Is the energy actually lost, though? E.g. imagine an ideal filter made of just capacitors and inductors - the energy cannot be dissipated... it kind of has to be present in the output, even if it doesn't look the same as it did going in... ?
Edit: If the filter is an RC filter, I can see that energy can be dissipated in the resistive part of the filter. That dissipation must be frequency dependent (or the filter wouldn't work as a filter). I am beginning to see how the shape of the response to a single pulse can tell you what the filter has to look like. In the case of the ideal LC filter above, the response would presumably be a sine wave at the resonant frequency that goes on forever... with an amplitude proportional to the energy in the original pulse. Wish I was better at math, all this is probably obvious if you understand the underlying equations...
That's correct, with the mention that voltage is not the same thing as energy. Also, the area of a voltage in time (Volt*second) is not the same thing as energy (Joule).
But, yes, energy is never lost, can not be destroyed, nor created. That is what the Energy Conservation Law says. Energy can only be moved from one place to another, or transformed from one form to another.
Speaking about energy in an LC circuit, indeed, if you somehow manage to put a "blob" of energy into an ideal LC tank, that blob of energy will start to "slosh", back and forth, between the capacitor and the inductor, forever, as you said.
That sloshing forever comes with another mention. Even if the coil and the capacitor and the connections between them have no resistance (are all ideal, so no rezistive losses), we will still have some losses because the circuit will radiate energy as radio waves. So, to keep the energy in the LC tank forever, we will need to somehow isolate the LC tank from the surrounding space, in order to prevent energy losses in the form of radio waves photons.
Such an isolation will be hard to imagine for a wire as a coil and a pair of plates as a capacitor, let's say we will just use an ideal resonant cavity ended in perfect mirrors that will reflect back all the radio waves photons, so the oscillations will last forever.
But even so, with all those ideal parts and materials, our Universe continuously expands with time. It is said that space itself expands, therefore the mirrors of our LC tank will keep going further and further apart. The further away an object is, the more speed that object will gain because of space expansion, and at some point the speed of the distant object will become faster than light.
To our resonant cavity ended in mirrors, that will mean that at some moment the photons will bounce on one mirror, but in their way to the other mirror, they will never reach the other mirror, because the other mirror is speeding away faster than light. So, our lump of energy will reflect for the last time, and that energy will never come back. The oscillations will end.
Whatever we do, it seems like it won't oscillate forever and ever, not even in the ideal case of no energy dissipation.
Nothing last forever.