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<h2 class="hd hd-2 unit-title">Introduction to Symmetry</h2>
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<p>
We will continue with a discussion of <i class="itshape">symmetry</i>, and the usefulness of a symmetry matrix for calculations. In particular, a system that exhibits a particular symmetry will share the same eigenvectors as that symmetry matrix. </p><p>
If the symmetry of a system is known, then one can simply use the the eigenvectors of the symmetry matrix to help solve for the normal mode frequencies (eigenfrequencies) of the system. We will exploit this useful fact in the lesson after this one. </p>
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<h2 class="hd hd-2 unit-title">L10Q1: Identifying Symmetry I</h2>
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Which Symmetry?
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There are three main types of symmetry: reflection, translational, and rotational. </p>
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Systems with <i class="itshape">reflection symmetry</i> maintain the same properties (look identical) when "reflected" about one axis (like looking at a mirror image). This image of two rows of columns is characterized by reflection symmetry. </p>
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Systems with <i class="itshape">rotational symmetry</i> maintain the same properties (appear identical) when rotated through an angle. This image of a 6-pointed snowflake has [mathjaxinline]60[/mathjaxinline] degree rotational symmetry. </p>
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Systems with <i class="itshape">translational symmetry</i> maintain the same properties (appear identical) when shifted in space. An infinite row of fence posts has translational symmetry. </p>
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References:<br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;columns: <a href="https://commons.wikimedia.org/wiki/File:Great_Mosque_of_Kairouan,_west_portico_of_the_courtyard.jpg" target="_blank">image link</a>; attribution: James Rose [<a href="https://creativecommons.org/licenses/by-sa/2.0" target="_blank">CC BY-SA</a>] <br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;snowflake: <a href="https://commons.wikimedia.org/wiki/File:Snowflake_macro_photography_1.jpg" target="_blank">image link</a>; attribution: Alexey Kljatov [<a href="https://creativecommons.org/licenses/by-sa/4.0" target="_blank">CC BY-SA</a>] <br/>&#8195;&#8195;&#8195;[mathjaxinline]\bullet[/mathjaxinline] &#8194;fence: <a href="https://ccsearch.creativecommons.org/photos/49df1bb5-eb9b-4dc5-b9e9-7b3b0f98053a" target="_blank">image link</a>; attribution: NRCS Montana [<a href="https://creativecommons.org/publicdomain/mark/1.0/?ref=ccsearchatype=rich" target="_blank">CC PDM 1.0</a>] <br/></p>
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Now, we leave it to you to identify the symmetry related to each of the following systems. </p>
<p><b class="bfseries">(Part a)</b> A system of three identical masses, connected by identical springs, constrained to move on a frictionless table top. </p>
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<p><b class="bfseries">(Part b)</b> A system of two coupled, identical pendula (assume that [mathjaxinline]m_1=m_2[/mathjaxinline]). </p>
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<p><b class="bfseries">(Part c)</b> An infinite array of coupled, identical masses. </p>
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<h2 class="hd hd-2 unit-title">L10v1: Overview of Symmetry Matrix</h2>
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<h2 class="hd hd-2 unit-title">L10Q2: Identifying Symmetry II</h2>
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The symmetry matrix [mathjaxinline]\textbf{S}[/mathjaxinline] transforms a system to new set of coordinates, which represent a new, physically valid way to represent the system. </p><p>
For instance, let's consider the case of reflection symmetry. If we define a set of coordinates as </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{X}= \begin{pmatrix} x_{1}\\ x_{2} \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
then these coordinates would be defined in the following way, in the reflected version of the system: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\tilde{\textbf{X}}= \begin{pmatrix} -x_{2}\\ -x_{1} \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
The matrix that transforms the vector [mathjaxinline]\textbf{X}[/mathjaxinline] to [mathjaxinline]\tilde{\textbf{X}}[/mathjaxinline] is the symmetry matrix: </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{S}\textbf{X}=\tilde{\textbf{X}}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In this case, </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{S}= \begin{pmatrix} 0 & & -1\\ -1 & & 0 \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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<h3 class="hd hd-3 problem-header" id="lect_08_01_a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_01_a-problem-progress" tabindex="-1">
Finding the Symmetry Matrix I
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<p><b class="bfseries">(Part a)</b> Identify the symmetry matrix associated with the reflection of the following system: </p>
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<p>
where the transformed coordinates are related by: </p>
<table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{X}= \begin{pmatrix} x_{1}\\ x_{2}\\ x_{3} \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\tilde{\textbf{X}}= \begin{pmatrix} -x_{3}\\ -x_{2}\\ -x_{1} \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p style="display:inline">[mathjaxinline]\textbf{S}=[/mathjaxinline]</p>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
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<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h3 class="hd hd-3 problem-header" id="lect_08_01_b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_01_b-problem-progress" tabindex="-1">
Finding the Symmetry Matrix II
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<p><b class="bfseries">(Part b)</b> Identify the symmetry matrix associated with the rotation of the following system: three identical beads on a circular ring, attached to identical springs: </p>
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where the transformed coordinates are related by (note that each mass shifts one position over): </p>
<table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]{\pmb {\Theta }}= \begin{pmatrix} \theta _{1}\\ \theta _{2}\\ \theta _{3} \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<table id="a0000000012" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\tilde{\pmb {\Theta }}= \begin{pmatrix} \theta _{3}\\ \theta _{1}\\ \theta _{2} \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p style="display:inline">[mathjaxinline]\textbf{S}=[/mathjaxinline]</p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L10Q3: Commutation Properties of the Symmetry Matrix</h2>
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Commutation Properties
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Consider a dynamical system of the type we have been studying, with matrices [mathjaxinline]\textbf{M}[/mathjaxinline] and [mathjaxinline]\textbf{K}[/mathjaxinline]. If [mathjaxinline]\textbf{S}[/mathjaxinline] is the symmetry matrix of this system, which of the following relations are valid: </p>
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<text>[mathjaxinline]\textbf{S}\textbf{K}=\textbf{K}\textbf{S}[/mathjaxinline]</text>
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<text>[mathjaxinline]\textbf{S}\textbf{M}=\textbf{M}\textbf{S}[/mathjaxinline]</text>
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<text>[mathjaxinline]\textbf{S}\textbf{M}^{-1}\textbf{K}=\textbf{M}^{-1}\textbf{S}\textbf{K}[/mathjaxinline]</text>
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<text>[mathjaxinline]\textbf{S}\textbf{M}^{-1}\textbf{K}=\textbf{M}^{-1}\textbf{K}\textbf{S}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L10Q4: Is it a Symmetry Matrix</h2>
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Identifying Symmetry Matrix of System
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Consider the following systems, where the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix is given. Which of the following symmetry matrices, [mathjaxinline]\textbf{S}_{1}[/mathjaxinline] or [mathjaxinline]\textbf{S}_{2}[/mathjaxinline], are actual symmetries of the system? [HINT: Explicitly calculate the commutator [mathjaxinline]\left[\textbf{S}, \textbf{M}^{-1}\textbf{K} \right][/mathjaxinline]. If the matrices commute, then [mathjaxinline]\textbf{S}[/mathjaxinline] represents a symmetry of the system.] </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\textbf{S}_{1}= \begin{pmatrix} 0 &amp; &amp; 1 &amp; &amp; 0\\ 0 &amp; &amp; 0 &amp; &amp; 1\\ 1 &amp; &amp; 0 &amp; &amp; 0 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<td class="equation" style="width:80%; border:none">[mathjax]\textbf{S}_{2}= \begin{pmatrix} 0 &amp; &amp; 0 &amp; &amp; 1\\ 0 &amp; &amp; 1 &amp; &amp; 0\\ 1 &amp; &amp; 0 &amp; &amp; 0 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p><b class="bfseries">(Part a)</b> Which matrix, [mathjaxinline]\textbf{S}_{1}[/mathjaxinline], [mathjaxinline]\textbf{S}_{2}[/mathjaxinline], both, or neither, represents a symmetry of the system with [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\textbf{M}^{-1}\textbf{K}= \begin{pmatrix} \frac{k}{m} &amp; &amp; -\frac{k}{m} &amp; &amp; 0\\ -\frac{k}{2m} &amp; &amp; \frac{k}{m} &amp; &amp; -\frac{k}{2m}\\ 0 &amp; &amp; -\frac{k}{m} &amp; &amp; \frac{k}{m} \end{pmatrix}[/mathjax]</td>
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<p><b class="bfseries">(Part b)</b> Which matrix, [mathjaxinline]\textbf{S}_{1}[/mathjaxinline], [mathjaxinline]\textbf{S}_{2}[/mathjaxinline], both, or neither, represents a symmetry of the system with [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\textbf{M}^{-1}\textbf{K}= \begin{pmatrix} \frac{2k}{m} &amp; &amp; -\frac{k}{m} &amp; &amp; -\frac{k}{m}\\ -\frac{k}{m} &amp; &amp; \frac{2k}{m} &amp; &amp; -\frac{k}{m}\\ -\frac{k}{m} &amp; &amp; -\frac{k}{m} &amp; &amp; \frac{2k}{m} \end{pmatrix}[/mathjax]</td>
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<h3 class="hd hd-2">L10v3: Use Symmetry Eigenvectors to Solve for Oscillation Eigenfrequency</h3>
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<h2 class="hd hd-2 unit-title">L10Q5: Solving Eigenmodes from S Matrix</h2>
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Symmetry Matrix Eigenvectors
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We have shown that the eigenvectors of the symmetry matrix [mathjaxinline]\textbf{S}[/mathjaxinline] are identical to the eigenvectors of the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix, when [mathjaxinline]\textbf{S}[/mathjaxinline] is a symmetry of the system. Let these eigenvectors be called [mathjaxinline]\textbf{A}[/mathjaxinline]. </p>
<p>
Importantly, one can use the eigenvectors [mathjaxinline]\textbf{A}[/mathjaxinline] to solve for the eigenvalues (i.e. the normal mode frequencies) of the physical system, through the following straightforward calculation. </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{M}^{-1}\textbf{K}\textbf{A}=\omega ^{2}\textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Let's do this with an example! Remember the system from the previous lesson, consisting of two coupled masses, attached to springs and a string of constant tension [mathjaxinline]T=\kappa L[/mathjaxinline], constrained to move vertically like pistons? </p>
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<p>
To simplify things, let [mathjaxinline]m_{1} = m_{2} = m[/mathjaxinline] and [mathjaxinline]k_{1} = k_{2} = k[/mathjaxinline] so that the system has reflection symmetry. Then the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix is: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{M}^{-1}\textbf{K}= \begin{pmatrix} \frac{k + \kappa }{m} &amp; &amp; - \frac{\kappa }{m} \\ -\frac{\kappa }{m} &amp; &amp; \frac{k + \kappa }{m} \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Since the system has reflection symmetry, it shares eigenvalues with the matrix: </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{S}= \begin{pmatrix} 0 &amp; &amp; 1\\ 1 &amp; &amp; 0 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Find the symmetric, [mathjaxinline]\textbf{A}_{\mathrm{sym}}[/mathjaxinline], and antisymmetric, [mathjaxinline]\textbf{A}_{\mathrm{asym}}[/mathjaxinline], eigenvectors of [mathjaxinline]\textbf{S}[/mathjaxinline], i.e. the eigenvectors in which the two masses move in the same or opposite directions, respectively. Normalize your eigenvectors so that the first element is [mathjaxinline]+1[/mathjaxinline]. </p>
<p style="display:inline">[mathjaxinline]\textbf{A}_{\mathrm{sym}}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\textbf{A}_{\mathrm{asym}}=[/mathjaxinline]</p>
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enter <code> 2+3*2 </code> for 8 </td>
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<td class="formulainput">enter (english) name of letter</td>
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enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
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Symmetry Matrix Eigenfrequencies
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<p><b class="bfseries">(Part b)</b> Determine the eigenvalues of the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] by explicitly computing [mathjaxinline]\textbf{M}^{-1}\textbf{K}\textbf{A}[/mathjaxinline]. Do these match what you derived previously? </p>
<p>
Express your answer in terms of <code>m</code>, <code>k</code>, and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\omega _{\mathrm{sym}} =[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\omega _{\mathrm{asym}} =[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
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<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
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<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<th class="formulainput" scope="row">Basic functions</th>
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<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
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<code>sin, cos, tan, sec, csc, cot</code>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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