# Two-body problem

In classical mechanics, the **two-body problem** is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other (a binary star), and a classical electron orbiting an atomic nucleus (although to solve the electron/nucleus 2-body system correctly a quantum mechanical approach must be used).

The two-body problem can be re-formulated as two **one-body problems**, a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. 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.

## Contents

## Reduction to two independent, one-body problems

Let **x**_{1} and **x**_{2} be the positions of the two bodies, and *m*_{1} and *m*_{2} be their masses. The goal is to determine the trajectories **x**_{1}(*t*) and **x**_{2}(*t*) for all times *t*, given the initial positions **x**_{1}(*t* = 0) and **x**_{2}(*t* = 0) and the initial velocities **v**_{1}(*t* = 0) and **v**_{2}(*t* = 0).

When applied to the two masses, Newton's second law states that

where **F**_{12} is the force on mass 1 due to its interactions with mass 2, and **F**_{21} is the force on mass 2 due to its interactions with mass 1.

Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. *Adding* equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, *subtracting* equation (2) from equation (1) results in an equation that describes how the vector **r** = **x**_{1} − **x**_{2} between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories **x**_{1}(*t*) and **x**_{2}(*t*).

### Center of mass motion (1st one-body problem)

Addition of the force equations (1) and (2) yields

where we have used Newton's third law **F**_{12} = −**F**_{21} and where

- is the position of the center of mass (barycenter) of the system.

The resulting equation:

shows that the velocity **V** = *d***R**/*dt* of the center of mass is constant, from which follows that the total momentum *m*_{1} **v**_{1} + *m*_{2} **v**_{2} is also constant (conservation of momentum). Hence, the position **R** (*t*) of the center of mass can be determined at all times from the initial positions and velocities.

## Two-body motion is planar

The motion of two bodies with respect to each other always lies in a plane (in the center of mass frame). Defining the linear momentum **p** and the angular momentum **L** by the equations

the rate of change of the angular momentum **L** equals the net torque **N**

and using the property of the vector cross product that **v** × **w** = **0** for any vectors * v* and

*pointing in the same direction,*

**w**with **F** = μ *d* ^{2}**r** / *dt* ^{2}.

Introducing the assumption (true of most physical forces, as they obey Newton's strong third law of motion) that the force between two particles acts along the line between their positions, it follows that **r** × **F** = **0** and the angular momentum vector **L** is constant (conserved). Therefore, the displacement vector **r** and its velocity **v** are always in the plane perpendicular to the constant vector **L**.

## Laws of Conservation of Energy for each of two bodies for arbitrary potentials

In system of the center of mass for arbitrary potentials

the value of energies of bodies do not change:

## Central forces

For many physical problems, the force **F**(**r**) is a central force, i.e., it is of the form

where r = |**r**| and **r̂** = **r**/r is the corresponding unit vector. We now have:

where *F(r)* is negative in the case of an attractive force.

## Work

The total work done in a given time interval by the forces exerted by two bodies on each other is the same as the work done by one force applied to the total relative displacement.

## See also

- Kepler orbit
- Energy drift
- Equation of the center
- Euler's three-body problem
- Gravitational two-body problem
- Kepler problem
*n*-body problem- Virial theorem
- Two-body problem (career)

## References

**^**David Betounes (2001).*Differential Equations*. Springer. p. 58; Figure 2.15. ISBN 0-387-95140-7.

## Bibliography

- Landau LD, Lifshitz EM (1976).
*Mechanics*(3rd. ed.). New York: Pergamon Press. ISBN 0-08-029141-4.

- Goldstein H (1980).
*Classical Mechanics*(2nd. ed.). New York: Addison-Wesley. ISBN 0-201-02918-9.

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