I System, potential energy, and nonconservative forces: The whole story

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Potential energy is defined as the energy of a system, focusing on internal and external forces. External forces can be conservative, allowing for potential energy calculations even when the source, like Earth, is outside the system. However, if the system is large enough to influence the source of the background potential, this assumption breaks down. The overall potential function of a system combines both external and internal conservative forces. Understanding these dynamics is crucial for accurately describing mechanical systems.
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[Mentors' note: This thread was split off from https://www.physicsforums.com/threads/system-potential-energy-and-nonconservative-forces.1009237/]

fog37 said:
However, I think that potential energy can only be the energy of the SYSTEM and not between the system and something else.
This is not a particularly helpful way to think about things;

For a general mechanical system, you usually split the specified forces into external specified forces ##\mathbf{F}_a## and internal specified forces ##\mathbf{G}_{ab}## (indices enumerate particles). If the external forces are conservative, ##\mathbf{F}_a = - \nabla \phi_a##, then\begin{align*}
\sum_a \int_1^2 \mathbf{F}_a \cdot \mathbf{v}_a dt = \sum_a (\phi_a(1) - \phi_a(2)) = \Phi(1) - \Phi(2)
\end{align*}and ##\Phi \equiv \displaystyle{\sum_a} \phi_a## is the external potential function. If the internal forces are conservative, that is, if ##\mathbf{G}_{ab} = g_{ab}(r_{ab}) \hat{\mathbf{r}}_{ab}## where ##\mathbf{r}_{ab} = \mathbf{r}_a - \mathbf{r}_b## and the function ##g_{ab}(r_{ab})## depends only on the inter-particle separation, then using ##\mathbf{G}_{ab} + \mathbf{G}_{ba} = \mathbf{0}## you have\begin{align*}
\sum_a \sum_{b \neq a} \int_1^2 \mathbf{G}_{ab} \cdot \mathbf{v}_a dt &=\sum_a \sum_{b>a} \int_{1}^2 (\mathbf{G}_{ab} \cdot \mathbf{v}_a + \mathbf{G}_{ba} \cdot \mathbf{v}_b)dt \\
&= \sum_a \sum_{b>a} \int_{1}^2 (\mathbf{G}_{ab} \cdot (\mathbf{v}_a - \cdot \mathbf{v}_b) )dt
\end{align*}Work on the term \begin{align*}
\mathbf{G}_{ab} \cdot (\mathbf{v}_a - \mathbf{v}_b) &= \dfrac{g_{ab}(r_{ab})}{r_{ab}} \mathbf{r}_{ab} \cdot \dfrac{d\mathbf{r}_{ab}}{dt} \\
&=\dfrac{1}{2} \dfrac{g_{ab}(r_{ab})}{r_{ab}} \dfrac{d}{dt}(r_{ab}^2) \\
&= g_{ab}(r_{ab}) \dfrac{dr_{ab}}{dt}
\end{align*}Then the integral becomes\begin{align*}
\sum_a \sum_{b>a} \int_1^2 g_{ab}(r_{ab}) \dfrac{dr_{ab}}{dt} dt &= \sum_a \int_1^2 g_{ab}(r_{ab}) dr_{ab} \\
&= \sum_a \sum_{b>a} [\psi_{ab}(r_{ab}(1)) - \psi_{ab}(r_{ab}(2)) ] \\
&= \Psi(1) - \Psi(2)
\end{align*}where the function ##\psi_{ab}## is such that ##g_{ab} = -\dfrac{d\psi_{ab}}{dr_{ab}}##, and where ##\Psi## is the internal potential function. The sum ##U = \Phi + \Psi## is then the overall potential function of the system, taking into account both the external and internal conservative specified forces.

If you have some non-conservative specified forces ##\mathbf{H}_a##, then\begin{align*}
\sum_a \int_1^2 (\mathbf{F}_a + \mathbf{H}_a) \cdot \mathbf{v}_a dt + \sum_a \sum_{b \neq a} \int_1^2 \mathbf{G}_{ab} \cdot \mathbf{v}_a dt&= \sum_a \int_1^2 m_a\ddot{\mathbf{r}}_a \cdot \dot{\mathbf{r}}_a dt \\
&= \sum_a \int_1^2 m_a \dfrac{1}{2} \dfrac{d}{dt} (v_a^2) dt \\
&= \sum_a \dfrac{1}{2}m_a(v_a(2)^2 - v_a(1)^2)
\end{align*}so that\begin{align*}-\Delta(\Phi + \Psi) + \sum_a \int_1^2 \mathbf{H}_a \cdot \mathbf{v}_a dt &= \Delta T \\
\implies \sum_a \int_1^2 \mathbf{H}_a \cdot \mathbf{v}_a dt &= \Delta (\Phi + \Psi + T) = \Delta E
\end{align*}Here ##\displaystyle{\sum_a \int_1^2 \mathbf{H}_a \cdot \mathbf{v}_a dt}## is the work done by any other non-conservative specified forces on the system.
 
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ergospherical said:
[Mentors' note: This thread was split off from https://www.physicsforums.com/threads/system-potential-energy-and-nonconservative-forces.1009237/]This is not a particularly helpful way to think about things;

For a general mechanical system, you usually split the specified forces into external specified forces ##\mathbf{F}_a## and internal specified forces ##\mathbf{G}_{ab}##

<< Rest of quote snipped by the Mentors for brevity>>
Hello ergospherical.

Thanks for this post. A little over my head. So, I guess I am incorrect to think that a force that is external (not internal to the system) cannot be conservative. For example, if the system is an object and planet Earth is not included inside the system, we can still define the potential energy between the system and planet Earth even if Earth is outside of the system...

Thank you!
 
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In a restricted sense. The assumption is that the external forces are derived from background potentials which depend only on time and not on the co-ordinates of the particles in your chosen system.

If the system is "large" enough to have an influence on the source of the background potential, then this approximation no longer holds and you can no longer consider the system & source to be independent of each other.
 
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