Lets look at this example

[![enter image description here][1]][1]

The particle energy in S frame is

$$E_s=\frac 12 m \mathbf v_s\cdot \mathbf v_s+m\,g\,y=\text{const.}$$

where $~\mathbf v_s=\frac{d}{dt}\mathbf R~$

The particle energy in S' frame is

$$E_s'=\frac 12 m \mathbf v_s'\cdot \mathbf v_s'+m\,g\,y'$$

where 

$$\mathbf v_s'=\frac{d}{dt}\begin{bmatrix}
  R'_x+\Delta x \\
   R'_y\\
\end{bmatrix}$$
thus, the particle energy $~E_s'~$ is conserved (constant) only if
$~\Delta x=v_x\,t~$ where $~v_x~$ constant, in this case is also the energy differenz (relative energy)  conserved.

is the energy invariant ? invariant mean that $~E_s=E_s'~$ this is not the case . 



 


  [1]: https://i.sstatic.net/JwTtF.png