Two body problem Definition and 12 Discussions

In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored.
The most prominent case of the classical two-body problem is the gravitational case (see also Kepler problem), arising in astronomy for predicting the orbits (or escapes from orbit) of objects such as satellites, planets, and stars. A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions.
A simpler "one body" model, the "central-force problem", treats one object as the immobile source of a force acting on other. One then seeks to predict the motion of the single remaining mobile object. Such an approximation can give useful results when one object is much more massive than the other (as with a light planet orbiting a heavy star, where the star can be treated as essentially stationary).
However, the one-body approximation is usually unnecessary except as a stepping stone. For many forces, including gravitational ones, the general version of the two-body problem can be reduced to a pair of one-body problems, allowing it to be solved completely, and giving a solution simple enough to be used effectively.
By contrast, the three-body problem (and, more generally, the n-body problem for n ≥ 3) cannot be solved in terms of first integrals, except in special cases.

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  1. antythingyani

    Averaging the cube of semimajor axis to position ratio wrt to time

    Summary:: Averaging (a power of) semimajor axis to position ratio wrt to time - celestial mechanics I evaluated it this far, but i don't know how to change the dt to d theta ... the final solution is supposedly (1-e^2)^-(3/2) . Any help will be appreciated. [Image re-inserted with correct...
  2. juardilag

    Help with a modified Kepler potential

    I have tried to solve the problem through the use of a rotating reference frame, since I should have as a solution an orbit given by the Kepler potential, but I haven't come up with anything. Any ideas ?
  3. pobro44

    Angular and orbital speed at perihelion

    Hello to all good people of physics forums. I just wanted to ask, whether the angular and linear (orbital) speed in perihelion of eliptical orbit are related the same way as in circular orbit (v = rw). If we take a look at the angular momentum (in polar coordinates) of reduced body moving in...
  4. Y

    A Capture in a Two-Body System by Gravitational Radiation?

    Is it possible for enough energy to be dissipated in the form of gravitational radiation in a two-body system to allow for capture? From what I remember, you would need extremely massive bodies passing extremely close to each other: I'd like to know how massive and how close. It has been a few...
  5. D

    Asteroid deflected by Earth -- effect on Earth

    Homework Statement Suppose the asteroid of [other problem] has a mass of 6 \times 10^{20} \textrm{kg} . Find the proportional change in the kinetic energy of the Earth in this encounter. What is the change in the semi-major axis of the Earth's orbit? By how much is its orbital period...
  6. K

    Orbital Perturbations

    Homework Statement Say I have some planet in a circular orbit around a star, and I give it a small radial push (directly toward or directly away from the star). How would I describe the new orbit? I.e. how would I determine the equations of motion? h Homework Equations Kepler orbital radius...
  7. R

    Doubts on angular momentum [master's degree exam]

    Two hollow spheres, both the mass and radius R M , which are rotating around a center of mass ( CM ) , with an initial period To, are kept distant from each other by an ideal wire with a distance of 8R. At a given instant a motor is driven by wrapping the wire and making the two spheres meet...
  8. 2

    How can you tell what kind of orbit a body will have?

    Is there a way of telling whether the orbit of a body around another, or rather of both around their centre of mass, will give the object in question a circular, elliptical, hyperbolic or parabolic orbit? Thank you!
  9. _Kenny_

    Euler-Lagrange Equations for Two Body Problem

    Homework Statement I'm trying to do a little review of Lagrangian Mechanics through studying the two-body problem for a radial force. I have the Lagrangian of the system L=\frac{1}{2}m_1\dot{\vec{r_1}}^{2}+\frac{1}{2}m_2\dot{\vec{r_2}}^{2}-V(|{\vec{r_1}-\vec{r_2}}|) . Now I'm trying to find...
  10. U

    Jacobian for CoM coordinates

    I am computing matrix elements of a two body quantum-mechanical potential, which take the form V_{k l m n} = \int d^3 r_1 d^3 r_2 e^{-i k \cdot r_1} e^{-i l \cdot r_2} V( | r_1-r_2 | ) e^{i m \cdot r_1} e^{i n \cdot r_2} To do this integral, I make the change of coordinates...
  11. D

    Twin Paradox In Empty Space

    Whenever the twin paradox in GR seems to be discussed, it always seems to be done in the presence of a large mass such that the twins can be considered as test particles moving in some metric. I was wondering whether the same problem could be generalised and be proposed in completely empty...
  12. J

    Frictionless Multiple Body Problem with Inclined planes

    Homework Statement An m1 = 7.6 kg block and an m2 = 10.7 kg block, connected by a rope that passes over a frictionless peg, slide on frictionless incline. Find acceleration of boxes and tension of the rope. Homework Equations F=ma I'm not sure what else The Attempt at a Solution I'm really...