11.4: Equations of Motion
- Page ID
- 9617
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The equations of motion for two bodies interacting via a conservative two-body central force can be determined using the center of mass Lagrangian, \(L_{cm},\) given by equation \((11.3.3)\). For the radial coordinate, the operator equation \(\Lambda _{r}L_{cm}=0\) for Lagrangian mechanics leads to
\[\frac{d}{dt}\left( \mu \dot{r}\right) -\mu r\dot{\psi}^{2}+\frac{\partial U}{ \partial r}=0\]
But
\[\dot{\psi}=\frac{l}{\mu r^{2}}\]
therefore the radial equation of motion is
\[\mu \ddot{r}=-\frac{\partial U}{\partial r}+\frac{l^{2}}{\mu r^{3}}\]
Similarly, for the angular coordinate, the operator equation \(\Lambda _{\psi }L_{cm}=0\) leads to equation \((11.3.5)\). That is, the angular equation of motion for the magnitude of \(p_{\psi }\) is \[p_{\psi }=\frac{\partial L}{\partial \dot{\psi}}=\mu r^{2}\dot{\psi}=l\]
Lagrange’s equations have given two equations of motion, one dependent on radius \(r\) and the other on the polar angle \(\psi\). Note that the radial acceleration is just a statement of Newton’s Laws of motion for the radial force \(F_{r}\) in the center-of-mass system of \[F_{r}=-\frac{\partial U}{\partial r}+\frac{l^{2}}{\mu r^{3}}\]
This can be written in terms of an effective potential
\[U_{eff}(r)\equiv U(r)+\frac{l^{2}}{2\mu r^{2}}\label{11.33}\]
which leads to an equation of motion
\[F_{r}=\mu \ddot{r}=-\frac{\partial U_{eff}(r)}{\partial r}\label{11.34}\]
Since \(\frac{l^{2}}{\mu r^{3}}=\mu r\dot{\psi}^{2}\), the second term in Equation \ref{11.33} is the usual centrifugal force that originates because the variable \(r\) is in a non-inertial, rotating frame of reference. Note that the angular equation of motion is independent of the radial dependence of the conservative two-body central force.
Figure \(\PageIndex{1}\) shows, by dashed lines, the radial dependence of the potential corresponding to the attractive inverse square law force, that is \(U=-\frac{k }{r}\), and the potential corresponding to the centrifugal term \(\frac{l^{2}}{ 2\mu r^{2}}\) corresponding to a repulsive centrifugal force. The sum of these two potentials \(U_{eff}(r)\), shown by the solid line, has a minimum \(U_{\min }\) value at a certain radius similar to that manifest by the diatomic molecule discussed in example \((2.12.1)\).
It is remarkable that the six-dimensional equations of motion, for two bodies interacting via a two-body central force, has been reduced to trivial center-of-mass translational motion, plus a one-dimensional one-body problem given by \ref{11.34} in terms of the relative separation \(r\) and an effective potential \(U_{eff}(r)\).