3 ms·

So if the problem was first proposed by Newton in the 17th century, how did early investigators simulate the interactions between the bodies up until the electr

So if the problem was first proposed by Newton in the 17th century, how did early investigators simulate the interactions between the bodies up until the electronic computer became available. There are no closed form solutions A.F.A.I.K. so they must have calculated the body's state by hand, I guess. How tedious.

musgravepeter 9y ago
With some symmetry you can find a couple of solutions, which is what Laplace and Lagrange did. e.g. http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/ThreeBodyProblem.pdf http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/T... The scholarpedia article is a good meta-reference:http://www.scholarpedia.org/article/Three_body_problem http://www.scholarpedia.org/article/Three_body_problem
tr1ck5t3r 9y ago
It was probably an early form of spirograph that helped solve some of the trajectories.
rwmj 9y ago
My probably naive intuition says that if you had a system with two "suns" in the centre rotating around each other very closely, and one distant planet rotating about the centre of mass of the two suns, that would be stable surely?
- musgravepeter 9y ago
  Yes, that does work. In the literature this is called a "heirarchical system" The very simplest version is called the Euler problem. Two fixed masses and a third moving in the "dipole field". All the solutions can be explicitly determined (although only in terms of e.g. Jacobi elliptic functions and other elliptics). There's a book "Integrable Systems in Celestial Mechanics" by Mathuna. I recently added the Euler problem to my iOS app ThreeBody: https://appadvice.com/app/threebody-lite/951920756 https://appadvice.com/app/threebody-lite/951920756 Some day I'll get around to adding these 600 solutions...
  - adrianratnapala 9y ago
    Right, but this dipole solution will neglect the perturbation of the third body on whatever is creating the dipole. Fair enough, because we expect the effect to be tiny. But still you would have to unpack that dipole approximation to figure out if this perturbation will slowly change it in ways that do something significant in the long run.

musgravepeter 9y ago

With some symmetry you can find a couple of solutions, which is what Laplace and Lagrange did. e.g. http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/ThreeBodyProblem.pdf http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/T... The scholarpedia article is a good meta-reference:http://www.scholarpedia.org/article/Three_body_problem http://www.scholarpedia.org/article/Three_body_problem

tr1ck5t3r 9y ago

It was probably an early form of spirograph that helped solve some of the trajectories.

rwmj 9y ago

My probably naive intuition says that if you had a system with two "suns" in the centre rotating around each other very closely, and one distant planet rotating about the centre of mass of the two suns, that would be stable surely?

musgravepeter 9y ago
Yes, that does work. In the literature this is called a "heirarchical system" The very simplest version is called the Euler problem. Two fixed masses and a third moving in the "dipole field". All the solutions can be explicitly determined (although only in terms of e.g. Jacobi elliptic functions and other elliptics). There's a book "Integrable Systems in Celestial Mechanics" by Mathuna. I recently added the Euler problem to my iOS app ThreeBody: https://appadvice.com/app/threebody-lite/951920756 https://appadvice.com/app/threebody-lite/951920756 Some day I'll get around to adding these 600 solutions...
- adrianratnapala 9y ago
  Right, but this dipole solution will neglect the perturbation of the third body on whatever is creating the dipole. Fair enough, because we expect the effect to be tiny. But still you would have to unpack that dipole approximation to figure out if this perturbation will slowly change it in ways that do something significant in the long run.