In Landau-Lifshitz's "Course of Theoretical Physics - Mechanics" It is told that a lagrangian is a function $\mathcal{L}$ such that the action $S$, defined by: $$S=\int_{t_0}^{t_1}\mathcal{L}(q(t),\dot{q}(t),t) \mathrm{d}t$$ Takes a minimum value. But, up to where i read, the only kind of lagrangian is the closed-system lagrangian defined by $$\mathcal{L}=E_k-E_p$$ Where $E_k$ and $E_p$ are, respectively, the kinetic and potential energy of the particle on a given system.
Is there any different Lagrangian type? if there isn't, does this means that the closed-system lagrangian is the most general formulation for lagrangian mechanics?
