The Newtonian formalism offers a very simple way to understand mechanical systems, but it has the difficult that it is necessary to measure and calculate the three components of the position and velocity of each particle that makes up the system.
Later, with the development of Lagrangian mechanics (1788) and Hamiltonian mechanics (1833) and their generalization to symplectic mechanics, the usefulness of the geometrical description of classical mechanics becomes clear. As we all know, this description of autonomous systems is carried out by means of the so-called symplectic manifolds.
When we make the leap from classical mechanics to classical field theory, the use of differential geometry has been a tool of great interest. At the end of the '60s and the beginning of the '70s of the past century, there are some attempts to develop a convenient geometric framework to study classical field theories. The first difficulty in this area is the generalization of the notion of symplectic form. There are different geometrical settings that allow describing classical field theory: k-symplectic formulation, k-cosymplectic formulation, multisymplectic, k-contact, etc.
The aim of this talk is to present the simplest one geometric description of classical field theories: the k-symplectic framework. The notion of k-symplectic manifolds will be introduced and these geometric structures will be analyzed. We will describe a Darboux' theorem for these structures and we will analyze certain types of submanifolds, for instance, k-symplectic orthogonal subspaces or Lagrangian of k-symplectic manifolds such as orthogonal subspaces or Lagrangian submanifolds.
Finally we will comment some interesting applications of the k-symplectic structures.