I'm trying to understand work and potential energy, specifically using the example of a human climbing against gravity to a high point, then falling under the force of gravity.
My understanding is that energy ($E$) is transferred to a system $S_1$ by another system $S_2$ doing positive work ($+\Delta W$) on system $S_1$. Potential energy $U_1$ of $S_1$ increases $+\Delta U_1$ as system $S_2$ does work $+\Delta W_2$ on $S_1$, while $S_1$ does negative work $-\Delta W_1$ on itself which results in $-\Delta U_2$. Is this understanding correct?
I then assign the variables: $S_1 =$ Body, $S_2 =$ Muscles/Metabolism. For the climb, $\Delta U_1 = zJ, \Delta U_2 = -zJ$. For the fall opposite, $\Delta U_1 = -zJ, \Delta U_2 = zJ$.
Does this appear correct?