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Suppose force on a particle depends on both position of the particle and time.

In this case, how to find potential energy of the particle?

To be specific, force is $$ F(r,t) = (k/r^2) \exp(-\alpha t);$$ where $k$ and $\alpha$ are constants. $r$ is position of the particle from force centre and $t$ is time.

I have read that - Potential energy can not be defined for non-conservative forces. Force which depends on time is non-conservative force. Is this correct ?

BioPhysicist
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1 Answers1

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  1. A velocity-dependent generalized potential $U=U({\bf r},{\bf v},t)$ satisfies by definition $$ {\bf F}~=~\frac{d}{dt} \frac{\partial U}{\partial {\bf v}} - \frac{\partial U}{\partial {\bf r}}.\tag{1} $$

  2. If there is no velocity-dependence but possible explicit time dependence, this simplifies to the well-known gradient form $$ {\bf F}~=~ - \frac{\partial U}{\partial {\bf r}},\tag{2} $$ which can be readily applied to OP's example, essentially because $t$ and ${\bf r}$ don't talk to each other in eq. (2).

  3. Whether the definition of a conservative force allows for explicitly time-dependence is ultimately a matter of convention. On one hand, the notion of potential energy still exists in the presence of explicit time dependence. Note in particular that the definition could involve virtual work along virtual paths where time $t$ is fixed/frozen. On the other hand, no sane person would probably call a physical system with explicit time dependence for 'conservative'. See also this related Phys.SE post.

Qmechanic
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