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Lagrangian and Hamiltonian Dynamics$
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Peter Mann

Print publication date: 2018

Print ISBN-13: 9780198822370

Published to Oxford Scholarship Online: August 2018

DOI: 10.1093/oso/9780198822370.001.0001

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Lagrangian Field Theory

Lagrangian Field Theory

(p.345) 25 Lagrangian Field Theory
Lagrangian and Hamiltonian Dynamics

Peter Mann

Oxford University Press

In this chapter, Hamiltonian field theory is derived classically via a Hamiltonian density, using the zeroth component of a 4-momentum density. In field theory, space and time are considered to be on equal footing but, in the canonical formalism, time is treated as being special and therefore, by definition, it is not covariant. Consequently, most field theoretic models are built on Lagrangian formulations. A covariant canonical formalism is the subject of the de Donder–Weyl formalism, which is briefly discussed as a covariant Hamiltonian field theory. In addition, the chapter examines the case of a generalised Poisson bracket in the continuous form for two local smooth functionals of phase space.

Keywords:   Hamiltonian density, 4-momentum density, de Donder–Weyl, Hamiltonian field theory, Poisson bracket

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