Split equations inside the align environment

You can use split inside align:

\documentclass[11pt]{book}
\usepackage{amsmath,amsfonts,amssymb,amsthm, bm}

\begin{document}

\begin{align}
\begin{split}
  \dot{v}     & = \dfrac{1}{m}[T\cos\beta\cos(\alpha+\epsilon)-D]-g\sin\gamma \\
              &\qquad +\dot{w}_{x} \cos\gamma\cos\chi+\dot{w}_{y}\cos\gamma\sin\chi
\end{split}
\\[2ex]
\begin{split}
\dot{\gamma}  & = \dfrac{1}{m}[(Y-T\sin\beta\cos(\alpha+\epsilon))\sin\mu \\
              &\qquad + (T\sin(\alpha +\epsilon)+L)\cos\mu] - \dfrac{g}{v}\cos\gamma \\
              &\qquad +\dfrac{1}{v}[\dot{w}_{x}\sin\gamma\cos\chi + \dot{w}_{y}\sin\gamma\sin\chi]
\end{split}
\\[2ex]
\begin{split}
\dot{\chi}    & = \dfrac{1}{mv\cos\gamma}[(L+T\sin(\alpha+\epsilon) )\sin\mu \\
              &\qquad + (T\sin\beta \cos(\alpha  + \epsilon) - Y)\cos\mu] \\
              &\qquad + \dfrac{1}{v\cos\gamma}[\dot{w}_{x}\sin\chi-\dot{w}_{y}\cos\chi]
\end{split}
\end{align}

\end{document} 

Choose manually some suitable points at which the equations can break and add the \notag or \nonumber command to remove the equation label at that line.

\documentclass[11pt]{book}
\usepackage{amsmath,amsfonts,amssymb,amsthm, bm}

\begin{document}

\begin{align}
\dot{v}        ={} &\dfrac{1}{m}[T\cos\beta\cos(\alpha+\epsilon)-D]-g\sin\gamma +\dot{w}_{x} \cos\gamma\cos\chi+\dot{w}_{y}\cos\gamma\sin\chi \\[2pt]
\dot{\gamma}   ={} &\dfrac{1}{m}[(Y-T\sin\beta\cos(\alpha+\epsilon))\sin\mu + (T\sin(\alpha +\epsilon)+L)\cos\mu] - \dfrac{g}{v}\cos\gamma \notag\\
                   &{}+\dfrac{1}{v}[\dot{w}_{x}\sin\gamma\cos\chi+ \dot{w}_{y}\sin\gamma\sin\chi] \\[2pt]
\dot{\chi}     ={} &\dfrac{1}{mv\cos\gamma}[(L+T\sin(\alpha+\epsilon) )\sin\mu + (T\sin\beta \cos(\alpha  + \epsilon) - Y)\cos\mu] \notag\\ 
                   &{}+\dfrac{1}{v\cos\gamma}[\dot{w}_{x}\sin\chi-\dot{w}_{y}\cos\chi] 
\end{align}

\end{document}

You can keep using the align environment. At suitable locations, insert \notag \\ &\qquad directives to (a) introduce line breaks and (b) provide an alignment anchor for the remaining stub.

\documentclass[11pt]{book}
\usepackage{amsmath}

\begin{document}
\begin{align}
\dot{v}       
&= \frac{1}{m}\bigl[T\cos\beta\cos(\alpha+\epsilon)-D\bigr]-g\sin\gamma \notag\\
&\qquad +\dot{w}_x \cos\gamma\cos\chi+\dot{w}_y\cos\gamma\sin\chi \\[2pt]
\dot{\gamma} 
&= \frac{1}{m}\bigl[(Y-T\sin\beta\cos(\alpha+\epsilon))\sin\mu + (T\sin(\alpha +\epsilon)+L)\cos\mu\bigr]\notag\\
&\qquad -\frac{g}{v}\cos\gamma +\frac{1}{v}(\dot{w}_x\sin\gamma\cos\chi + \dot{w}_y\sin\gamma\sin\chi) \\[2pt]
\dot{\chi}    
&= \frac{1}{mv\cos\gamma}\bigl[(L+T\sin(\alpha+\epsilon) )\sin\mu + (T\sin\beta \cos(\alpha  + \epsilon) - Y)\cos\mu\bigr] \notag\\
&\qquad +\frac{1}{v\cos\gamma}(\dot{w}_x\sin\chi-\dot{w}_y\cos\chi)
\end{align}

\end{document}