Work function and potential energy in a Lagrangian

Question

I would like to know the difference between work function and potential energy . Below attached are 3 questions:

1. In question 1 the potential energy is used to calculate the Lagrangian whereas in question 5 work function is used. Why is it so?
2. Again in question 5 a constant C is added to work function W but NO such constant is added to work function W of question 7. I would like to know the reason for the same.

Answer 1

For holonomic systems the d'Alembert-Lagrange equation is written in the form

$$ \frac{d}{dt}\frac{\partial T}{\partial \dot q_k}-\frac{\partial T}{\partial q_k} =Q_k,\ \ \ \{k,1,\cdots,n\} $$

in which $T$ is the kinetic energy and $q_k$ are the configuration variables. When $Q_k$ forces have a potential $U(q)$ and $Q_k = \frac{\partial U}{\partial q_k}$ then the d'Alembert-Lagrange equations can be written as

$$ \frac{d}{dt}\frac{\partial (T-U)}{\partial \dot q_k}-\frac{\partial (T-U)}{\partial q_k} =0,\ \ \ \{k,1,\cdots,n\} $$

or calling $L = T - U$ we have then

$$ \frac{d}{dt}\frac{\partial L}{\partial \dot q_k}-\frac{\partial L}{\partial q_k} =0,\ \ \ \{k,1,\cdots,n\} $$

Note that $\frac{\partial U}{\partial \dot q_k} = 0$. This is the case for all the examples shown in the OP. Also note that $U$ and $U+C_0$ in which $C_0$ is a constant, are equivalent.