We consider the classical perturbation theory for the equations of motion of a field theory Lagrangian. We consider a scalar field with canonical kinetic term and a potential that contains interactions, and we describe the general formalism. In the case of a polynomial potential, we describe the formal solution and how we can self-consistently solve it in perturbation theory, considering that the potential interaction is small. We construct a diagrammatic procedure for solving it iteratively – that is, the classical limit of the Feynman diagram procedure in quantum field theory, but here it is just a mathematical trick.

We consider the Skyrmion solution of classical field theory. We define the Skyrme model as the extension of the nonlinear sigma model of QCD by the addition of a new “Skyrme term”. We analyze the model and define a topological “winding number” for the scalars in spacetime. The Skyrmion solution is found by imposing a “hedgehog” ansatz for the scalars. Generalizations of the model are studied, the Skyrme–Faddeev model and the DBI–Skyrme model, for which we identify the solution, the DBI–Skyrmion.

We examine Poisson brackets in field theory and the symplectic formulation of Hamiltonian dynamics. We start by describing the symplectic formulation of classical mechanics. Then we generalize it and Poisson brackets to field theory. As examples of the formalism, we consider a scalar field with canonical kinetic term and the nonlinear sigma model.