De Donder–Weyl theory
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In
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, the De Donder–Weyl theory is a generalization of the
Hamiltonian formalism Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
in the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
and
classical field theory A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
over
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
which treats the space and time coordinates on equal footing. In this framework, the
Hamiltonian formalism Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
in
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
is generalized to field theory in the way that a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
is represented as a system that varies both in space and in time. This generalization is different from the
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...
Hamiltonian formalism Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
in field theory which treats space and time variables differently and describes classical fields as infinite-dimensional systems evolving in time.


De Donder–Weyl formulation of field theory

The De Donder–Weyl theory is based on a change of variables known as
Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of ...
. Let ''xi'' be
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
coordinates, for ''i'' = 1 to ''n'' (with ''n'' = 4 representing 3 + 1 dimensions of space and time), and ''ya'' field variables, for ''a'' = 1 to ''m'', and ''L'' the
Lagrangian density Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
:L = L(y^,\partial_i y^,x^) With the polymomenta ''pia'' defined as :p^_a = \partial L / \partial (\partial_i y^) and the De Donder–Weyl Hamiltonian function ''H'' defined as :H = p^_a \partial_i y^ - L the De Donder–Weyl equations are: :\partial p^_a / \partial x^ = -\partial H / \partial y^ \, , \, \partial y^ / \partial x^ = \partial H / \partial p^_a This De Donder-Weyl Hamiltonian form of field equations is covariant and it is equivalent to the Euler-Lagrange equations when the Legendre transformation to the variables ''pia'' and ''H'' is not singular. The theory is a formulation of a
covariant Hamiltonian field theory In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory. ...
which is different from the
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...
Hamiltonian formalism Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
and for ''n'' = 1 it reduces to
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
(see also action principle in the calculus of variations).
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
in 1935 has developed the Hamilton-Jacobi theory for the De Donder–Weyl theory. Similarly to the
Hamiltonian formalism Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
in mechanics formulated using the
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...
of
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
the De Donder-Weyl theory can be formulated using the multisymplectic geometry or polysymplectic geometry and the geometry of
jet bundle In differential topology, the jet bundle is a certain construction that 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. Je ...
s. A generalization of the
Poisson brackets In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
to the De Donder–Weyl theory and the representation of De Donder–Weyl equations in terms of generalized
Poisson brackets In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
satisfying the
Gerstenhaber algebra In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring an ...
was found by Kanatchikov in 1993.


History

The formalism, now known as De Donder–Weyl (DW) theory, was developed by
Théophile De Donder Théophile Ernest de Donder (; 19 August 1872 – 11 May 1957) was a Belgian mathematician and physicist famous for his work (published in 1923) in developing correlations between the Newtonian concept of chemical affinity and the Gibbsian concept ...
and
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
. Hermann Weyl made his proposal in 1934 being inspired by the work of
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
, which in turn was founded on the work of
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in Anc ...
. The work of De Donder on the other hand started from the theory of integral invariants of
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
. The De Donder–Weyl theory has been a part of the calculus of variations since the 1930s and initially it found very few applications in physics. Recently it was applied in theoretical physics in the context of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
and
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
. In 1970, Jedrzej Śniatycki, the author of ''Geometric quantization and quantum mechanics'', developed an invariant geometrical formulation of
jet bundle In differential topology, the jet bundle is a certain construction that 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. Je ...
s, building on the work of De Donder and Weyl. In 1999 Igor Kanatchikov has shown that the De Donder–Weyl covariant Hamiltonian field equations can be formulated in terms of Duffin–Kemmer–Petiau matrices.Igor V. Kanatchikov
''On the Duffin–Kemmer–Petiau formulation of the covariant Hamiltonian dynamics in field theory''
arXiv:hep-th/9911175 (submitted on 23 November 1999)


See also

*
Hamiltonian field theory In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory. ...
*
Covariant Hamiltonian field theory In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory. ...


Further reading

* Selected papers on GEODESIC FIELDS, Translated and edited by D. H. Delphenich. Part

Part

* H.A. Kastrup, Canonical theories of Lagrangian dynamical systems in physics, Physics Reports, Volume 101, Issues 1–2, Pages 1-167 (1983). * Mark J. Gotay, James Isenberg, Jerrold E. Marsden, Richard Montgomery: "Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory" * Cornelius Paufler, Hartmann Römer
''De Donder–Weyl equations and multisymplectic geometry''
Reports on Mathematical Physics, vol. 49 (2002), no. 2–3, pp. 325–334 * Krzysztof Maurin: ''The Riemann legacy: Riemannian ideas in mathematics and physics'', Part II, Chapter 7.16 ''Field theories for calculus of variation for multiple integrals'', Kluwer Academic Publishers, , 1997
p. 482 ff.


References

{{DEFAULTSORT:De Donder-Weyl theory Calculus of variations Mathematical physics