18: Driven Oscillator
- Page ID
- 29518
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Michael Fowler (closely following Landau para 22)
Consider a one-dimensional simple harmonic oscillator with a variable external force acting, so the equation of motion is
\[\ddot{x}+\omega^{2} x=F(t) / m\]
which would come from the Lagrangian
\[L=\dfrac{1}{2} m \dot{x}^{2}-\dfrac{1}{2} k x^{2}+x F(t)\]
(Landau “derives” this as the leading order non-constant term in a time-dependent external potential.)
The general solution of the differential equation is \(x=x_{0}+x_{1}, \text { where } x_{0}=a \cos (\omega t+\alpha)\), the solution of the homogeneous equation, and \(x_{1}\) is some particular integral of the inhomogeneous equation.
An important case is that of a periodic driving force \(F(t)=f \cos (\gamma t+\beta)\). A trial solution \(x_{1}(t)=b \cos (\gamma t+\beta) \text { yields } b=f / m\left(\omega^{2}-\gamma^{2}\right) \text { so }\]
\[x(t)=a \cos (\omega t+\alpha)+\dfrac{f}{m\left(\omega^{2}-\gamma^{2}\right)} \cos (\gamma t+\beta)\]
But what happens when \(\gamma=\omega ?\) To find out, take part of the first solution into the second, that is,
\[x(t)=a^{\prime} \cos \left(\omega t+\alpha^{\prime}\right)+\dfrac{f}{m\left(\omega^{2}-\gamma^{2}\right)}[\cos (\gamma t+\beta)-\cos (\omega t+\beta)]\)
The second term now goes to \(0 / 0 \text { as } \gamma \rightarrow \omega\), so becomes the ratio of its first derivatives with respect to \(\omega\) (or, equivalently, \(\gamma\)).
\[x(t)=a^{\prime} \cos \left(\omega t+\alpha^{\prime}\right)+\dfrac{f}{2 m \omega} t \sin (\omega t+\beta)\]
The amplitude of the oscillations grows linearly with time. Obviously, this small oscillations theory will crash eventually.
But what if the external force frequency is slightly off resonance?
Then (real part understood)
\[x=A e^{i \omega t}+B e^{i(\omega+\varepsilon) t}=\left(A+B e^{i \varepsilon t}\right) e^{i \omega t}, \quad A=a e^{i \alpha}, \quad B=b e^{i \beta}\]
with \(a, b, \alpha, \beta\) real.
The wave amplitude squared
\[C^{2}=\left|A+B e^{i \varepsilon t}\right|^{2}=a^{2}+b^{2}+2 a b \cos (\varepsilon t+\beta-\alpha)\]
We’re seeing beats, with beat frequency \(\varepsilon\). Note that if the oscillator begins at the origin, \(x(t=0)=0\), then \(A+B=0\) and the amplitude periodically goes to zero, this evidently only occurs when \(|A|=|B|\).
Energy is exchanged back and forth with the driving external force.