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Fix typo
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henry2004y committed May 27, 2024
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Expand Up @@ -354,7 +354,7 @@ If the effective fields @eq-gc-effective-em were replaced by the standard fields
The guiding-center equations of motion @eq-gc-u-time-derivative and @eq-gc-X-time-derivative can be used to derive the guiding-center energy equation
$$
\frac{\mathrm{d}H_\text{gc}}{\mathrm{d}t} = \frac{\partial H_\text{gc}}{\partial t} + \dot{\mathbf{X}}\cdot\nabla H_\text{gc} + \dot{u}\frac{\partial H_\text{gc}}{\partial u} = e\frac{\partial\Phi^ast}{\partial t} -\frac{e}{c}\frac{\partial\mathbf{A}^\ast}{\partial t}\cdot\dot{\mathbf{X}}
\frac{\mathrm{d}H_\text{gc}}{\mathrm{d}t} = \frac{\partial H_\text{gc}}{\partial t} + \dot{\mathbf{X}}\cdot\nabla H_\text{gc} + \dot{u}\frac{\partial H_\text{gc}}{\partial u} = e\frac{\partial\Phi^\ast}{\partial t} -\frac{e}{c}\frac{\partial\mathbf{A}^\ast}{\partial t}\cdot\dot{\mathbf{X}}
$$ {#eq-gc-energy-time-derivative}
which implies that the guiding-center energy $E_\text{gc}\equiv mu^2/2+e\Phi^\ast$ is a constant of the motion for time-independent fields.
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