Sfdwhf

With this code

documentclass[a4paper,12pt]{article}

usepackage{mathtools, mathrsfs}

makeatletter

letsave@mathaccentmathaccent

newcommand*if@single[3]{%

  setbox0hbox{${mathaccent"0362{#1}}^H$}%

  setbox2hbox{${mathaccent"0362{kern0pt#1}}^H$}%

  ifdimht0=ht2 #3else #2fi

  }

%The bar will be moved to the right by a half of macc@kerna, which is computed by amsmath:

newcommand*rel@kern[1]{kern#1dimexprmacc@kerna}

%If there's a superscript following the bar, then no negative kern may follow the bar;

%an additional {} makes sure that the superscript is high enough in this case:

newcommand*widebar[1]{@ifnextchar^{{wide@bar{#1}{0}}}{wide@bar{#1}{1}}}

%Use a separate algorithm for single symbols:

newcommand*wide@bar[2]{if@single{#1}{wide@bar@{#1}{#2}{1}}{wide@bar@{#1}{#2}{2}}}

newcommand*wide@bar@[3]{%

  begingroup

  defmathaccent##1##2{%

%Enable nesting of accents:

    letmathaccentsave@mathaccent

%If there's more than a single symbol, use the first character instead (see below):

    if#32 letmacc@nucleusfirst@char fi

%Determine the italic correction:

    setboxz@hbox{$macc@style{macc@nucleus}_{}$}%

    setboxtw@hbox{$macc@style{macc@nucleus}{}_{}$}%

    dimen@wdtw@

    advancedimen@-wdz@

%Now dimen@ is the italic correction of the symbol.

    dividedimen@ 3

    @tempdimawdtw@

    advance@tempdima-scriptspace

%Now @tempdima is the width of the symbol.

    divide@tempdima 10

    advancedimen@-@tempdima

%Now dimen@ = (italic correction / 3) - (Breite / 10)

    ifdimdimen@>z@ dimen@0ptfi

%The bar will be shortened in the case dimen@<0 !

    rel@kern{0.6}kern-dimen@

    if#31

      overline{rel@kern{-0.6}kerndimen@macc@nucleusrel@kern{0.4}kerndimen@}%

      advancedimen@0.4dimexprmacc@kerna

%Place the combined final kern (-dimen@) if it is >0 or if a superscript follows:

      letfinal@kern#2%

      ifdimdimen@<z@ letfinal@kern1fi

      iffinal@kern1 kern-dimen@fi

    else

      overline{rel@kern{-0.6}kerndimen@#1}%

    fi

  }%

  macc@depth@ne

  letmath@bgroup@empty letmath@egroupmacc@set@skewchar

  mathsurroundz@ frozen@everymath{mathgroupmacc@grouprelax}%

  macc@set@skewcharrelax

  letmathaccentVmacc@nested@a

%The following initialises macc@kerna and calls mathaccent:

  if#31

    macc@nested@arelax111{#1}%

  else

%If the argument consists of more than one symbol, and if the first token is

%a letter, use that letter for the computations:

    defgobble@till@marker##1endmarker{}%

    futureletfirst@chargobble@till@marker#1endmarker

    ifcatnoexpandfirst@char Aelse

      deffirst@char{}%

    fi

    macc@nested@arelax111{first@char}%

  fi

  endgroup

}

makeatother

begin{document}

[begin{aligned}

mathscr F(bar{r}(t))&=& int_a^{b} L dt& =&int_a^{b} left[L frac{dt}{dtau}right]dtau=\

=int_a^{b} Bigl[-mc^2-qvarphidfrac{1}{sqrt{1-dfrac{u^{2}}{c^{2}}}}+qfrac{bar{u}cdot widebar{A}}{sqrt{1-dfrac{u^{2}}{c^{2}}}}Bigr]  dtau=& &&&\

=int_a^{b} left[-mc^2+q,boldsymbol{mathcal{U}}cdot boldsymbol{mathcal{A}}right] dtau&&&&\

end{aligned}

]

end{document}

I have this output:

and I would like to have the following alignment as the image below:

In the last few days I have done several tests but I have not succeeded. For other formulas the alignment on the left is very good. With this formula I can't do it.

Lastly, how can I improve the shape of the integral of the second line where the two green rectangles are highlighted? The square brackets do not respect the length of the integral symbol.

My mathematician's eyes bleed when I see something like

frac{<whatever}{sqrt{1-dfrac{u^2}{c^2}}

appearing more than once; I guess in your document it appears many times.

documentclass[a4paper,12pt]{article}

usepackage{mathtools, mathrsfs,bm}



makeatletter

%<...long code omitted for brevity...>

makeatother



begin{document}

[

begin{aligned}

mathscr F(bar{r}(t))

&= int_a^{b} L dt = int_a^{b} left[L frac{dt}{dtau}right],dtau=\

&=int_a^{b} [-mc^2-qvarphigamma(u)+qbar{u}cdot widebar{A}gamma(u)] ,dtau=\

&=int_a^{b} [-mc^2+q,bm{mathcal{U}}cdot bm{mathcal{A}}] ,dtau\

end{aligned}

]

where

[

gamma(u)=left(1-frac{u^2}{c^2}right)^{-1/2}

]

end{document}

There should be a single & per line.

I also fixed the usage of left and right and loaded bm that performs better than amsbsy and its boldsymbol command, replaced by bm (but boldsymbol works as well).

Here are some suggestions:

documentclass{article}



usepackage{mathtools,mathrsfs,bm,bigints}



begin{document}



[

  begin{aligned}

    mathscr{F}(bar{r}(t)) &= int_a^b L ,mathrm{d}t = int_a^b left[L dfrac{mathrm{d}t}{mathrm{d}tau} right] ,mathrm{d}tau = \

      &= bigint_a^b left[ -m c^2 - q varphi dfrac{1}{sqrt{1 - dfrac{u^2}{c^2}}} + 

        q frac{bar{u} cdot bar{A}}{sqrt{1 - dfrac{u^2}{c^2}}} right] ,mathrm{d}tau = \

      &= int_a^b bigl[ -m c^2 + q,bm{mathcal{U}} cdot bm{mathcal{A}} bigr] mathrm{d}tau

  end{aligned}

]



begin{align*}

  mathscr{F}(bar{r}(t)) &= int_a^b L ,mathrm{d}t = int_a^b left[L dfrac{mathrm{d}t}{mathrm{d}tau} right] ,mathrm{d}tau = \

    &= int_a^b bigl( -m c^2 - q varphi / sqrt{1 - u^2 / c^2} + 

      q (bar{u} cdot bar{A}) / sqrt{1 - u^2 / c^2} ,bigr) ,mathrm{d}tau = \

    &= int_a^b bigl( -m c^2 + q,bm{mathcal{U}} cdot bm{mathcal{A}} bigr) ,mathrm{d}tau

end{align*}



end{document}

The first suggestion uses an extended integral from bigints, but it places far too big an emphasis visually. Hence the second suggestion, and to use a less-intrusive fraction of the form a / b.

First you had unnecessary ampersands, and others were missing.

To have some equations left aligned, the simplest way is to use the fleqn environment from nccmath. Further, I improved the layout of the square roots, using the mfrac command (medium-sized fractions) instead of dfrac.

documentclass[a4paper,12pt]{article}

usepackage{mathtools, mathrsfs, nccmath}

usepackage[showframe]{geometry}

makeatletter

letsave@mathaccentmathaccent

newcommand*if@single[3]{%

  setbox0hbox{${mathaccent"0362{#1}}^H$}%

  setbox2hbox{${mathaccent"0362{kern0pt#1}}^H$}%

  ifdimht0=ht2 #3else #2fi

  }

%The bar will be moved to the right by a half of macc@kerna, which is computed by amsmath:

newcommand*rel@kern[1]{kern#1dimexprmacc@kerna}

%If there's a superscript following the bar, then no negative kern may follow the bar;

%an additional {} makes sure that the superscript is high enough in this case:

newcommand*widebar[1]{@ifnextchar^{{wide@bar{#1}{0}}}{wide@bar{#1}{1}}}

%Use a separate algorithm for single symbols:

newcommand*wide@bar[2]{if@single{#1}{wide@bar@{#1}{#2}{1}}{wide@bar@{#1}{#2}{2}}}

newcommand*wide@bar@[3]{%

  begingroup

  defmathaccent##1##2{%

%Enable nesting of accents:

    letmathaccentsave@mathaccent

%If there's more than a single symbol, use the first character instead (see below):

    if#32 letmacc@nucleusfirst@char fi

%Determine the italic correction:

    setboxz@hbox{$macc@style{macc@nucleus}_{}$}%

    setboxtw@hbox{$macc@style{macc@nucleus}{}_{}$}%

    dimen@wdtw@

    advancedimen@-wdz@

%Now dimen@ is the italic correction of the symbol.

    dividedimen@ 3

    @tempdimawdtw@

    advance@tempdima-scriptspace

%Now @tempdima is the width of the symbol.

    divide@tempdima 10

    advancedimen@-@tempdima

%Now dimen@ = (italic correction / 3) - (Breite / 10)

    ifdimdimen@>z@ dimen@0ptfi

%The bar will be shortened in the case dimen@<0 !

    rel@kern{0.6}kern-dimen@

    if#31

      overline{rel@kern{-0.6}kerndimen@macc@nucleusrel@kern{0.4}kerndimen@}%

      advancedimen@0.4dimexprmacc@kerna

%Place the combined final kern (-dimen@) if it is >0 or if a superscript follows:

      letfinal@kern#2%

      ifdimdimen@<z@ letfinal@kern1fi

      iffinal@kern1 kern-dimen@fi

    else

      overline{rel@kern{-0.6}kerndimen@#1}%

    fi

  }%

  macc@depth@ne

  letmath@bgroup@empty letmath@egroupmacc@set@skewchar

  mathsurroundz@ frozen@everymath{mathgroupmacc@grouprelax}%

  macc@set@skewcharrelax

  letmathaccentVmacc@nested@a

%The following initialises macc@kerna and calls mathaccent:

  if#31

    macc@nested@arelax111{#1}%

  else

%If the argument consists of more than one symbol, and if the first token is

%a letter, use that letter for the computations:

    defgobble@till@marker##1endmarker{}%

    futureletfirst@chargobble@till@marker#1endmarker

    ifcatnoexpandfirst@char Aelse

      deffirst@char{}%

    fi

    macc@nested@arelax111{first@char}%

  fi

  endgroup

}

makeatother



begin{document}



begin{fleqn}

begin{align*}

mathscr F(bar{r}(t))&= int_a^{b} L dt =int_a^{b} left[L frac{dt}{dtau}right]dtau=\

 & =int_a^{b} Bigl[-mc^2-qvarphidfrac{1}{sqrt{1-mfrac{u^{2}mathstrut}{c^{2}}}}+qfrac{bar{u}cdot widebar{A}}{sqrt{1-mfrac{u^{2}}{c^{2}}}}Bigr] dtau=& &&&\

 & =int_a^{b} left[-mc^2+q,boldsymbol{mathcal{U}}cdot boldsymbol{mathcal{A}}right] dtau\

end{align*}

end{fleqn}



end{document}