# What is the potential of the field?

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.

-
 The Lagrange equation is wrong, the $V$-identification has a bug, the "In the dynamics of..." sentence is dubious and if I interpret it correctly, I think it's a tautology. Regarding the question, in contrast to the kinetic term $(\partial\phi)^2$, the term $V(\phi)$ is certainly a functional of the value of the field (not it's change in space or time). It's instructive to consider the Hamiltonian instead - you then see that different potentials mean different possible contributions of the slopes of the field (kinetic energy). Values and slopes of $x(t)^n$ for $x'>0 , t=t_0$ grow with $n$. – Nick Kidman May 30 '12 at 18:55