trinidad Lane, Foster City, CA 94404
grl@charm.UUCP (George Lake) (09/30/85)
The two-body problem is exceedlingly simple. You can solve the equations of both particles by considering their motion in the center of mass frame. Conservation of momentume insures that the center of mass moves at a constant velocity. Then the two particles perform symmetric motions about this point. The entire motion lies in a plane in this frame. In some deep sense this occurs because the "1-body" problem in a central force is so very simple. Most potentials have 3 distinct invariants of motion. The central force has 4, the Kepler problem has 5. This super-integrability makes the next higher problem tractable. At three it goes away. One way to see it is in terms of resonance. Two particles orbit one another with a single frequency. Three particles have multiple frequencies. When the frequencies are equal, resonance makes the motion wild and difficult to calculate. There are infinitely many resonances that all have be calculated. Each resonance is a singularity and perturbation calculations won't go through them.
john@frog.UUCP (John Woods) (10/02/85)
> One often hears that the two-body problem (two bodies interacting > gravitationally) is completely solvable (I guess that means that > one can completely describe the motions and ineteractions of these > two bodies in an isolated system), but when the problem involves any > more tha two bodies (3 or "many"), then there does not exist a known > solution for describing the system completely. About this I have several > questions: > What "they" really mean to say is that the two body problem is completely *solved*, in that someone has taken the differential equations (which are quite easy to write down for any arbitrary number of bodies) and integrated them to produce equations of position as a function of time. One can easily write the differential equations for the 3-body problem, but of the many titanic intellects who have spent time on the problem (Gauss, for instance), not one has been able to arrive at the integrated equations for Xn(t). This is usually used, in Differential Equations courses, as the motivation for numerical approximation. So, > 1. Why is the three- or many-bodied problem unsolvable? It is not *known* to be unsolvable (yet). It is known to be extremely tough. If I remember correctly, there is a proven theorem in DE that *all* differential equations have existant, unique solutions (unique after boundary conditions are applied) -- which makes the lack of a solution to the three- body problem even more irritating. :-) > 2. Do the three-body problems apply for systems where the mass of > one of the bodies is vanishingly small compared to the others > (such as in a Voyager/Jupiter/Sun system)? > Yes, but it turns out that the special case of one mass being "zero" is as easy to solve as the two body problem (in fact, it looks much like the two body problem). > 3. Since general relativity seems to approach gravitation not as > a force acting over a distance, but more as a deformation in the > geometry of space-time (a wild simplification, I agree), can the > three- (or many-) bodied problem be solved as a geometry problem? > In other words, is the difficulty associated with a Newtonian > view of gravity and the attendant mechanisms, or does general > relativity suffer the same way? General relativity, if anything, has more complicated equations. As a wild guess, solving the geometrical equations would be even worse a nightmare than solving the Newtonian equations (otherwise, one would think that someone would have already solved them). > 4. Is the solution to all this merely one of computational > fortitude? (Has JPL solved the problem simply by brute > force, or has the brute force merely made their approximations > less approximate?) Even the brute force available in a TI-59 calculator is sufficient to get a "good" approximation (a "feel" for the orbits) in the 3 body problem. Real live brute force will get you (or your spacecraft) a *loooooong* way... For those who have had calculus, I recommend any standard text on Differential Equations for further study (sorry, I don't remember the name of the text I used at MIT). -- John Woods, Charles River Data Systems, Framingham MA, (617) 626-1101 ...!decvax!frog!john, ...!mit-eddie!jfw, jfw%mit-ccc@MIT-XX.ARPA "Out of my way, I'm a scientist!" - War of the Worlds
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (10/02/85)
Early in Vol. I of the Feynman Lectures on Physics (available in hardback and paperback; A-W I think), there is a nice example of computation of orbits including a trick for maintaining higher accuracy than one might obtain without knowing the trick.
wrf@ernie.BERKELEY.EDU (W. Randolph Franklin) (10/24/85)
I think that multi-body systems can become chaotic in the sense that small perturbations can grow to macroscopic size. Thus it is impossible to predict the exact state in the future even with an arbitrarily small delta-t iteration. wrf@ucbernie.arpa
bill@utastro.UUCP (William H. Jefferys) (10/24/85)
> I think that multi-body systems can become chaotic in the sense > that small perturbations can grow to macroscopic size. Thus it is > impossible to predict the exact state in the future even with an > arbitrarily small delta-t iteration. Absolutely correct. It's even true for n=3. It is the real reason why the three body problem is not solvable (although this fact was not appreciated until recently). Note that for a system to be chaotic, the small perturbations have to grow exponentially fast. In the two-body problem, for example, a small error in the initial conditions eventually produces a macroscopic perturbation, but it only grows linearly. The two-body problem is not chaotic. -- Glend. I can call spirits from the vasty deep. Hot. Why, so can I, or so can any man; But will they come when you do call for them? -- Henry IV Pt. I, III, i, 53 Bill Jefferys 8-% Astronomy Dept, University of Texas, Austin TX 78712 (USnail) {allegra,ihnp4}!{ut-sally,noao}!utastro!bill (UUCP) bill@astro.UTEXAS.EDU. (Internet)