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I think about position, its time derivative speed, and its second time derivative, acceleration. I would like to identify a single word that can be used as a handle for the second time derivative of energy (i.e., the time derivative of power). If there is a widely used term, I'd prefer to use it. If not, I'd like to get your suggestions as to what a term might be. Any ideas?

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"Power-up rate"? :) –  Lagerbaer Dec 15 '11 at 22:57
    
A heuristical argument why this is not really a physically relevant quantity: the nontrivial physical solutions to second-order differential equations are essentially oscillatory, but if you study nontrivial energy changes these are typically due to thermodynamic processes involving entropy increase and therefore not reversible. –  leftaroundabout Dec 16 '11 at 2:16
    
Within the full systems studied by proper physics, the single word you're looking for is "zero". The reason is called the energy conservation law. ;-) Well, a more accurate term has several words: "a silly awkward way to write zero as a time derivative of yet another zero". :-) –  Luboš Motl Dec 16 '11 at 17:03

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Within power systems such as regional or national electricity grids, $\frac{\mathrm{d}^2E}{\mathrm{d}t^2}$ is called the slew rate: it's used to denote the rate of change of power demanded from, or supplied to, electricity grids. It's typically either expressed as MW/s or GW/h, being two time periods of interest in balancing electricity grids.

Inconveniently, you might find that (in some contexts) slew rate is also used to refer to rate of change of voltage, or of current, with respect to time (in units of V/s or A/s respectively).

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The quantity itself, $\frac{\mathrm{d}^2E}{\mathrm{d}t^2}$, is not widely used (as a matter of fact I can't think of any equation in which it appears off the top of my head), so correspondingly there is no widely used term for it. You can just say "second time derivative of energy" or "rate of change of power" or some such thing and it will get the point across.

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Thanks David, this explains why searching Google turned up nothing. If you are curious, we are considering the second time derivative of energy as it applies to the rate of change of metabolic rate incurred by different activities. –  Sipp Dec 16 '11 at 14:21

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