In recent years, there has been renewed interest in covariant classical field theory. Here, classical fields are represented by sections of fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

s and their dynamics is phrased in the context of a finite-dimensional space of fields

Field (physics)

In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

. Nowadays, it is well known that jet bundle

Jet bundle

In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...

s and the variational bicomplex

Variational bicomplex

In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations...

 are the correct domain for such a description. The Hamiltonian variant of covariant classical field theory is the covariant Hamiltonian field theory

Covariant Hamiltonian field theory

Applied to classical field theory, the familiar symplectic Hamiltonian formalism takes the form of instantaneous Hamiltonian formalism on an infinite-dimensional phase space, where canonical coordinates are field functions at some instant of time...

 where momenta correspond to derivatives of field variables with respect to all world coordinates. Non-autonomous mechanics

Non-autonomous mechanics

Non-autonomous mechanics describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics isa fiber bundle Q\to...

 is formulated as covariant classical field theory on fiber bundles over the time axis .

Derivation and proof

The worksheet provides some of the geometric structure to the covariant formalism of first-order classical field theories.
Talk:Covariant classical field theory/workpage

See also

• Classical field theory
  Classical field theory
  A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
  
  
• Exterior algebra
  Exterior algebra
  In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
  
  
• Variational bicomplex
  Variational bicomplex
  In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations...
  
  
• Quantum field theory
  Quantum field theory
  Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...