difficulty in numbering the equation properly when used in subequation along with split

Question

Don't understand why there is some extra equation number which is coming in the last line of the equation

 \begin{subequations}
    \label{eq: non dimensional lagrangian}
     \begin{align}
     \mathcal{L}&=\mathcal{T}-\mathcal{V} \nonumber \\
       \mathcal{V}&=\frac{EI_{zz}}{2L}\, \int_{0}^{1} \hspace{-0.05in} \left(\frac{\partial^2\psi(\eta)}{\partial \eta^2}\right)^2 \mathrm{d}\eta +\frac{L^2}{2}\sum_{j=1}^{n}\,k_{1,j}\left(\psi(\eta)\right)^2 \nonumber\\
        \mathcal{T}&=\frac{\rho A\omega^2L^3}{2}\, \int_{0}^{1}  \hspace{-0.05in} \left( \psi(\eta)\right)^2 \mathrm{d}\eta \nonumber \\
        \begin{split}
     \mathcal{\bar{L}}&=\frac{\beta^4}{2}\, \int_{0}^{1}  \hspace{-0.05in} \left( \psi(\eta)\right)^2 \mathrm{d}\eta-\frac{1}{2}\, \int_{0}^{1} \hspace{-0.05in} \left(\frac{\partial^2\psi(\eta)}{\partial \eta^2}\right)^2 \mathrm{d}\eta\\ &\quad{}-\frac{1}{2}\sum_{j=1}^{n}\,K_{1,j}\left(\psi(\eta)\right)^2    
     \end{split}\\ \nonumber \\
     \end{align}
     \end{subequations}

Answer 1

You have wrong \\ in your input.

If you want the number to sit next to the last line, don't use split:

\documentclass[twocolumn]{article}
\usepackage{amsmath}

\newcommand{\diff}{\mathop{}\!\mathrm{d}}

\begin{document}

\begin{subequations}\label{eq: non dimensional lagrangian}
\begin{align}
\mathcal{L} &= \mathcal{T}-\mathcal{V} \nonumber \\
\mathcal{V} &= \frac{EI_{zz}}{2L} \int_{0}^{1} \left(
                 \frac{\partial^2\psi(\eta)}{\partial \eta^2}
               \right)^2 \diff\eta +
               \frac{L^2}{2}\sum_{j=1}^{n} k_{1,j}(\psi(\eta))^2 \nonumber \\
\mathcal{T} &= \frac{\rho A\omega^2L^3}{2} \int_{0}^{1} (\psi(\eta))^2 \diff\eta \nonumber \\
\bar{\mathcal{L}} &= \frac{\beta^4}{2} \int_{0}^{1} (\psi(\eta))^2 \diff\eta -
                     \frac{1}{2} \int_{0}^{1} \left(
                       \frac{\partial^2\psi(\eta)}{\partial \eta^2}
                     \right)^2 \diff\eta \nonumber \\
                  &\quad-\frac{1}{2}\sum_{j=1}^{n}K_{1,j}(\psi(\eta))^2    
\end{align}
\end{subequations}

\end{document}

I've made some changes to your input, in particular removing useless \left and \right. Also \mathcal{\bar{L}}, while yielding the expected result, should be \bar{\mathcal{L}} which is sounder on a logical basis.

Use \diff instead of explicit \mathrm{d} (there are many reasons to). I removed the explicit spacings you used.

You may find split more logical and I can agree. In this case, add the tbtags option when calling amsmath and all numbers will be aligned with the bottom line in splits.