Would three galaxies, held together by dark matter, be an example of a three body problem?
No, because they are too loose constructed. The Sun, Earth, and Moon are a good three body problem. Galaxies are a many body problem.
Galaxies have so many bodies that they can probably be treated as a fluid dynamical problem of the non-ideal gas variety: gravity being analogous to VanderWaals forces.
Back in the bad old days, AFAIK, modelling of galaxies was done analytically ... with some success (e.g. spiral density wave). These days the preferred approach is simulation; the Toomre brothers did some of the first numerical modelling and showed nice tidal tails from collisions. These days the number of point masses ('stars') is in the millions.
If the galaxies are far enough apart (i.e. compared to their sizes), then their relative motions could be looked at as a three body problem, as long as there are no other galaxies nearby.
... or dark matter concentrations
The classic 3 body problem is the homework equivalent of violating the Geneva convention. Solving it for 3 galaxies is a physics 101 assignment at Hades University.
I wonder how many more schools exist down there? The University of Hell, Brimstone Technical School...
I'ld like to propose a relatively simple solution to this 3 body problem. Suppose we had a 'satellite' orbitting around the earth. It is affected by both the earth andthe sun.
Now, there are 4 forces which affect the firmament's orbit around
1. Fge - earth's gravity
2. Fgs - sun's gravity
3. Fae - acceleration around the earth
4. Fas - acceleration around the sun
Fge + Fas = Fgs + Fae --- equation 1
The standard physics textbook 'Geostationary satellite orbit' simply
Fge = Fae
So, substituting this into equation 1, we get:
Fge + Fas = Fgs + Fge
and we are left with:
Fas = Fgs
When this is done, the orbit is like 20,000 to 35,000 miles in space
--- depending on the speed (and thus the period). However, what if,
the Satellite's orbit was designed so that the Earth's gravity was way
larger than the satellite's acceleration around the earth? -- that is:
Fge >> Fae
Well, consider equation 1:
Fge + Fas = Fgs + Fae
Fge - Fae = Fgs - Fas
Since, Fge >> Fae, this reduces to:
Fge = Fgs - Fas --- equation 2
Does this jive so far?
I know there is no solution to the 3-body problem except numeric integration. But does this just mean nobody has found the solution yet? I know many prominent mathimaticians from years ago tried to solve it. Were they wasting their time? Has it been proven that the 3-body problem has no solution? Or is it possible that I might wake tommorow to find on page 9 of my newspaper "Mathamatician solves 3-body problem".
How is that? I thought van der Waals forces were an electric dipole thing.
Yes, people have solved it.
That's how you find your Lagrange points, for one thing. One of the simple ones would be directly between the Sun and Earth. Your satellite's angular velocity around the Sun is dependent upon the strength of the Sun's gravitational attraction at that distance. The closer to the Sun, the greater the angular velocity. If the satellite is directly between the Earth and Sun, the net gravitational attraction would be the the Sun's minus the Earth's. In essence, you've given the satellite a weaker gravitational attraction to the Sun than it would normally have at that distance - weak enough that the satellite's angular velocity matches the Earth's angular velocity around the Sun giving you a satellite that always stays between you and the Sun (like ACE, for example).
Others can get much more difficult to do by hand, since your angles and distances are always changing, but not so much so for a computer which can do repetitive tasks very well.
But a computer doing this as a repetitive task is an example on a numerical integration. And numerical integrations are subject to truncation errors (if you make the time step too fast) and round-off errors.
I was referring to an analytical solution to the 3-body problem. A solution where the problem was not divided into n time steps, solve for each time step and add together. For example: How far will a car moving at 60 mph travel in 200 hours? D = V * T = 12000 miles. This is an anaytical solution that would work equally well for any value of T or V. There's no need to break this problem into 1-second time intervals, solve for each interval and add them together, although you could do it that way. But doing it that way does introduce error into the problem that isn't there in a purely analytical solution.
In the case of planetary motion, there are analytical methods that provide good approximations. VSOP87 is an example of one. This method will predict the position of a planet as viewed from Earth to an acrsecond of accuracy for hundreds of years. A similar method is used for the Moon. This is how solar eclipses and planetary occulations are predicted so well. But these methods fail miserably if trying to determine the long-term state of the solar system over millions or billions of years. That is because these solutions must have all known pertubations spelled out exactly in mathamatical terms. There may be pertabutions that we don't know about that don't have any significant effect in the short term. And these methods are only effective because the planets' orbits are close to circular, and not very chaotic in the short term.
Take for example this diagram http://www.orbitsimulator.com/gravity/capture.jpg of the Earth's Moon capturing an asteroid into Earth orbit. An analytical solution should be able to produce a position for the asteroid as a function of time. This diagram was created using the time-step method.
Numerically only. There aren't enough constants to solve it analytically.
The only reason we're able to solve the 2 body problem analytically is because you can make the assumption that 'body 2's mass' << 'body 1's mass'.
I might be wrong, but I think the 2-body is also analytically solveable even if both bodies have significant mass. They would both be tracing perfect unperturbed ellipses around their barycenter. A 3rd body introduces pertubations and messes the whole thing up.
But do you know if a 3-body analytical solution is an impossiblilty, or just a solution that nobody's discovered yet?
The two body problem can be solved analytically. The three body problem can be solved in special cases, such as the Lagrange points of the restricted three body problem (note the word: restricted). The general three body problem cannot be solved analytically, there are two few constants and too many variables:
but I've never personally seen the proof.
Here's an idea for solving the three body problem:
Instead of three masses have two masses for which the force of attraction
varies in magnitude which is what would happen as the third mass moves around them.So Newton's law Gm1m2/r^2 would not be valid but would be
replaced by a time dependent function e.g if r = t^2 + t + 1
we would get Gm1m2 / (t^2 + t + 1)^2
We could find this function by asking "how do we minimise r ?"
i.e by using Euler-Lagrange equations.
There's a little more about Brunn's theorem at
Proposals that try to solve the three body problem without addressing the roadblock of this theorem are just ignoring the math.
Note that in some sense there is a "solution", though not an analytic one, to the planar three body problem, as there is a series that converges due to Sundman. But it's convergence is very very slow
talks about the need for 10^8,000,000 terms of this series to achieve the accuracy needed for astronomy.
If one of the bodies in a three body problem has a large mass - like a galaxy -
and this mass is in reality made of many masses,what determines when two much smaller masses become part of the many body probelm that the galaxy already is in its own right? Where is the borderline?
As long as you have more than 2 bodies, and 2 of your bodies have mass, then you have an n-body problem. In certain instances, you can treat an n-body problem like a bunch of 2 body problems and get good results, but they'll just be approximations. So a galaxy can not be considered a body because it is many bodies.
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