Aligning multiple lines in long equations and arranging the line breaks in `\align` environment

Question

I have the following code sample:

\begin{align}
    \pi^{(1)}_{i j} \enspace & = \enspace -\frac{1}{4}\eta^{\lambda \sigma} \left( \eta^{\mu \alpha}\eta^{\nu \beta} + \eta^{\mu \beta}\eta^{\nu \alpha}  \right) \biggl\{ \frac{1}{2}\delta_{\lambda 0}( \delta_{\mu i}\delta_{\nu j} + \delta_{\mu j}\delta_{\nu i}  )   \biggr\} \partial_{\sigma} h_{\alpha \beta}
    \nonumber
    \\
    -\frac{1}{4}\eta^{\lambda \sigma} \left( \eta^{\mu \alpha}\eta^{\nu \beta} + \eta^{\mu \beta}\eta^{\nu \alpha}  \right) \biggl\{ \frac{1}{2}\delta_{\sigma 0}( \delta_{\alpha i}\delta_{\beta j} + \delta_{\alpha j}\delta_{\beta i} )  \biggr\} \partial_{\lambda} h_{\mu \nu}
    \mspace{-458mu}
    \nonumber
    \\
    & = \enspace -\frac{1}{4}\partial^{\lambda} h^{\mu \nu}\delta_{\lambda 0}( \delta_{\mu i}\delta_{\nu j} + \delta_{\mu j}\delta_{\nu i} ) -\frac{1}{4}\partial^{\sigma} h^{\alpha \beta}\delta_{\sigma 0}( \delta_{\alpha i}\delta_{\beta j} + \delta_{\alpha j}\delta_{\beta i} )
    \nonumber
    \\
    & = \enspace -\frac{1}{2}\partial^0 h^{i j} - \frac{1}{2}\partial^0 h^{i j}
    \nonumber
    \\
    & = \enspace \dot{h}_{i j}
\end{align}

which gives the output:

As you see, I have a problem in aligning the lines properly which I tried to handle by using \mspace{}. My first question is: What is the correct or proper way of doing this in these kind of lengthy equations?

I also have another problem. When I use separate \align environments repeatedly, at some moment I reach to the end of the page and the last \align environment automatically starts from a new page. In such a case all vertical spaces of the previous page changes and they ruin the proper format of the text. So, my second question is: How to properly arrange line breaks or spacing in repeated \align environments consisting lengthy equations?

Answer 1

You should use split inside equation for this:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation}
\begin{split}
\pi^{(1)}_{i j}
={}&  -\frac{1}{4}\eta^{\lambda \sigma} ( \eta^{\mu \alpha}\eta^{\nu \beta} + \eta^{\mu \beta}\eta^{\nu \alpha}  ) \biggl\{ \frac{1}{2}\delta_{\lambda 0}( \delta_{\mu i}\delta_{\nu j} + \delta_{\mu j}\delta_{\nu i}  )   \biggr\} \partial_{\sigma} h_{\alpha \beta}
\\
   & -\frac{1}{4}\eta^{\lambda \sigma} ( \eta^{\mu \alpha}\eta^{\nu \beta} + \eta^{\mu \beta}\eta^{\nu \alpha}  ) \biggl\{ \frac{1}{2}\delta_{\sigma 0}( \delta_{\alpha i}\delta_{\beta j} + \delta_{\alpha j}\delta_{\beta i} )  \biggr\} \partial_{\lambda} h_{\mu \nu}
\\
={}& -\frac{1}{4}\partial^{\lambda} h^{\mu \nu}\delta_{\lambda 0}( \delta_{\mu i}\delta_{\nu j} + \delta_{\mu j}\delta_{\nu i} ) -\frac{1}{4}\partial^{\sigma} h^{\alpha \beta}\delta_{\sigma 0}( \delta_{\alpha i}\delta_{\beta j} + \delta_{\alpha j}\delta_{\beta i} )
\\
={}& -\frac{1}{2}\partial^0 h^{i j} - \frac{1}{2}\partial^0 h^{i j}
\\
={}& \dot{h}_{i j}
\end{split}
\end{equation}

\end{document}

The trick is to put = on the left side, following it with {} for ensuring correct spacing.

Avoid useless \left and \right. Also, no \enspace should be used around =.

The number should go in the middle, in my opinion. However, the tbtags option to amsmath will put it at the bottom of a split.

Answer 2

You can add few {} and \phatoms to align.

\documentclass{article}
\usepackage{amsmath}
\allowdisplaybreaks
\begin{document}
  \begin{align}
    \pi^{(1)}_{i j} & = {}-\frac{1}{4}\eta^{\lambda \sigma} \left( \eta^{\mu \alpha}\eta^{\nu \beta} + \eta^{\mu \beta}\eta^{\nu \alpha}  \right) \biggl\{ \frac{1}{2}\delta_{\lambda 0}( \delta_{\mu i}\delta_{\nu j} + \delta_{\mu j}\delta_{\nu i}  )   \biggr\} \partial_{\sigma} h_{\alpha \beta}
    \nonumber
    \\
    &\phantom{{}={}}{}-\frac{1}{4}\eta^{\lambda \sigma} \left( \eta^{\mu \alpha}\eta^{\nu \beta} + \eta^{\mu \beta}\eta^{\nu \alpha}  \right) \biggl\{ \frac{1}{2}\delta_{\sigma 0}( \delta_{\alpha i}\delta_{\beta j} + \delta_{\alpha j}\delta_{\beta i} )  \biggr\} \partial_{\lambda} h_{\mu \nu}
    \nonumber
    \\
    & = {}-\frac{1}{4}\partial^{\lambda} h^{\mu \nu}\delta_{\lambda 0}( \delta_{\mu i}\delta_{\nu j} + \delta_{\mu j}\delta_{\nu i} ) -\frac{1}{4}\partial^{\sigma} h^{\alpha \beta}\delta_{\sigma 0}( \delta_{\alpha i}\delta_{\beta j} + \delta_{\alpha j}\delta_{\beta i} )
    \nonumber
    \\
    & ={} -\frac{1}{2}\partial^0 h^{i j} - \frac{1}{2}\partial^0 h^{i j}
    \nonumber
    \\
    & ={}\dot{h}_{i j}
\end{align}
\end{document}

For your page breaking problem, use \allowdisplaybreaks.