3.3: Linearity and Superposition
- Page ID
- 9574
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)An important aspect of linear systems is that the solutions obey the Principle of Superposition, that is, for the superposition of different oscillatory modes, the amplitudes add linearly. The linearly-damped linear oscillator is an example of a linear system in that it involves only linear operators, that is, it can be written in the operator form (appendix \(19.6.2\))
\[ \label{eq:3.6} \Big ( \frac{d^2}{dt^2} + \Gamma \frac{d}{dt} + \omega_0^2 \Big ) x (t) = A \cos \omega t \]
The quantity in the brackets on the left hand side is a linear operator that can be designated by \(\mathbb{L}\) where
\[ \label{eq:3.7} \mathbb{L} x (t) = F (t) \]
An important feature of linear operators is that they obey the principle of superposition. This property results from the fact that linear operators are distributive, that is
\[ \label{eq:3.8} \mathbb{L} ( x_1 + x_2 ) = \mathbb{L} ( x_1 ) + \mathbb{L}(x_2) \]
Therefore if there are two solutions \( x_1 (t) \) and \( x_2 (t) \) for two different forcing functions \( F_1 (t) \) and \( F_2 ( t) \)
\[ \label{eq:3.9} \begin{align*} \mathbb{L}x_1(t) & = & F_1(t) \\ \mathbb{L}x_1(t) & = & F_2(t) \end{align*} \]
then the addition of these two solutions, with arbitrary constants, also is a solution for linear operators.
\[ \label{eq:3.10} \mathbb{L} ( \alpha_1 x_! + \alpha _2 x_2 ) = \alpha_1 F_1 (t) + \alpha_2 F_2 (t) \]
In general then
\[ \label{eq:3.11} \mathbb{L} \Bigg ( \sum_{n=1}^N \alpha_n x_n (t) \Bigg ) = \Bigg ( \sum_{n=1}^N \alpha_n F_n (t) \Bigg ) \]
The left hand bracket can be identified as the linear combination of solutions
\[ \label{eq:3.12} x(t) = \sum_{n=1}^N \alpha_n x_n (t) \]
while the driving force is a linear superposition of harmonic forces
\[ \label{eq:3.13} F(t) = \sum_{n=1}^N \alpha_n F_n (t) \]
Thus these linear combinations also satisfy the general linear equation
\[ \label{eq:3.14} \mathbb{L} x(t) = F(t) \]
Applicability of the Principle of Superposition to a system provides a tremendous advantage for handling and solving the equations of motion of oscillatory systems.