Why does this alignment not work with this “cases” environment

casesvertical alignment

I'm trying to align a short equation and a long one.

enter image description here

Could you explain why the highlighted parts are not properly aligned?

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\[
\left\{
\begin{aligned}
d x_t^i &= v_t^i d t, \\
\begin{split}
d v_t^i &= - (\nabla U (X_t^{i, n}) - \nabla U (\bar X_t^i)) d t \\
& \qquad + \bigg [ \bigg ( \frac{1}{n} \sum_{j=1}^n K (X_t^{i, n} - X_t^{j, n}) \bigg )^m - (K* \mu_t)^m (\bar X_t^i) \bigg ] d B_t^i. 
\end{split}
\end{aligned}
\right.
\]

\end{document}

Best Answer

The whole line will get a single equation number (if you decide to have it numbered), so I personally don't see any (too important) reason why you should use split here. You should get the desired output by the following.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[
    \left\{
    \begin{aligned}
        d x_t^i ={}& v_t^i d t, \\
        d v_t^i ={}& - (\nabla U (X_t^{i, n}) - \nabla U (\bar X_t^i)) d t \\
        & + \biggl [ \biggl ( \frac{1}{n} \sum_{j=1}^n K (X_t^{i, n} - X_t^{j, n}) \biggr )^m - (K* \mu_t)^m (\bar X_t^i) \biggr ] d B_t^i.
    \end{aligned}
    \right.
\]
\end{document}

Result

I deliberately changed the position of alignment to be after the equal sign, such that - and + are aligned.

Personally I'd also like to have the exponent after the biggr parenthesis to be closer to the parenthesis, so I'd prefer \biggr)^{\mkern-3mu m} or similar.

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