In field theory, the key word is the Lagrangian $L(\phi(x^{\mu}), \frac{\partial \phi (x^{\mu}) }{\partial x^\mu}) $. The equations of motion can be written as $\frac{\partial L}{\partial \phi} - \frac{\partial L}{\partial \frac{\partial \phi }{\partial x^\mu}}=0$.
Typical Lagrangian for scalar field, which satisfies the field equationsб has the form : $L = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - V(\phi)$, where $V(\phi) = - \frac{\partial L}{\partial \phi}$ - commonly referred to as the field potential (potential energy). In the dynamics of discrete systems under the potential (potential energy) has always understood the potential energy of a particle in a some potential field, nothing more. The question is what is meant by the potential of the field? Maybe there are some simple physical systems, to understand the concept.
