Two-body Problem - Reduction To Two Independent, One-body Problems

Reduction To Two Independent, One-body Problems

Let x1 and x2 be the positions of the two bodies, and m1 and m2 be their masses. The goal is to determine the trajectories x1(t) and x2(t) for all times t, given the initial positions x1(t = 0) and x2(t = 0) and the initial velocities v1(t = 0) and v2(t = 0).

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

\mathbf{F}_{12}(\mathbf{x}_{1},\mathbf{x}_{2}) = m_{1} \ddot{\mathbf{x}}_{1} \quad \quad \quad (\mathrm{Equation} \ 1)
\mathbf{F}_{21}(\mathbf{x}_{1},\mathbf{x}_{2}) = m_{2} \ddot{\mathbf{x}}_{2} \quad \quad \quad (\mathrm{Equation} \ 2)

where F12 is the force on mass 1 due to its interactions with mass 2, and F21 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 = x1 − x2 between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories x1(t) and x2(t).

Read more about this topic:  Two-body Problem

Famous quotes containing the words reduction and/or problems:

    “The reduction of nuclear arsenals and the removal of the threat of worldwide nuclear destruction is a measure, in my judgment, of the power and strength of a great nation.”
    —Jimmy Carter (James Earl Carter, Jr.)

    “The problems of victory are more agreeable than the problems of defeat, but they are no less difficult.”
    —Winston Churchill (1874–1965)