documentclass{article}
usepackage{amsmath}

begin{document}


begin{alignat}{2}
&dot{x}_i^1 = x_i^2 &&nonumber\
&dot{x}_i^2 = x_i^3&&left(k_i^2 + frac{k_i^3}{left(1 +
left(x_i^3right)^2right)^{frac{1}{2}}}left(x_i^2 - frac{k_i^1}{left(1
+ left(x_i^3right)^2right)^{frac{1}{2}}}x_i^4right)^2 + frac{k_i^1}
{left(1 + x_3^2right)^{frac{3}{2}}}right.nonumber\
&&left.left (x_i^4right)^2vphantom{left(x_i^2 - frac{k_i^1}{left(1 +
left(x_i^3right)^2right)^{frac{1}{2}}}x_i^4right)^2}right)nonumber\
&dot{x}_i^3 = x_i^4 &&nonumber\
&dot{x}_i^4 = v_i &&
end{alignat}

end{document}