align: using aligned within align troubles

Question

I would like the first two pieces in align look as though they are the only two items in the align environment and the rest to align at the equals signs and separated by some space for the different equality.

Right now it compiles looking like:

So the first two I would like to align as such:

and the last equalities I would like to align as

\documentclass{article}
\usepackage{mathtools}
\usepackage{amssymb}
\begin{document}
\begin{alignat*}{2}
    \vec{x}_1(t) &=
    \begin{pmatrix}
      x_1^{(1)}(t)\\
      x_2^{(1)}(t)
    \end{pmatrix} &&{}=
    \begin{pmatrix}
      X_1^{(1)}\cos(\omega_1t + \phi_1)\\
      r_1X_1^{(1)}\cos(\omega_1t + \phi_1)
    \end{pmatrix}\\
    \vec{x}_2(t) &=
    \begin{pmatrix}
      x_1^{(2)}(t)\\
      x_2^{(2)}(t)
    \end{pmatrix} &&{}=
    \begin{pmatrix}
      X_1^{(2)}\cos(\omega_2t + \phi_2)\\
      r_2X_1^{(2)}\cos(\omega_2t + \phi_2)
    \end{pmatrix}\\
    r_1 &=
    \begin{aligned}
      \frac{3k - m\omega_1^2}{2k} && \qquad
      X_1^{(1)} &&{}= \frac{1}{r_2 - r_1}\bigg[(r_2x_1(0) - x_2(0))^2 +
      \frac{(-r_2\dot{x}_1(0) + \dot{x}_2(0))^2}{\omega_1^2}\bigg]^{1/2}
    \end{aligned}\\
    r_2 &=
    \begin{aligned}
      \frac{3k - m\omega_2^2}{2k} && \qquad
      X_1^{(2)} &&{}= \frac{1}{r_2 - r_1}\bigg[(-r_1x_1(0) + x_2(0))^2 +
      \frac{(r_1\dot{x}_1(0) - \dot{x}_2(0))^2}{\omega_2^2}\bigg]^{1/2}
    \end{aligned}\\
    \phi_1 &=
    \begin{aligned}
      \arctan\bigg[\frac{-r_2\dot{x}_1(0) + \dot{x}_2(0)}
      {\omega_1^2(r_2x_1(0) - x_2(0))}\bigg] && \qquad
      \phi_2 &&{}= \arctan\bigg[\frac{r_1\dot{x}_1(0) - \dot{x}_2(0)}
      {\omega_2^2(-r_1x_1(0) + x_2(0))}\bigg]
    \end{aligned}
  \end{alignat*}
\end{document}

Answer 1

\documentclass{article}
\usepackage{mathtools}
\usepackage{amssymb}
\begin{document}
\begin{alignat*}{2}
    \vec{x}_1(t) &=
    \begin{pmatrix}
      x_1^{(1)}(t)\\
      x_2^{(1)}(t)
    \end{pmatrix} =
    \mathrlap{\begin{pmatrix}
      X_1^{(1)}\cos(\omega_1t + \phi_1)\\
      r_1X_1^{(1)}\cos(\omega_1t + \phi_1)
    \end{pmatrix}}\\
    \vec{x}_2(t) &=
    \begin{pmatrix}
      x_1^{(2)}(t)\\
      x_2^{(2)}(t)
    \end{pmatrix} =
    \mathrlap{\begin{pmatrix}
      X_1^{(2)}\cos(\omega_2t + \phi_2)\\
      r_2X_1^{(2)}\cos(\omega_2t + \phi_2)
    \end{pmatrix}}\\
    r_1 &=
      \frac{3k - m\omega_1^2}{2k} &
      X_1^{(1)} &= \frac{1}{r_2 - r_1}\bigg[(r_2x_1(0) - x_2(0))^2 +
      \frac{(-r_2\dot{x}_1(0) + \dot{x}_2(0))^2}{\omega_1^2}\bigg]^{1/2}
\\
    r_2 &=
      \frac{3k - m\omega_2^2}{2k} &
      X_1^{(2)} &= \frac{1}{r_2 - r_1}\bigg[(-r_1x_1(0) + x_2(0))^2 +
      \frac{(r_1\dot{x}_1(0) - \dot{x}_2(0))^2}{\omega_2^2}\bigg]^{1/2}
\\
    \phi_1 &=
      \arctan\bigg[\frac{-r_2\dot{x}_1(0) + \dot{x}_2(0)}
      {\omega_1^2(r_2x_1(0) - x_2(0))}\bigg] &
\phi_2 &= \arctan\bigg[\frac{r_1\dot{x}_1(0) - \dot{x}_2(0)}
      {\omega_2^2(-r_1x_1(0) + x_2(0))}\bigg]
  \end{alignat*}
\end{document}

Answer 2

I don't understand what's supposed to be achieved by the proposed layout and the attendant alignment of various = symbols. Rather than create the impression that various equations are related in ways that are probably not intended, I'd simplify the layout and get by with a single align* environment and use \qquad statements to insert some horizontal whitespace where needed. Observe that it's important to use \biggl[ instead of just \bigg[ following the two \arctan statements; if you omit the l (mathopen) specifiers, LaTeX will insert an inappropriate whitespace.

\documentclass{article}
\usepackage{amsmath} % for "align*" and "pmatrix" environments
\usepackage{newpxtext}             % Palatino text font
\usepackage[euler-digits]{eulervm} % Euler math font
\begin{document}
\begin{align*}
    \vec{x}_1(t) &=
    \begin{pmatrix}
      x_1^{(1)}(t)\\
      x_2^{(1)}(t)
    \end{pmatrix} =
    \begin{pmatrix}
      X_1^{(1)}\cos(\omega_1t + \phi_1)\\
      r_1X_1^{(1)}\cos(\omega_1t + \phi_1)
    \end{pmatrix}\\
    \vec{x}_2(t) &=
    \begin{pmatrix}
      x_1^{(2)}(t)\\
      x_2^{(2)}(t)
    \end{pmatrix} =
    \begin{pmatrix}
      X_1^{(2)}\cos(\omega_2t + \phi_2)\\
      r_2X_1^{(2)}\cos(\omega_2t + \phi_2)
    \end{pmatrix}\\[2ex]  % additional vertical space
    r_1 &=
      \frac{3k - m\omega_1^2}{2k} \qquad
      X_1^{(1)} = \frac{1}{r_2 - r_1}\biggl[(r_2x_1(0) - x_2(0))^2 +
      \frac{(-r_2\dot{x}_1(0) + \dot{x}_2(0))^2}{\omega_1^2}\biggr]^{1/2}\\
    r_2 &=
      \frac{3k - m\omega_2^2}{2k}  \qquad
      X_1^{(2)} = \frac{1}{r_2 - r_1}\biggl[(-r_1x_1(0) + x_2(0))^2 +
      \frac{(r_1\dot{x}_1(0) - \dot{x}_2(0))^2}{\omega_2^2}\biggr]^{1/2}\\[2ex] % additional vertical space
    \phi_1 &=
      \arctan\biggl[\frac{-r_2\dot{x}_1(0) + \dot{x}_2(0)}
      {\omega_1^2(r_2x_1(0) - x_2(0))}\biggr] \qquad
      \phi_2 = \arctan\biggl[\frac{r_1\dot{x}_1(0) - \dot{x}_2(0)}
      {\omega_2^2(-r_1x_1(0) + x_2(0))}\biggr]
  \end{align*}
\end{document}