Methods of Differential Geometry in Classical Field Theories

The following sections are included:

• Introduction
• Contents
• List of Tables

In this chapter we present a brief review of Hamiltonian and Lagrangian mechanics; firstly on the cotangent bundle of an arbitrary manifold Q (the Hamiltonian formalism) and then on the tangent bundle (the Lagrangian formalism). Finally, we consider the general theory on an arbitrary symplectic manifold.

In the last part of this chapter we give a review of the non-autonomous mechanics using cosymplectic structures.

A complete description of Hamiltonian and Lagrangian mechanics can be found in [Abraham and Marsden (1978); Arnold (1978, 1998); Godbillon (1969); Godstein, Poole Jr. and Safko (2001); Holm, Schmah and Stoica (2009); Holm (2008); Libermann and Marle (1987); de León and Rodrigues (1989)]. There exists an alternative description of the Lagrangian and Hamiltonian dynamics using the notion of Lagrangian submanifold, this description can be found in [Tulczyjew (1976,b)].

The symplectic geometry allows us to give a geometric description of classical mechanics (see chapter 1). On the contrary, there exist several alternative models for describing geometrically first-order classical field theories. From a conceptual point of view, the simplest one is the k-symplectic formalism, which is a natural generalization to field theories of the standard symplectic formalism…

The k-symplectic formulation is based on the so-called k-symplectic geometry. In this chapter we introduce the k-symplectic structure which is a generalization of the notion of symplectic structure.

We first describe the geometric model of the k-symplectic manifolds, that is the cotangent bundle of k¹-covelocities and we introduce the notion of canonical geometric structures on this manifold. The formal definition of the k-symplectic manifold is given in section 2.2.

In this chapter we shall describe the k-symplectic formalism. As we shall see in the following chapters, using this formalism we can study classical field theories in the Hamiltonian and Lagrangian cases.

One of the most important elements in the k-symplectic approach is the notion of k-vector field. Roughly speaking, it is a family of k vector fields. In order to introduce this notion in section 3.1, we previously consider the tangent bundle of k¹-velocities of a manifold, i.e., the Whitney sum of k copies of its tangent bundle with itself. In section 6.1 we shall describe this manifold with more details.

Here we shall introduce a geometric equation, called the k-symplectic Hamiltonian equation, which allows us to describe classical field theories when the k-symplectic manifold is the cotangent bundle of k¹-covelocities or its Lagrangian counterpart under some regularity condition satisfied by the Lagrangian function.

In this chapter we shall study Hamiltonian classical field theories, that is, we shall discuss the Hamilton-De Donder-Weyl equations (these equations will also be called the HDW equations for short) which have the following local expression

The usefulness of Hamilton-Jacobi theory in classical mechanics is well-known, giving an alternative procedure to study and, in some cases, to solve the evolution equations [Abraham and Marsden (1978)]. The use of symplectic geometry in the study of classical mechanics has permitted to connect the Hamilton-Jacobi theory with the theory of Lagrangian submanifolds and generating functions [Barbero-Liñan, de León and Martín de Diego (2013)]…

The aim of this chapter is to give a geometric description of the Euler-Lagrange field equations

In this chapter we describe several physical examples using the k-symplectic formulation developed in this part of the book. In [Muñoz-Lecanda, Salgado and Vilariño (2009)] one can find several of these examples. Previously, we recall the geometric version of the Hamiltonian and Lagrangian approaches for classical field theories and its correspondence with the case k = 1…

Part 2 of this book has been devoted to give a geometric description of certain kinds of classical field theories. The purpose of Part 3 is to extend the above study to classical field theories involving the independent parameters, i.e., the “space-time” coordinates (x¹, … , x^k) in an explicit way. In others words, in this part we shall give a geometrical description of classical field theories whose Lagrangian and Hamiltonian functions are of the form and …

The k-cosymplectic formulation is based on the so-called k-cosymplectic geometry. In this chapter we introduce these structures which are a generalization of the notion of cosymplectic forms.

Firstly, we describe the model of the k-cosymplectic manifolds, that is the stable cotangent bundle of k¹-covelocities and introduce the canonical structures living there. Using this model we introduce the notion of k-cosymplectic manifold. A complete description of these structures can be found in [de León, Merino, Oubiña, Rodrigues and Salgado (1998); de León, Merino and Salgado (2001)].

In this chapter we describe the k-cosymplectic formalism. As we shall see in the following chapters, using this formalism we can study classical field theories that explicitly involve the space-time coordinates on the Hamiltonian and Lagrangian. This is the principal difference with the k-symplectic approach. As in previous case, in this formalism the notion of k-vector field is fundamental; let us recall that this notion was introduced in section 3.1 for an arbitrary manifold M.

In this chapter we shall study Hamiltonian classical field theories when the Hamiltonian function involves the space-time coordinates, that is, H is a function defined on . Therefore, we shall discuss the Hamilton-De Donder-Weyl equations (HDW equations for short) which have the following local expression

There are several attempts to extend the Hamilton-Jacobi theory for classical field theories. In [de León, Marrero, Martín de Diego, Salgado and Vilariño (2010)] we have described this theory in the framework of the so-called k-symplectic formalism [Awane (1992); Günther (1987); de León, Méndez and Salgado (1988, 1991)]. In this section we consider the k-cosymplectic framework. Another attempts in the framework of the multisymplectic formalism [Cantrijn, Ibort and de León (1999); Kijowski and Tulczyjew (1979)] have been discussed in [de León, Marrero and Martín de Diego (2009); Paufler and Römer (2002,b)]…

In a similar way to that developed in chapter 6, we now give a description of the Lagrangian classical field theories using two different approaches: a variational principle and a k-cosymplectic approach…

In this chapter we shall present some physical examples which can be described using the k-cosymplectic formalism (see [Muñoz-Lecanda, Salgado and Vilariño (2009)] for more details).

In this book we are presenting two different approaches to describe first-order classical field theories: first, when the Lagrangian and Hamiltonian do not depend on the base coordinates, and, later, when the Lagrangian and Hamiltonian also depend on the “space-time” coordinates. However, if we observe the corresponding descriptions we see that in local coordinates they give a geometric description of the same system of partial differential equations. Therefore the natural question is: Is there any relationship between k-symplectic and k-cosymplectic systems? In this section we give an affirmative answer to this question. Naturally, this relation will be established only when the Lagrangian and Hamiltonian do not depend on the base coordinates.

Along this section we work over the geometrical models of k-symplectic and k-cosymplectic manifolds, that is and and the Lagrangian counterparts and . However the following results and comments can be extended to the case ℝ^k × M and M, being M an arbitrary k-symplectic manifold.

Following a similar terminology to that in mechanics, we introduce the following definition.

In this book, we have developed a framework to describe classical field theories using k-symplectic and k-cosymplectic manifolds. An alternative geometric framework is the multisymplectic formalism [Cariñena, Crampin and Ibort (1991); Gotay, Isenberg, Marsden and Montgomery (2004); Gotay, Isenberg, Marsden (2004); Marsden and Shkoller (1999); Román-Roy (2009)], first introduced in [Kijowski (1973); Kijowski and Szczyrba (1975); Kijowski and Tulczyjew (1979); Sniatycki (1970)], which is based on the use of multisymplectic manifolds. In particular, jet bundles are the appropriate domain to develop the Lagrangian formalism [Saunders (1989)], and different kinds of multimomentum bundles are used for developing the Hamiltonian description [Echeverría-Enríquez, Muñoz-Lecanda and Román-Roy (2000); Hélein and Kouneiher (2004); de León, Marrero and Martín de Diego (2009)]. In these models, the field equations can also be obtained in terms of multivector fields [Echeverría-Enríquez, Muñoz-Lecanda and Román-Roy (1998, 1999); Paufler and Römer (2002b)]…

The following sections are included:

• Appendix A Symplectic Manifolds
• Appendix B Cosymplectic Manifolds
• Appendix C Glossary of Symbols
• Bibliography
• Index