21.2: Motion Under a Central Force
- Page ID
- 8539
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.