The differential form $\delta w$ for infinitesimal work (I switched notation because I don't want to draw that silly bar each time) is defined on the space of Lagrangian mechanics, i.e. the space spanned by the $(q,\dot{q})$. When you form $\mathrm{d}U$ as $$ \mathrm{d}U = \partial_{q^i} U \mathr...

If the force is time dependent, you do not have energy conservation: $$m \frac{\mathrm dV}{\mathrm dt}= F(t)$$ $$\mathrm d\left(\frac{1}{2}mV^2\right)=F V~\mathrm dt=F~\mathrm dx$$ For a time dependent potential: $$\mathrm dU=\mathbb{grad}(U(t))~\mathrm dx + \frac{\partial U}{\partial t}~\mat...

Seems that spinning black holes are capable of complex quantum information processes https://arxiv.org/abs/1608.06822 - Photonic Bell states creation around rotating black holes It might be possible that rotating black hole to "write" quantum algorithms?

The differential form $\delta w$ for infinitesimal work (I switched notation because I don't want to draw that silly bar each time) is defined on the space of Lagrangian mechanics, i.e. the space spanned by the $(q,\dot{q})$. When you form $\mathrm{d}U$ as $$ \mathrm{d}U = \partial_{q^i} U \mathr...

The definition of work function on the basis of $$U:= U(q_1,q_2,\ldots,q_n)\tag{17.6}$$ is too restricted. We have forces in nature which are derivable from a time-dependent work function $U(q_1,q_2,\ldots,q_n; \,t)\,.$ the equation $$\overline{\mathrm dw} = \mathrm dU\tag{17.5}$$ still holds,...