- #1
paradoxymoron
- 21
- 1
In my intro to Quantum Mechanics course, my professor gave a little aside while exploring the analogy between the Schrodinger Equation and Newton's second law: in classical physics, energy is conserved when the potential energy is not a function of time.
I wanted to try to answer this my self, and I thought I had arrived at an answer but then I got stuck again. Can someone explain why this is true? Here is my thought process (for the sake of the question, I leave the velocity as only a function of time and explore its position dependence when its relevant).
If energy is to be conserved, then its time-derivative must be zero, i.e
##
\begin{align}
E(x,t)&=\frac{1}{2}mv^2+V(x,t)\\
\frac{\partial E}{\partial t}&=mv\frac{dv}{dt}+\frac{\partial V}{\partial x}\frac{dx}{dt}+\frac{\partial V}{\partial t}\\
&=mv\frac{dv}{dt}+\frac{\partial V}{\partial x}v+\frac{\partial V}{\partial t}\\
&=v(m\frac{dv}{dt}+\frac{\partial V}{\partial x})+\frac{\partial V}{\partial t}\\
\end{align}
##If all external forces are conservative, then so is the net force, and the potential would be time-independent, so the energy rate would be zero since ##m\frac{dv}{dt}=-\frac{\partial V}{\partial x}##.
But, in the general case, if non-conservative forces exist, then would the above statement be true, or would the net force have to be split up as ##m\frac{dv}{dt}=F_{cons}+F_{non}## and then cancel?
##
\begin{align}
\frac{\partial E}{\partial t}&=v(F_{cons}+F_{non}+\frac{\partial V}{\partial x})+\frac{\partial V}{\partial t}\\
&=vF_{non}+\frac{\partial V}{\partial t}
\end{align}
##
It looks like the non-conservative force is a source (sink, I guess) of power, and it's somewhat obvious that the potential energy changing would affect the total energy. But what is the nature of the potential? Like, how is it changing? Can you give me a simple physical example?
Also, a little bonus question: What's the difference between power, and the time-derivative of the total energy?
Thanks in advance for your answer! Sorry if it was lengthy.
I wanted to try to answer this my self, and I thought I had arrived at an answer but then I got stuck again. Can someone explain why this is true? Here is my thought process (for the sake of the question, I leave the velocity as only a function of time and explore its position dependence when its relevant).
If energy is to be conserved, then its time-derivative must be zero, i.e
##
\begin{align}
E(x,t)&=\frac{1}{2}mv^2+V(x,t)\\
\frac{\partial E}{\partial t}&=mv\frac{dv}{dt}+\frac{\partial V}{\partial x}\frac{dx}{dt}+\frac{\partial V}{\partial t}\\
&=mv\frac{dv}{dt}+\frac{\partial V}{\partial x}v+\frac{\partial V}{\partial t}\\
&=v(m\frac{dv}{dt}+\frac{\partial V}{\partial x})+\frac{\partial V}{\partial t}\\
\end{align}
##If all external forces are conservative, then so is the net force, and the potential would be time-independent, so the energy rate would be zero since ##m\frac{dv}{dt}=-\frac{\partial V}{\partial x}##.
But, in the general case, if non-conservative forces exist, then would the above statement be true, or would the net force have to be split up as ##m\frac{dv}{dt}=F_{cons}+F_{non}## and then cancel?
##
\begin{align}
\frac{\partial E}{\partial t}&=v(F_{cons}+F_{non}+\frac{\partial V}{\partial x})+\frac{\partial V}{\partial t}\\
&=vF_{non}+\frac{\partial V}{\partial t}
\end{align}
##
It looks like the non-conservative force is a source (sink, I guess) of power, and it's somewhat obvious that the potential energy changing would affect the total energy. But what is the nature of the potential? Like, how is it changing? Can you give me a simple physical example?
Also, a little bonus question: What's the difference between power, and the time-derivative of the total energy?
Thanks in advance for your answer! Sorry if it was lengthy.