If I have the time history of the energy, Does it make sense to do the fourier transform of this energy? or if I want to see the energy in frequency the PSD is the only tool? and Why
It could makes sense to do a Fourier Transform for the kinetic energy vs time (or of any parameter vs time) for evaluation of the power spectral density when the process is wide-sense-stationary, or at least stationary over the block of time used in the FFT (which specifically is the FT of the autocorrelation function and can be estimated from the magnitude squared of the DFT, normalized by the resolution bandwidth). JasonR further explains this here. For all other cases, the PSD does not exist and you will get a very different result for any shift in time or change in duration. (This is a motivation to instead use the STFT or wavelet transform to observe both time and frequency characteristics).
To assess if the process is sufficiently stationary, I recommend using the Allan Deviation which I detail in other posts such as here for that purpose.