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-<title>Key Notes in Plasma Physics - 23&nbsp; Guiding Center Approximations</title>
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-<h1 class="title"><span id="sec-GCA" class="quarto-section-identifier"><span class="chapter-number">23</span>&nbsp; <span class="chapter-title">Guiding Center Approximations</span></span></h1>
+<h1 class="title"><span id="sec-GC" class="quarto-section-identifier"><span class="chapter-number">23</span>&nbsp; <span class="chapter-title">Guiding Center</span></span></h1>
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@@ -559,7 +559,7 @@ <h2 data-number="23.4" class="anchored" data-anchor-id="derivations-of-gc-equati
 x_\mu = \xi_\mu x + \eta_\mu y + x_\mu^\prime + X_\mu
 \tag{23.22}\]</span></span> where <span class="math inline">\(x_\mu\)</span> is the space-time position of the particle and <span class="math inline">\(X_\mu\)</span> is the position of the guiding center. We associate two variables <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> to describe the periodic motion in the <span class="math inline">\((\xi, \eta)\)</span>-plane, and the term <span class="math inline">\(x_\mu^\prime\)</span> will account for the periodic motions which do not lie in the same plane. To more easily describe the gyration, we also choose a new pair of variables <span id="eq-gc-new-vars"><span class="math display">\[
 \zeta = \frac{1}{\sqrt{2}}\left( x+ iy \right)\quad\text{and}\quad \zeta^\ast = \frac{1}{\sqrt{2}}\left( x - iy \right)
-\tag{23.23}\]</span></span> so that <span class="math inline">\(\xi_\mu x + \eta_\mu y = \delta_\mu\zeta + \sigma_\mu\zeta^\ast\)</span>. The new variables will allow us to choose <span class="math inline">\(\zeta_\mu\)</span> and <span class="math inline">\(\delta_\mu\)</span> in a more convenient way and let some factor <span class="math inline">\(e^{i\phi}\)</span> account for the initial conditions of the problem. We rewrite the particle coordinate using <span class="quarto-unresolved-ref">?eq-gc-new-variables</span> <span id="eq-eq-gc-new-vars2"><span class="math display">\[
+\tag{23.23}\]</span></span> so that <span class="math inline">\(\xi_\mu x + \eta_\mu y = \delta_\mu\zeta + \sigma_\mu\zeta^\ast\)</span>. The new variables will allow us to choose <span class="math inline">\(\zeta_\mu\)</span> and <span class="math inline">\(\delta_\mu\)</span> in a more convenient way and let some factor <span class="math inline">\(e^{i\phi}\)</span> account for the initial conditions of the problem. We rewrite the particle coordinate using <a href="#eq-gc-new-vars" class="quarto-xref">Equation&nbsp;<span>23.23</span></a> <span id="eq-eq-gc-new-vars2"><span class="math display">\[
 x_\mu = \delta_\mu\zeta + \sigma_\mu\zeta^\ast + x_\mu^\prime + X_\mu
 \tag{23.24}\]</span></span></p>
 <p>The next step is substituting this expression into <a href="#eq-charged-particle-motion-general" class="quarto-xref">Equation&nbsp;<span>23.11</span></a> and expand the second derivative, which leads to a lengthy expression containing eight unkowns: <span class="math inline">\(\zeta,\zeta^\ast\)</span>, two components of <span class="math inline">\(x_\mu\)</span> (the condition with the <span class="math inline">\((\sigma, \delta)\)</span>-plane sets the other two), and four components of <span class="math inline">\(X_\mu\)</span>.</p>
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 \tag{23.26}\]</span></span> where <span class="math inline">\(\Omega a \zeta\)</span> contains all the linear terms in <span class="math inline">\(\zeta\)</span>. Consequently, the corresponding solution <span class="math inline">\(\zeta\)</span>, by analogy to the <em>Wentzel–Kramers–Brillouin (WKB)</em> approximation in quantum mechanics, is <span id="eq-gc-wkb"><span class="math display">\[
 \zeta = \rho_0 \sqrt{\frac{\omega_0}{\Omega}}\exp\left[ -i\left( \Phi + \int_{\tau_0}^\tau \sigma_1 \mathrm{d}\tau \right) \right]
 \tag{23.27}\]</span></span> to the 0th-order, being <span id="eq-gc-wkb2"><span class="math display">\[
-\Omega = \omega - 2i\sigma_\mu \frac{\mathrm{d}\delta_\mu}{\mathrm{d}\tau},\quad \Phi = \int_{\tau_0}^{\tau}\Omega\mathrm{d}\tau - \phi,\quad\text{and} \sigma = a - \frac{a^2}{\Omega}
+\Omega = \omega - 2i\sigma_\mu \frac{\mathrm{d}\delta_\mu}{\mathrm{d}\tau},\quad \Phi = \int_{\tau_0}^{\tau}\Omega\mathrm{d}\tau - \phi,\quad\text{and}\quad \sigma = a - \frac{a^2}{\Omega}
 \tag{23.28}\]</span></span> The subscripts 0, 1 indicate the associated order of approximation for quantities defined above.</p>
 <p>Since now we know the expression for <span class="math inline">\(\zeta\)</span>, we can write the 1st−order EOM for <span class="math inline">\(X_\mu\)</span>. For this purpose, we require that in the left-end side of <a href="#eq-gc-expansion" class="quarto-xref">Equation&nbsp;<span>23.25</span></a> the terms containing the 1st− and 2nd−order derivatives of <span class="math inline">\(X_\mu\)</span> are balanced by the remaining nonoscillatory terms, that is <span id="eq-gc-balance"><span class="math display">\[
 \frac{\mathrm{d}^2 X_\mu}{\mathrm{d}\tau^2} - F_{\mu\nu}\frac{\mathrm{d}X_\nu}{\mathrm{d}\tau} = \frac{\partial F_{\mu\nu}}{\partial x_\xi}\left( \delta_\xi \sigma_\nu \frac{\mathrm{d}\zeta^\ast}{\mathrm{d}\tau} + \delta_\nu \sigma_\xi \frac{\mathrm{d}\zeta}{\mathrm{d}\tau} \right)

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