Kinetic Energy & Angular Momentum

Skip to main content

\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}} \renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}} \let\VF=\vf \let\HAT=\Hat \newcommand{\Prime}{{}\kern0.5pt'} \newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}} \newcommand{\Partial}[2]{{\partial#1\over\partial#2}} \newcommand{\tr}{{\mathrm tr}} \newcommand{\CC}{{\mathbb C}} \newcommand{\HH}{{\mathbb H}} \newcommand{\KK}{{\mathbb K}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HR}{{}^*{\mathbb R}} \renewcommand{\AA}{\vf{A}} \newcommand{\BB}{\vf{B}} \newcommand{\CCv}{\vf{C}} \newcommand{\EE}{\vf{E}} \newcommand{\FF}{\vf{F}} \newcommand{\GG}{\vf{G}} \newcommand{\HHv}{\vf{H}} \newcommand{\II}{\vf{I}} \newcommand{\JJ}{\vf{J}} \newcommand{\KKv}{\vf{Kv}} \renewcommand{\SS}{\vf{S}} \renewcommand{\aa}{\VF{a}} \newcommand{\bb}{\VF{b}} \newcommand{\ee}{\VF{e}} \newcommand{\gv}{\VF{g}} \newcommand{\iv}{\vf{imath}} \newcommand{\rr}{\VF{r}} \newcommand{\rrp}{\rr\Prime} \newcommand{\uu}{\VF{u}} \newcommand{\vv}{\VF{v}} \newcommand{\ww}{\VF{w}} \newcommand{\grad}{\vf{\nabla}} \newcommand{\zero}{\vf{0}} \newcommand{\Ihat}{\Hat I} \newcommand{\Jhat}{\Hat J} \newcommand{\nn}{\Hat n} \newcommand{\NN}{\Hat N} \newcommand{\TT}{\Hat T} \newcommand{\ihat}{\Hat\imath} \newcommand{\jhat}{\Hat\jmath} \newcommand{\khat}{\Hat k} \newcommand{\nhat}{\Hat n} \newcommand{\rhat}{\HAT r} \newcommand{\shat}{\HAT s} \newcommand{\xhat}{\Hat x} \newcommand{\yhat}{\Hat y} \newcommand{\zhat}{\Hat z} \newcommand{\that}{\Hat\theta} \newcommand{\phat}{\Hat\phi} \newcommand{\LL}{\mathcal{L}} \newcommand{\DD}[1]{D_{\textrm{$#1$}}} \newcommand{\bra}[1]{\langle#1|} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\braket}[2]{\langle#1|#2\rangle} \newcommand{\LargeMath}[1]{\hbox{\large$#1$}} \newcommand{\INT}{\LargeMath{\int}} \newcommand{\OINT}{\LargeMath{\oint}} \newcommand{\LINT}{\mathop{\INT}\limits_C} \newcommand{\Int}{\int\limits} \newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}} \newcommand{\tint}{\int\!\!\!\int\!\!\!\int} \newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}} \newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int} \newcommand{\Bint}{\TInt{B}} \newcommand{\Dint}{\DInt{D}} \newcommand{\Eint}{\TInt{E}} \newcommand{\Lint}{\int\limits_C} \newcommand{\Oint}{\oint\limits_C} \newcommand{\Rint}{\DInt{R}} \newcommand{\Sint}{\int\limits_S} \newcommand{\Item}{\smallskip\item{$\bullet$}} \newcommand{\LeftB}{\vector(-1,-2){25}} \newcommand{\RightB}{\vector(1,-2){25}} \newcommand{\DownB}{\vector(0,-1){60}} \newcommand{\DLeft}{\vector(-1,-1){60}} \newcommand{\DRight}{\vector(1,-1){60}} \newcommand{\Left}{\vector(-1,-1){50}} \newcommand{\Down}{\vector(0,-1){50}} \newcommand{\Right}{\vector(1,-1){50}} \newcommand{\ILeft}{\vector(1,1){50}} \newcommand{\IRight}{\vector(-1,1){50}} \newcommand{\Partials}[3] {\displaystyle{\partial^2#1\over\partial#2\,\partial#3}} \newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}} \newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}} \newcommand{\LLv}{\vf{L}} \newcommand{\OOb}{\boldsymbol{O}} \newcommand{\PPv}{\vf{P}_\text{cm}} \newcommand{\RRv}{\vf{R}_\text{cm}} \newcommand{\ff}{\vf{f}} \newcommand{\pp}{\vf{p}} \newcommand{\tauv}{\vf{\tau}} \newcommand{\Lap}{\nabla^2} \newcommand{\Hop}{H_\text{op}} \newcommand{\Lop}{L_\text{op}} \newcommand{\Hhat}{\hat{H}} \newcommand{\Lhat}{\hat{L}} \newcommand{\defeq}{\overset{\rm def}{=}} \newcommand{\absm}{\vert m\vert} \newcommand{\ii}{\ihat} \newcommand{\jj}{\jhat} \newcommand{\kk}{\khat} \newcommand{\dS}{dS} \newcommand{\dA}{dA} \newcommand{\dV}{d\tau} \renewcommand{\ii}{\xhat} \renewcommand{\jj}{\yhat} \renewcommand{\kk}{\zhat} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 21.9 Kinetic Energy & Angular Momentum

In Section 21.7, we showed that the position and velocity vectors in polar coordinates are given by

\begin{align} \vv \amp = \dot{r}\,\rhat+ r\dot\phi\,\phat\tag{21.9.1}\\ \aa \amp = \left( \ddot{r} - r\dot\phi^2 \right) \rhat + \left( r\ddot\phi + 2\dot{r}\dot\phi \right) \phat\tag{21.9.2} \end{align}

Activity 21.4. Find the Kinetic Energy and Angular Momentum in Polar Coordinates.

Show that the kinetic energy \(T\) of the reduced mass in polar coordinates is given by:

\begin{equation} T = \frac12 \mu \,(\dot{r}^2 + r^2 \dot\phi^2)\tag{21.9.3} \end{equation}

Similarly, show that the magnitude of the angular momentum \(|\LLv|=\ell\) of the reduced mass \(\mu\) is given in polar coordinates by:

\begin{equation} \ell = \mu r^2 \dot\phi\tag{21.9.4} \end{equation}

Hint.

Don’t forget that \(v^2=\vv\cdot\vv\text{.}\) Use the product rule.

Since the angular momentum is a constant in central force problems, it’s magnitude \(\ell\) is also constant. Therefore (21.9.4) can be used to rewrite differential equations, getting rid of \(\dot\phi\)’s in favor of the variable \(r\) and the constant \(\ell\text{.}\)

Kepler’s second law says that the areal velocity of a planet in orbit is constant in time. This is equivalent to equation (21.9.4). To see why, read in section 8.3 of Marion and Thornton, page 294, from equation 8.10 to the bottom of the page.