[...] the infinitesimal work $\overline{\mathrm dw}$ comes out as a linear differential form of the variables $q_i$: $$\overline{\mathrm dw}= F_1~\mathrm dq_1 +F_2~\mathrm dq_2+ \ldots + F_n~\mathrm dq_n\,.\tag{17.3}$$ This differential form is of prime-importance for the analytical treatment of force. .... the coefficients $$F_1, ~ F_2, \ldots, ~F_n \tag{17.4}$$ are the components of a vector, but it is a vector of $n$-dimensional configuration space and not one of ordinary space. We call this vector the "generalised force" and the quantities $f_i$ the "components of generalised force".
[...] Let us assume that the infinitesimal work $\overline{\mathrm dw}$ is the true differential of a certain function so that the dash above $\mathrm dw$ may be omitted. The function is called the "work function" and is frequently denoted by the letter $U\,.$ Hence, we put $$\overline{\mathrm dw}= \mathrm dU,\tag{17.5}$$ where $$U:= U(q_1,q_2, \ldots, q_n)\,.\tag{17.6}$$ We thus have $$\sum F_i~\mathrm dq_i = \sum \frac{\partial U}{\partial q_i}~\mathrm dq_i\tag{17.7}$$ which gives $$F_i = \frac{\partial U}{\partial q_i}\,.\tag{17.8}$$ The present practice is to use the negative work function and call this quantity $V:$ $$V= -~U\,.\tag{17.9}$$ ... The equation $(17.8)$ can be now re-written in the form $$F_i = -~ \frac{\partial V }{\partial q_i}\,, \tag{17.10}$$ where $V$ is a function of the position coordinates: $$V:= V(q_1,q_2,\ldots,q_n)\tag{17.11}$$
[...] The definition of work function on the basis of the equation $$U:= U(q_1,q_2,\ldots,q_n)\tag{17.6}$$ 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, with the understanding that in forming the differential $\mathrm dU$ the time $t$ is considered as a constant. The equations $(17.7)$ and $(17.8)$ remain valid, but the conservation of energy is lost.
This is excerpted from The Variational principles of Mechanics' Work function and generalised force by Cornelius Lanczos; the author asserts that for a $U$ which explicitly depends on time $t,$ the law of conservation of energy doesn't apply it is "lost".
Why is it so? How did he conclude that?
Also, he said, even though $U$ depends explicitly on $t,$ the equations $(17.7)$ and $(17.8)$ are still true.
Can I say, $(17.9)$ i.e. $V= -~U$ is also true when $U$ depends explicitly on $t$?
Could anyone thus explain me why the conservation of energy doesn't hold true for $U:= U(q_1,q_2,\ldots, q_n; ~t)$ and also whether in this case, $V= -~U$ is still true?
