Sstrhsrtj

This is my code:

The Lagrangian eqref{eq:14} differentiated according to eqref{eq:15} gives the equation of motion in terms of linear acceleration $ddot{x}$:

begin{align} label{eq:17}

frac{d}{dt}left(frac{partial L}{partial dot{x}}right) =frac{partial L}{partial x} Rightarrow qquad qquad frac{d}{dt}(mdot{x})=m(l+x)^2dot{theta}^2-kx+mgcostheta nonumber \ 

mddot{x}=m(l+x)^2dot{theta}^2-kx+mgcostheta nonumber \

ddot{x} = (l+x)^2dot{theta}^2-frac{k}{m}x+gcostheta.

end{align} 

The same process is done according to eqref{eq:16} to generate the equation of motion with respect to $ddot{theta}$:

begin{align} label{eq:18}

frac{d}{dt}left(frac{partial L}{partial dot{theta}}right) =frac{partial L}{partial theta} Rightarrow qquad qquad frac{d}{dt}[m(l+x)^2dot{theta}] = -mg(l+x)sintheta nonumber \

m(l+x)^2ddot{theta}+2m(l+x)dot{x}dot{theta} = -mg(l+x)sintheta nonumber \

(l+x)ddot{theta}+2dot{x}dot{theta} = -gsintheta nonumber \

ddot{theta}=frac{-gsintheta-2dot{x}dot{theta}}{l+x}

end{align}

which gives this output:

I wish for the systems of equations on the right to be aligned (not necessarily with the other system, but within its own) and to not vertically overlap with the equation on the left (as the second system currently does). Would someone please show me how to do this? Apologies if unclear.

Welcome to TeX.SE! You already dialed the right environment for that, align, but are not really using it. To use it, you need to set alignment points with &. So to first approximation I'd like to suggest.

documentclass{article}

usepackage{mathtools}

begin{document}

The Lagrangian eqref{eq:14} differentiated according to eqref{eq:15} gives the equation of motion in terms of linear acceleration $ddot{x}$:

begin{align} label{eq:17}

frac{d}{dt}left(frac{partial L}{partial dot{x}}right)& =frac{partial

L}{partial x} &Rightarrow qquad qquad frac{d}{dt}(mdot{x})&=m(l+x)^2dot{theta}^2-kx+mgcostheta nonumber \ 

&& mddot{x}&=m(l+x)^2dot{theta}^2-kx+mgcostheta nonumber \

&& ddot{x} &= (l+x)^2dot{theta}^2-frac{k}{m}x+gcostheta.

end{align} 

The same process is done according to eqref{eq:16} to generate the equation of motion with respect to $ddot{theta}$:

begin{align} label{eq:18}

frac{d}{dt}left(frac{partial L}{partial dot{theta}}right)&

=frac{partial L}{partial theta} &Rightarrow qquad qquad

frac{d}{dt}[m(l+x)^2dot{theta}] &= -mg(l+x)sintheta nonumber \

&&m(l+x)^2ddot{theta}+2m(l+x)dot{x}dot{theta}& = -mg(l+x)sintheta nonumber \

&& (l+x)ddot{theta}+2dot{x}dot{theta} &= -gsintheta nonumber \

&& ddot{theta}&=frac{-gsintheta-2dot{x}dot{theta}}{l+x}

end{align}

end{document}