The Two-Body Problem and Radial Reduction

From Two Bodies to One Particle

Imagine two masses, \( m_1 \) and \( m_2 \), interacting via a force that depends only on the distance between them. Newton’s third law tells us the forces are equal and opposite. If we look at the system from a frame moving with the centre of mass, the total momentum is zero. The motion of the two bodies is perfectly correlated: as one moves, the other moves in a specific way to keep the centre of mass fixed. Instead of tracking two separate positions, we can define a single vector \( \mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1 \) that points from mass 1 to mass 2. The dynamics of this relative vector are mathematically identical to the dynamics of a single particle with a specific "reduced mass" \( \mu \) moving in a potential field created by the other mass. This is the essence of the reduction: we trade a complex two-body interaction for a simpler one-body problem in a central field.

The Reduced Mass

To perform this reduction rigorously, we introduce the reduced mass \( \mu \), defined by the relation \( \frac{1}{\mu} = \frac{1}{m_1} + \frac{1}{m_2} \). This quantity has units of mass and captures the inertial resistance of the relative coordinate. If one mass is much larger than the other, \( \mu \) approaches the smaller mass, recovering the familiar case of a test particle in a fixed potential. The kinetic energy of the two-body system can be decomposed into the kinetic energy of the centre of mass plus the kinetic energy of the relative motion, \( T = \frac{1}{2} M \dot{\mathbf{R}}^2 + \frac{1}{2} \mu \dot{\mathbf{r}}^2 \), where \( M = m_1 + m_2 \). Since the potential energy \( V \) depends only on the separation \( r = |\mathbf{r}| \), the Lagrangian for the relative motion is \( L = \frac{1}{2} \mu \dot{\mathbf{r}}^2 - V(r) \). This is formally identical to the Lagrangian of a single particle of mass \( \mu \) in a central potential.

Radial Reduction and the Effective Potential

Because the potential is central, the angular momentum \( \ell \) is conserved. In polar coordinates \( (r, \phi) \), the kinetic energy is \( T = \frac{1}{2} \mu (\dot{r}^2 + r^2 \dot{\phi}^2) \). The angular momentum is \( \ell = \mu r^2 \dot{\phi} \). We can eliminate \( \dot{\phi} \) from the energy equation by substituting \( \dot{\phi} = \ell / (\mu r^2) \). The total energy \( E \) becomes:

This equation looks exactly like the energy of a one-dimensional particle with kinetic energy \( \frac{1}{2} \mu \dot{r}^2 \) moving in a potential \( V_{\mathrm{eff}}(r) \). The effective potential is defined as:

The term \( \frac{\ell^2}{2 \mu r^2} \) is the centrifugal barrier. It arises from the conservation of angular momentum and prevents the particle from reaching \( r=0 \) if \( \ell \neq 0 \). The radial motion is governed by the equation \( \mu \ddot{r} = -\frac{dV_{\mathrm{eff}}}{dr} \). This reduction is crucial because it allows us to analyse the radial dynamics using one-dimensional techniques, such as finding turning points where \( E = V_{\mathrm{eff}}(r) \).

Interpreting the Effective Potential

The shape of \( V_{\mathrm{eff}}(r) \) determines the allowed regions of motion. For an attractive inverse-square potential, \( V(r) = -\frac{k}{r} \), the effective potential is \( V_{\mathrm{eff}}(r) = -\frac{k}{r} + \frac{\ell^2}{2 \mu r^2} \). This function has a minimum at \( r_0 = \frac{\ell^2}{\mu k} \). If the total energy \( E \) is less than zero, the motion is bounded, corresponding to an elliptical orbit. The particle oscillates between a minimum radius \( r_{\min} \) and a maximum radius \( r_{\max} \), which are the roots of \( E = V_{\mathrm{eff}}(r) \). If \( E \) is greater than zero, the motion is unbounded, corresponding to a hyperbolic trajectory. The centrifugal term dominates at small \( r \), creating a repulsive barrier, while the attractive term dominates at large \( r \). This interplay dictates the stability of circular orbits and the existence of bound states.