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21.2: Motion Under a Central Force

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  • I consider the two-dimensional motion of a particle of mass \(m\) under the influence of a conservative central force \(F(r)\), which can be either attractive or repulsive, but depends only on the radial coordinate \(r\). Recalling the formula \( \ddot{r} - r \dot{ \theta }^2 \) \)for acceleration in polar coordinates (the second term being the centripetal acceleration), we see that the equation of motion is

    \[ m \ddot{r} - m r \dot{ \theta }^2 = F(r). \tag{21.2.1}\label{eq:21.2.1}\]

    This describes, in polar coordinates, two-dimensional motion in a plane. But since there are no transverse forces, the angular momentum \( m^2 \dot{ \theta }^2 \) is constant and equal to \(L\), say. Thus we can write Equation \(\ref{eq:21.2.1}\) as

    \[ m \ddot{r} = F(r) + \frac{L^2}{mr^3}. \tag{21.2.2}\label{eq:21.2.2}\]

    This has reduced it to a one-dimensional equation; that is, we are describing, relative to a co-rotating frame, how the distance of the particle from the centre of attraction (or repulsion) varies with time. In this co-rotating frame it is as if the particle were subject not only to the force \(F(r)\), but also to an additional force \( \frac{L^2}{mr^3} \). In other words the total force on the particle (referred to the co-rotating frame) is

    \[ F'(r) = F(r) + \frac{L^2}{mr^3}. \tag{21.2.3}\label{eq:21.2.3}\]

    Now \(F(r)\), being a conservative force, can be written as minus the derivative of a potential energy function, \( F = - \frac{dV}{dr}\). Likewise, \( \frac{L^2}{m^3}\) is minus the derivative of \( \frac{L^2}{2mr^2} \). Thus, in the co-rotating frame, the motion of the particle can be described as constrained by the potential energy function \(V'\), where

    \[ V' = V + \frac{L^2}{2mr^2}. \tag{21.2.4}\label{eq:21.2.4}\]

    This is the equivalent potential energy. If we divide both sides by the mass m of the orbiting particle, this becomes

    \[ \Phi' = \Phi + \frac{h^2}{2r^2}. \tag{21.2.5}\label{eq:21.2.5}\]

    Here \(h\) is the angular momentum per unit mass of the orbiting particle, \( \Phi \) is the potential in the inertial frame, and \( \Phi ' \)is the equivalent potential in the corotating frame.