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\documentclass[12pt]{article}
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\usepackage{amsmath}
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\begin{document}
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\begin{align*}
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\dot{\mathbf r}_i &= \mathbf v_i, \\
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\dot{\mathbf v}_i &= \frac{\mathbf f_i}{m_i} + \frac{\mathbf g_i}{m_i}.
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\end{align*}
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\end{document}
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\documentclass[12pt]{article}
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\usepackage{amsmath}
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\begin{document}
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\begin{equation*}
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\mathbf g_i =
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\begin{cases} \frac{m_i}{2} \frac{ F_{\Gamma_{k(\mathbf r_i)}}}{ K_{\Gamma_{k(\mathbf r_i)}}}
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\left(\mathbf v_i - \mathbf v_{\Gamma_{k(\mathbf r_i)}} \right) & \mbox{$k(\mathbf r_i)> 0$ (inside a reservoir),} \\
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0 & \mbox{otherwise, }
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\end{cases}
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\end{equation*}
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\end{document}
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