Topic
Jet bundle
About: Jet bundle is a research topic. Over the lifetime, 210 publications have been published within this topic receiving 2725 citations.
Papers published on a yearly basis
Papers
1 Jan 1999
TL;DR: In this paper, the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations was developed and applied for wave propagation problems, particularly the instability of water waves.
Abstract: This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unies the classical Hamiltonian formalism of particle mechanics as well as the many pre-symplectic 2-forms used by Bridges. In this theory, solutions of a partial differential equation are sections of a bre bundle Y over a base manifold X of dimension n+1, typically taken to be spacetime. Given a connection onY , a covariant Hamiltonian density H is then intrinsically dened on the primary constraint manifoldPL, the image of the multisymplectic version of the Legendre transformation. One views PL as a subbundle of J 1 (Y ) ? , the afne dual of J 1 (Y ), the rst jet bundle of Y . A canonical multisymplectic (n+2)-form H is then dened, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original partial differential equation as well as the Euler{Lagrange equations of the corresponding Lagrangian. Furthermore, we show that the n+1 2-forms ! () dened by Bridges are a particular coordinate representation for a single multisymplectic (n+2)-form and, in the presence of symmetries, can be assembled into H. A generalized Hamiltonian Noether theory is then constructed which relates the action of the symmetry groups lifted to PL with the conservation laws of the system. These conservation laws are dened by our generalized Noether’s theorem which recovers the vanishing of the divergence of the vector of n+1 distinct momentum mappings dened by Bridges and, when applied to water waves, recovers Whitham’s conservation of wave action. In our view, the multisymplectic structure provides the natural setting for studying dispersive wave propagation problems, particularly the instability of water waves, as discovered by Bridges. After developing the theory, we show its utility in the study of periodic pattern formation and wave instability.
145 citations
TL;DR: In this paper, a multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations is proposed, which is equivalent to both the original PDE as well as the Euler-Lagrange equations of the corresponding Lagrangians.
Abstract: This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations.
In this theory, solutions of a PDE are sections of a fiber bundle $Y$ over a base manifold $X$ of dimension $n$$+$1, typically taken to be spacetime. Given a connection on $Y$, a covariant Hamiltonian density ${\mathcal H}$ is then intrinsically defined on the primary constraint manifold $P_{\mathcal L}$, the image of the multisymplectic version of the Legendre transformation. One views $P_{\mathcal L}$ as a subbundle of $J^1(Y)^\star$, the affine dual of $J^1(Y)$, the first jet bundle of $Y$. A canonical multisymplectic ($n$$+$2)-form $\Omega_{\mathcal H}$ is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original PDE as well as the Euler-Lagrange equations of the corresponding Lagrangian. We show that the $n$$+$1 2-forms $\omega^{(\mu)}$ defined by Bridges [1997] are a particular coordinate representation for a single multisymplectic ($n$$+$2)-form, and in the presence of symmetries, can be assembled into $\Omega_{\mathcal H}$. A generalized Hamiltonian Noether theory is then constructed which recovers the vanishing of the divergence of the vector of $n$$+$1 distinct momentum mappings defined in Bridges [1997] and, when applied to water waves, recovers Whitham's conservation of wave action. We also show the utility of this theory in the study of periodic pattern formation and wave instability.
143 citations
TL;DR: In this article, a Lagrangian geometric formulation for first-order field theories using the canonical structures of firstorder jet bundles, which are taken as the phase spaces of the systems in consideration, is presented.
Abstract: We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a firstorder jet bundle and, using them, we develop the lagrangian formalism, defining the canonical forms associated with a lagrangian density and the density of lagrangian energy, obtaining the Euler-Lagrange equations in two equivalent ways: as the result of a variational problem and developing the jet field formalism (which is a formulation more similar to the case of mechanical systems). A statement and proof of Noether’s theorem is also given, using the latter formalism. Finally, some classical examples are briefly studied.
105 citations
TL;DR: In this paper, a Lagrangian geometric formulation for first-order field theories using the canonical structures of firstorder jet bundles, which are taken as the phase spaces of the systems in consideration, is presented.
Abstract: We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a first-order jet bundle and, using them, we develop the lagrangian formalism, defining the canonical forms associated with a lagrangian density and the density of lagrangian energy, obtaining the {\sl Euler-Lagrange equations} in two equivalent ways: as the result of a variational problem and developing the {\sl jet field formalism} (which is a formulation more similar to the case of mechanical systems). A statement and proof of Noether's theorem is also given, using the latter formalism. Finally, some classical examples are briefly studied.
85 citations
Journal Article•
TL;DR: In this article, a tensor analysis on the first jet extension of the configuration space-time, based on a suitable linear connection, determined entirely by the dynamics of the system, is presented.
Abstract: Some aspects of the geometry of jet-bundles, especially relevant for the formulation of Classical Mechanics, are investigated. The main result is the construction of a tensor analysis on the first jet extension of the configuration space-time, based on a suitable linear connection, determined entirely by the dynamics of the system. The significance of this \"dynamical connection\" in the geometrization of Classical Mechanics is discussed, paying a special attention to two particular aspects, namely the implementation of the concept of \"relative time derivative\" in the Lagrangian framework, and the derivation of the Helmholtz conditions for the inverse problem of Lagrangian Dynamics. Key-words : Lagrangian Dynamics, inverse problem, linear and affine connections. 02.40.+m, 03.20.+i 1991 Mathematical subject Classification : 70 D 10, 34 A 55, 53 B 05 This research was partly supported by the National Group for Mathematical Physics of the Italian Research Council (CNR), and by the Italian Ministry for Public Education, through the research project \"Metodi Geometrici e Probabilistici in Sistems Dinamici, Meccanica Statistica, Relativita e Teoria dei Campi\". Annales de l’Institut Henri Poincaré Physique theorique 0246-0211 Vol. 61/94/01/$ 4.00/@ Gauthier-Villars 18 E. MASSA AND E. PAGANI Dans cet article on etudie des aspects de la geometric des espaces des jets qui sont importants pour une formulation geometrique de la mecanique classique. On construit une « analyse tensorielle » sur la premiere extension des jets de l’espace-temps des configurations moyennant une connexion lineaire determinee completement par la dynamique du systeme. On met en evidence Ie role joue par cette « connexion dynamique » dans la geometrisation de la mecanique classique ; en particulier on introduit Ie concept de « derivee temporelle relative » dans Ie contexte lagrangien et les conditions de Helmholtz pour Ie probleme inverse de la dynamique lagrangienne.
77 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2020 | 6 |
| 2019 | 4 |
| 2018 | 9 |
| 2017 | 7 |
| 2016 | 12 |