Potential energy term in Lagrangian formulation

Question

Why in some cases there is a term of only kinetic energy in the Lagrangian while is some cases there are both the terms showing both the kinetic energy and the potential energy in the Lagrangian? why this is so?

Answer 1

If the potential energy is the same everywhere - like for an object on a horizontal surface, then you don't include it in the lagrangian (or more formally, you set it to 0 everywhere)

Answer 2

If a generalized force has a generalized potential, one can include it in Lagrange equations $$\frac{d}{dt}\frac{\partial (T-U)}{\partial \dot{q}^j}-\frac{\partial (T-U)}{\partial q^j}~=~Q_j,\qquad j~\in \{1,\ldots, n\}, \tag{L}$$ via $Q_j$ or via $U$, but of course not in both. That would be double counting. See also this related Phys.SE post.