What is Two body problem: Definition and 36 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. BEORTA39

    Calculating Force and Acceleration in a Two Block System with Varying Friction

    in the A we need to find the force and acceleration of the system when we pull the lower block and the upper mass starts to move in B we need to find the force and acceleration of the system when we pull the upper block and the upper mass starts to move the friction before the lower mass and...
  2. deuteron

    Solving two body central force motion using Lagrangian

    For the central force ##F=-\nabla U(r_r)## where ##\vec r_r=\vec r_1-\vec r_2##, and ##\vec r_1## and ##\vec r_2## denote the positions of the masses, we get the following kinetic energy using the definition of center of mass ##\vec r_{cm}= \frac{m_1\vec r_1+m_2\vec r_2}{m_1+m_2}##: $$T= \frac...
  3. K

    I Why don't we consider ##E_{com}## while solving a two body problem?

    While studying two interacting particles such as a Hydrogen atom, I learned how to reduce the problem into two independent parts by using center of mass coordinates and the relative coordinates. The resulting two independent energy eigenvalue equations give me two eigenvalues for energy as...
  4. 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...
  5. 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 ?
  6. patric44

    Solving the two body problem in Matlab

    hi guys i am trying to code an algorithm for computing the trajectory of a basic two body problem situation according to the equation $$\ddot{r} = \frac{-\mu}{r^3} \vec{r}$$ i am trying to use the Euler method , but the problem is in converting this problem into a 3 separate equations one for...
  7. 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...
  8. pobro44

    Two body problem, velocities of two bodies, relative velocity given

    When only force acting on body is a central force, angular momentum is constant and given by: L = mr^2 * w where r is distance from origin, and w is angular velocity. Angular momentum can also be written as following: L = r x mv = rmv * sin(theta) where v is tangential velocity, which is...
  9. Robin04

    Solving equation of motion for two body problem

    Homework Statement ##\ddot{r} = c \frac{1}{r^2}##, where ##c## is a constant, and ##r## is the position of one object with respect to the other. I need to find the function ##\dot{r}(r)## We are in one dimension. Homework EquationsThe Attempt at a Solution I don't really have any idea how to...
  10. Y

    A Capture by Gravitational Radiation in 2-Body System?

    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...
  11. tomwilliam2

    Two body problem, deriving energy expression

    Homework Statement This problem relates to the two body problem of two rotating point masses, where one is much larger than the other. Equate the orbital energy per unit mass ##\epsilon## with the moments of momenta at the apses to get: ##\epsilon = -\mu/2a## Homework Equations The orbital...
  12. 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...
  13. K

    Orbital Perturbations: Solving for Equations of Motion in Elliptical Orbits

    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...
  14. 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...
  15. T

    Classical Mechanics: Two Body Problem

    Homework Statement Consider two objects with masses ##m_1## and ##m_2## exerting forces on each other with magnitude ##F##. If no other net forces act on the objects, they obey the equations of motion ##m_1\ddot r_1=F##, ##m_2\ddot r_2=-F## Show that the corresponding equations of...
  16. V

    Lagrangian of two body problem with spin

    I know how to solve "typical" Kepler problem but I'm interested in a global view to "binary" systems. For example Earth - Moon. If I set lagrangian of system as ##L=\frac{1}{2}(m_1\dot{r}_1^2 + m_2\dot{r}_2^2)-V(|r_2-r_1|)## there isn't included a spin. My questions are: 1) If it is solved as...
  17. 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!
  18. _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...
  19. U

    How do I account for a Jacobian when changing coordinates for two particles?

    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...
  20. D

    Could the Twin Paradox in GR be Generalized to Completely 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...
  21. 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...
  22. A

    C/C++ Two Body problem with RK4 in C++

    I want to output an excel file with the results of the trajectories of a two body problem, with initial position and velocity. But my program is not compiling. Any suggestions/problems that you can see? #include <iostream> #include <vector> #include <fstream> using namespace std...
  23. J

    Two Body problem in python using RK4

    So I am writing a program in python to do RK4 for the two body problem. I want it to display a sphere moving around another. It currently displays one sphere for a split second and then it goes blank. Any suggestions? from __future__ import division from visual import * from visual.graph import...
  24. A

    The choice of origin in solving the two body problem

    In solving the two body problem we choose the center of mass of the system as the origin. But is the frame of reference we are setting up inertial? I don't think so. Suppose the central force is gravitation, A has mass 4kg and B has mass 1kg. If we put an object at the center of mass...
  25. Z

    Two Body problem using Verlet Algorithm Python

    Homework Statement Hi ! I'm trying to solve numerically two body problem using Verlet algorithm in Python. I wrote a code which looks like that : import numpy as np import scipy as sp rm=np.array([0.,0.]) r0=np.array([2.,0.]) p0=np.array([0.,0.1]) dt=0.001 m=0.1 G=0.01 M=500.0def r(dt)...
  26. D

    Two body problem with additional force

    Hey folks I'm trying to find some info on this scenario I'm working on. Consider a particle orbiting a star with eccentricity e, basically standard two body problem. I have an approximation for the particles distance from the sun at some time t. I would now like to introduce another force, F...
  27. P

    Is the Two Body Problem in GR Solvable Using Geodesics?

    In this thread I would like to discuss aspects of this separate from some recent related threads. In particular, I prefer that proposing that mathematical results of differential geometry and well known results in GR are wrong please not occur. I have a disussion question at the end of this...
  28. M

    Two-Body Problem: Angular Momentum Calculation

    Homework Statement Two gravitating particles with masses m1 and m2 start from rest a large distance apart. They are allowed to fall freely towards one another. The particles are given equal and opposite impulses I when they are a distance a apart, such that each impulse is perpendicular to the...
  29. M

    Two body problem - effective particle

    as for the center of mass, the position vector is Rcm=\frac{m1r1+m2r2}{m1+m2} and Vcm is determined in a similar way (it is simply the time derivative of Rcm). But what is the radius vector of the effective particle (with the reduced mass)? is it simply the vectoric sum of the two radius...
  30. T

    Two body problem of Hidrogen Atom

    In Hidrogen atom, why we choose the center of mass coordinate which is on the same direction with relative coordinate?
  31. T

    Two Body Problem in GR: Formulation & Solution

    How do we formulate the two body problem in Gr. Also if possible how do we solve it, or approximate said solution?
  32. M

    Two Body Problem: Exact Solutions?

    Do you think it is (in principle) possible to write down an exact solution, using the current geometric model of space-time, describing two idealized bodies (say two black holes or two stars) of a different mass orbiting each other?
  33. L

    Finding the inertial frame in the two-body problem in GR

    This weekend I played the game of the perihelion precession in GR. I started with the Schwarzschild geometry and used the hamilton-jacobi method. It was quite interresting to compare the integral with the classical counterpart. The full two-body problem may be more complicated to handle...
  34. G

    Can You Solve the Keplerian Two-Body Problem in Physics?

    Alright I'm really stuck on this question. I was wondering if anyone could help: (a) Show that the total energy (per unit mass) of a particle orbiting in an attractive Keplerian potential V(r) = -GM/r is E = (1/2)(dr/dt)^2 + (1/2)(J^2/r^2)-(GM)/r where J = |r x v| is the particle's angular...
  35. E

    Solution of the Two Body Problem

    I have been trying to understand why the three body problem is unsolvable for some time now. But I realized that it would probably help to see a closed form solution to the two body problem first. I was wondering if anyone can direct me to such a solution via link or just shed some light on the...
  36. Y

    Gravitational field, two body problem, lagrangian points

    I want to calculate the lagrangian points of the gravitational field in a 2-body problem. i want to superpose the two gravitational fields of the bodys and after that transform them into a rotating system which has the same angular velocity as the 2-body system does have. from this...