We know that kinetic energy is mathematically represented as $$E_k = \frac{1}{2} m v^2$$

Similarly, potential energy is defined as $$E_p = mg\Delta x$$

Considering these mechanical quantities in a system, is it possible to define them as multivariable functions such as

$$E(m, v)\ or \ E(m, \Delta x)$$

I know why this may be invalid, it is because they are simply equations defined physically. But, if we take either of the dependent variables as a non-constant, then I think that the multivariable definition is valid.

  • $\begingroup$ Do you have a way of varying mass at the same time that you vary velocity, for your object of interest? My first idea that would give a "yes" answer involves rockets in flight. $\endgroup$ – David White Aug 9 '18 at 16:36
  • $\begingroup$ @DavidWhite yeah, that's sort of where I got the idea. Rockets expel mass and have changing velocities, so a function of mass and velocity sounds appropriate. $\endgroup$ – PartialDifferentials Aug 9 '18 at 16:42
  • $\begingroup$ Yes, they are by definition multivariable functions. Why do you ask? $\endgroup$ – Qmechanic Aug 9 '18 at 17:24
  • $\begingroup$ Translational KE: $$K(\mathbf{p})=\frac{p^2}{2m}=\frac{p_x^2+p_y^2+p_z^2}{2m}$$ Hence, $$\mathbf{v}=\frac{\partial K}{\partial \mathbf{p}}$$ $\endgroup$ – Ng Chung Tak Aug 9 '18 at 17:26

You are right and the approach is feasible. The kinetic energy concept, where mass varies with time, is used for example in jet propulsion theory.

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