breaking lines of a lengthy equation in a multiline bracket using equation*

Question

I am trying to write an equation -first order conditions of a Lagrangian- in a multiline open bracket but due the length of the equation it doesn't look right. I am using equation* and cases from amsmath. How do I insert line breaks within each line so that the whole thing looks better?

\documentclass{article}

%packages
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}

\title{To be determined later}
\date{December 2024}

\begin{document}
\maketitle
\begin{equation*}
    \dfrac{\partial \mathcal{L}}{\partial \psi_k} = 
    \begin{cases}
        \frac{1}{n} - \delta_1 H_2(\psi_2,\psi_1) +\mu_2\delta_2 & \mathrm{if} \, k = 1    \\
        \frac{1}{n} - \delta_1 H_1(\psi_2,\psi_1) -\delta_2 H_2(\psi_3,\psi_2) - (\delta_1+\delta_2)\mu_2 + \delta_3\mu_3 &  \mathrm{if} \, k = 2    \\
        \frac{1}{n} - \delta_{k-1} H_1(\psi_k,\psi_{k-1}) -\delta_k H_2(\psi_{k+1},\psi_k) + \delta_{k-2}\mu_{k-1}- (\delta_k+\delta_{k-1})\mu_k + \delta_{k+1}\mu_{k+1} &  \mathrm{if} \, 3\le k \le n-2    \\
        \frac{1}{n} - \delta_{n-2}H_1(\psi_{n-1},\psi_{n-2}) -\delta_{n-1}H_2(\psi_n,\psi_{n-1})+\delta_{n-3}\mu_{n-2}-(\delta_{n-1}+\delta_{n-2})\mu_{n-1} & \mathrm{if} \, k = n-1\\
        \frac{1}{n} - \delta_{n-1}H_1(\psi_n,\psi_{n-1}) + \delta_{n-2}\mu_{n-1} & \mathrm{if} k = n
    \end{cases}
\end{equation*}

\end{document}

Edit: Here is the document's settings and the packages I use:

\documentclass{article}

%packages
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}

\title{To be determined later}
\author{Aref Sadeghi}
\date{December 2024}

\begin{document}

Here is a picture of what the equation looks like now:

Obviously 3 is the page number so the equation is eating into the right margin

Answer 1

You can nest aligned. Since material inside aligned is in display style, I use dcases from mathtools, but the 1/n common parts should be \tfrac in each equation.

With the help of \addlinespace from booktabs we can uniformly add space to separate off the various lines so as not to confuse readers in understanding what each line belongs to.

I replaced \mathrm{if}\, with a more proper `\text{if }.

\documentclass{article}

%packages
\usepackage{amsmath,mathtools}
\usepackage{amssymb}
\usepackage{booktabs}

\begin{document}

\begin{equation*}
  \frac{\partial \mathcal{L}}{\partial \psi_k} = 
  \begin{dcases}
    \tfrac{1}{n} - \delta_1 H_2(\psi_2,\psi_1) +\mu_2\delta_2
    & \text{if } k = 1
    \\ \addlinespace[1.5ex]
    \begin{aligned}[t]
      \tfrac{1}{n}
      &- \delta_1 H_1(\psi_2,\psi_1) -\delta_2 H_2(\psi_3,\psi_2)
      \\
      &- (\delta_1+\delta_2)\mu_2 + \delta_3\mu_3
    \end{aligned}
    &  \text{if } k = 2
    \\ \addlinespace[1.5ex]
    \begin{aligned}[t]
      \tfrac{1}{n}
      &- \delta_{k-1}H_1(\psi_k,\psi_{k-1}) -\delta_k H_2(\psi_{k+1},\psi_k)
      \\
      &+ \delta_{k-2}\mu_{k-1}- (\delta_k+\delta_{k-1})\mu_k + \delta_{k+1}\mu_{k+1}
    \end{aligned}
    & \text{if } 3\le k \le n-2 
    \\ \addlinespace[1.5ex]
    \begin{aligned}[t]
      \tfrac{1}{n}
      &- \delta_{n-2}H_1(\psi_{n-1},\psi_{n-2}) -\delta_{n-1}H_2(\psi_n,\psi_{n-1})
      \\
      &+ \delta_{n-3}\mu_{n-2}-(\delta_{n-1}+\delta_{n-2})\mu_{n-1}
    \end{aligned}
    & \text{if } k = n-1
    \\ \addlinespace[1.5ex]
    \tfrac{1}{n} - \delta_{n-1}H_1(\psi_n,\psi_{n-1}) + \delta_{n-2}\mu_{n-1}
    & \text{if } k = n
  \end{dcases}
\end{equation*}

\end{document}

If you remove the [t] option to aligned you get

Take your pick.

Answer 2

I would create a hanging indentation for broken lines. And, to cut down on visual clutter, I'd move the repeated \frac{1}{n}- part out of the cases environment.

\documentclass{article}
\usepackage{amssymb,amsmath}

\begin{document}

\begin{equation*}
\frac{\partial \mathcal{L}}{\partial \psi_k} 
= \frac{1}{n} - {}
\begin{cases}
    \delta_1 H_2(\psi_2,\psi_1) +\mu_2\delta_2 
        & \text{if $k = 1$} \\[\jot]
    \delta_1 H_1(\psi_2,\psi_1) -\delta_2 H_2(\psi_3,\psi_2)  
        & \text{if $k = 2$} \\
    \quad{}  - (\delta_1+\delta_2)\mu_2 + \delta_3\mu_3 \\[\jot]
    \delta_{k-1} H_1(\psi_k,\psi_{k-1}) -\delta_k H_2(\psi_{k+1},\psi_k)  
        & \text{if $3\le k \le n-2$} \\
    \quad{}+ \delta_{k-2}\mu_{k-1}- (\delta_k+\delta_{k-1})\mu_k \\
    \quad{}+ \delta_{k+1}\mu_{k+1}\\[\jot]
    \delta_{n-2}H_1(\psi_{n-1},\psi_{n-2}) -\delta_{n-1}H_2(\psi_n,\psi_{n-1}) 
        & \text{if $k = n-1$} \\
    \quad{}+\delta_{n-3}\mu_{n-2}-(\delta_{n-1}+\delta_{n-2})\mu_{n-1} \\[\jot]
    \delta_{n-1}H_1(\psi_n,\psi_{n-1}) + \delta_{n-2}\mu_{n-1} 
        & \text{if $k = n$}
    \end{cases}
\end{equation*}

\end{document}

Answer 3

This is an idea which makes use of just adding new cases. Alternatively, you may use, perhaps, an aligned environment, or an array.

\documentclass{article}
\usepackage{amsmath,showframe}
\begin{document}
\begin{equation*}
    \dfrac{\partial \mathcal{L}}{\partial \psi_k} = 
    \begin{cases}
        % 1
        \frac{1}{n} - \delta_1 H_2(\psi_2,\psi_1) +\mu_2\delta_2 & \mathrm{if} \, k = 1    \\
        % 2
        \frac{1}{n} - \delta_1 H_1(\psi_2,\psi_1) -\delta_2 H_2(\psi_3,\psi_2)\\
        - (\delta_1+\delta_2)\mu_2 + \delta_3\mu_3 &  \mathrm{if} \, k = 2    \\
        % 3
        \frac{1}{n} - \delta_{k-1} H_1(\psi_k,\psi_{k-1}) -\delta_k H_2(\psi_{k+1},\psi_k)\\
        + \delta_{k-2}\mu_{k-1}- (\delta_k+\delta_{k-1})\mu_k + \delta_{k+1}\mu_{k+1} &  \mathrm{if} \, 3\le k \le n-2    \\
        %4
        \frac{1}{n} - \delta_{n-2}H_1(\psi_{n-1},\psi_{n-2}) -\delta_{n-1}H_2(\psi_n,\psi_{n-1})\\
        +\delta_{n-3}\mu_{n-2}-(\delta_{n-1}+\delta_{n-2})\mu_{n-1} & \mathrm{if} \, k = n-1\\
        % 4
        \frac{1}{n} - \delta_{n-1}H_1(\psi_n,\psi_{n-1}) + \delta_{n-2}\mu_{n-1} & \mathrm{if} k = n
    \end{cases}
\end{equation*}
\end{document}