Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

Zatloukal, Vaclav

doi:10.1142/s0219887816500729

Cited by 4 publications

(2 citation statements)

References 14 publications

(35 reference statements)

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“…This 1-form transforms the integral curves of a vector field X dW H on Q into integral curves of the dynamical vector field XH on TQ (the latter satisfying Hamilton equations). This geometric procedure 2,7 has been extended to many other different contexts, such as nonholonomic systems, [7][8][9][10][11] singular Lagrangian systems, [12][13][14] higher-order systems, 15 field theories, [16][17][18][19][20][21][22] or contact systems. 23 An…”

Section: Introductionmentioning

confidence: 99%

Geometric Hamilton–Jacobi theory for systems with external forces

León

Valcázar

Asier

2022

Journal of Mathematical Physics

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In this paper, we develop a Hamilton–Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems, and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton–Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton–Jacobi problem for a Čaplygin system to the Hamilton–Jacobi problem for a forced Lagrangian system.

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Section: Introductionmentioning

confidence: 99%

Geometric Hamilton–Jacobi theory for systems with external forces

León

Valcázar

Asier

2022

Journal of Mathematical Physics

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“…This 1-form transforms the integral curves of a vector field X dW H on Q into integral curves of the dynamical vector field X H on T Q (the latter satisfying Hamilton equations). This geometric procedure [1,12] has been extended to many other different contexts, such as nonholonomic systems [12,13,31,40,55], singular Lagrangian systems [18,35,46], higher-order systems [16], field theories [7,20,37,48,50,59,61] or contact systems [45]. An unifying Hamilton-Jacobi theory for almost-Poisson manifolds was developed in reference [47].…”

Section: Introductionmentioning

confidence: 99%

Geometric Hamilton-Jacobi theory for systems with external forces

de León,

Lainz,

López-Gordón

2021

Preprint

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In this paper, we develop a Hamilton-Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton-Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton-Jacobi problem for a Čaplygin system to the Hamilton-Jacobi problem for a forced Lagrangian system.

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Classical Field Theories from Hamiltonian Constraint: Local Symmetries and Static Gauge Fields

Zatloukal

2018

Adv. Appl. Clifford Algebras

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We consider the Hamiltonian constraint formulation of classical field theories, which treats spacetime and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for non-invariance of the Hamiltonian under local transformations. It is a position-dependent linear mapping, which couples to the Hamiltonian by acting on the momentum multivector. We investigate symmetries of the ensuing gauged Hamiltonian, and propose a generic form of the gauge field strength. In examples we show how a generic gauge field can be specialized in order to realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is not discussed in this article.Throughout, we employ the mathematical language of geometric algebra and calculus. * Electronic address: zatlovac@gmail.com; URL: http://www.zatlovac.eu arXiv:1611.02906v1 [math-ph] 9 Nov 2016 3 generic local transformations that possess additional position dependence. To compensate for this non-invariance, we introduce in Sec. II a gauge field with appropriate transformation properties, which acts as a q-dependent linear map on the momentum multivector P . The ensuing gauged Hamiltonian acquires thereupon an additional q-dependency, and the canonical equations (2) are modified to Eqs. (15). The structure of the latter canonical equations suggests to define, by Eq. (19) in Sec. III, the field strength corresponding to the gauge field, which is a linear q-dependent function that maps grade-r multivectors to grade-r + 1 multivectors. In Appendix B, it is interpreted as torsion corresponding to the Weitzenböck connection on the configuration space C .In the present article, the gauge field and the field strength are static background quantities in the sense that they do not obey their own dynamical equations of motion, but rather are prescribed by some external body. This is also why we do not attempt to include them in the set of partial observables, but treat them separately. We relegate the study of gauge field dynamics (presumably implemented by means of a suitable kinetic term) to the future.Conservation laws maintain the form of Eq. (7), where the left-hand side features the gauged Hamiltonian. Whether a vector field v is a symmetry generator thus depends on the concrete form of the gauge field. The issue is discussed in Sec. IV.The general theory of Sections II, III and IV is much illuminated through the examples of Sec. V. In Examples V A and V B, without any reference to a concrete form of the Hamiltonian, the partial observables are divided into D spacetime coordinates and N field components, and two subgroups of the group of all configuration-space diffeomorphisms are considered. In the first example, we "gauge" spacetime diffeomorphisms by a gauge field equivalent to the tetrad (or vierbein) in the tetrad formulation of gravity. It acts nontrivially only on the spacetime, and therefore has fewer degrees of freedom than a generic gauge field. In the second example, we consider rotations in the field ...

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Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

Cited by 4 publications

References 14 publications

Geometric Hamilton–Jacobi theory for systems with external forces

Geometric Hamilton–Jacobi theory for systems with external forces

Geometric Hamilton-Jacobi theory for systems with external forces

Classical Field Theories from Hamiltonian Constraint: Local Symmetries and Static Gauge Fields

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