Electronic – Capacitor loop with no resistance

The question appears to be "How do DC Capacitors get UL recognized?"

Or more generally "How does a product get UL Recognized or Listed?"

If the product fall under existing standards (UL and non UL standards) then the manufacturer applies to UL.

If the product does not fit the above standards or a change to the standard is required then the manufacture contacts the relevant standards committee representative (this could be UL ). For example ISO (http://www.iso.ch) or ANSI (http://www.ansi.org). (The committees are composed of “interested parties” like UL, consumer groups, manufacturer’s, insurance representatives etc. )

Another route is to participate as a member of the UL Standards Technical Panel, I believe it takes substantial time and effort. But if you want to participate try here ... http://www.ul.com/global/eng/pages/solutions/standards/developstandards/participation/callformembers/index.jsp

The OP wanted specifically the history of how how the standards were developed for the capacitors. The work of the committees / panels is recorded in minutes / bulitins however I don't know how to get hold of these.

Is amplitude of the waves modulated by the amount of current?

The amplitude of the current signal is half of the difference between the maximum and minimum of the current in each cycle of the oscillation.

For example, if the current over time is described by the equation

$$i(t) = A \sin(\omega{}t+\phi)$$

then A is the amplitude of signal because the current oscillates between +A and -A (and A has units of amps).

In the circuit shown, there is nothing modulating the current. Modulation happens when some other signal (like an audio waveform) changes the amplitude (for example) of the current waveform. This would require a more complex circuit than what is shown in your example.

Frequency is modulated by the frequency of the capacitor release of energy, correct?

Again, your circuit shows no mechanism for modulating the frequency. If you used a variable capacitor and the capacitance was controlled by another signal, that would cause frequency modulation. But it would probably also cause some undesirable parasitic amplitude modulation. To see some typical frequency modulator circuits, you can do a google image search for "frequency modulator" and look at the schematics that are found.

That would mean that using a capacitor with a lower farad rating should produce a higher frequency?

Generally the resonance of an LC tank circuit like in your example is given by

$$\omega_0=\frac{1}{\sqrt{LC}}$$

So the oscillation frequency can be increased by reducing either the capacitance or inductance value.

How would one go about keeping a sustained current to the inductor since it's constantly losing current without messing up the cycle?

You can use an oscillator circuit. Generally this means adding some kind of amplifier to the circuit to add energy to the waveform as quickly as it is lost to parasitic resistance and to feeding the load.