Article records
https://feeds.library.caltech.edu/people/Weinstein-Alan-J-Math/article.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 15 Apr 2024 14:30:28 +0000A comparison theorem for Hamiltonian vector fields
https://resolver.caltech.edu/CaltechAUTHORS:20100910-101004513
Authors: {'items': [{'id': 'Weinstein-Alan-J-Math', 'name': {'family': 'Weinstein', 'given': 'Alan J.'}}, {'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'Jerrold E.'}}]}
Year: 1970
DOI: 10.1090/S0002-9939-1970-0273648-6
The question of completeness of Hamiltonian systems is investigated for a class of potentials not necessarily bounded below. The result generalizes previous work of W. Gordon and D. Ebin.https://authors.library.caltech.edu/records/k43ab-bns89Reduction of symplectic manifolds with symmetry
https://resolver.caltech.edu/CaltechAUTHORS:20100910-101924768
Authors: {'items': [{'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'Jerrold E.'}}, {'id': 'Weinstein-Alan-J-Math', 'name': {'family': 'Weinstein', 'given': 'Alan J.'}}]}
Year: 1974
DOI: 10.1016/0034-4877(74)90021-4
We give a unified framework for the construction of symplectic manifolds from systems with symmetries. Several physical and mathematical examples are given; for instance, we obtain Kostant's result on the symplectic structure of the orbits under the coadjoint representation of a Lie group. The framework also allows us to give a simple derivation of Smale's criterion for relative equilibria. We apply our scheme to various systems, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.https://authors.library.caltech.edu/records/8k5v6-4pa21The Hamiltonian structure of the Maxwell-Vlasov equations
https://resolver.caltech.edu/CaltechAUTHORS:20100910-112829181
Authors: {'items': [{'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'Jerrold E.'}}, {'id': 'Weinstein-Alan-J-Math', 'name': {'family': 'Weinstein', 'given': 'Alan J.'}}]}
Year: 1982
DOI: 10.1016/0167-2789(82)90043-4
Morrison [25] has observed that the Maxwell-Vlasov and Poisson-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. We derive another Poisson structure for these equations by using general methods of symplectic geometry. The main ingredients in our construction are the symplectic structure on the co-adjoint orbits for the group of canonical transformations, and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory. Our Poisson bracket satisfies the Jacobi identity, whereas Morrison's does not [37]. Our construction also shows where canonical variables can be found and can be applied to the Yang-Mills-Vlasov equations and to electromagnetic fluid dynamics.https://authors.library.caltech.edu/records/ytf6x-wsp44Hamiltonian systems with symmetry, coadjoint orbits and plasma physics
https://resolver.caltech.edu/CaltechAUTHORS:20100922-080914569
Authors: {'items': [{'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'Jerrold E.'}}, {'id': 'Ratiu-T-S', 'name': {'family': 'Ratiu', 'given': 'T.'}, 'orcid': '0000-0003-1972-5768'}, {'id': 'Schmid-R', 'name': {'family': 'Schmid', 'given': 'R.'}}, {'id': 'Spencer-R-G', 'name': {'family': 'Spencer', 'given': 'R. G.'}}, {'id': 'Weinstein-Alan-J-Math', 'name': {'family': 'Weinstein', 'given': 'Alan J.'}}]}
Year: 1983
The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the Lie-Poisson structure on the dual of a Lie algebra. These results are
applied to plasma physics. We show in three steps how the Maxwell-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. First, the Poisson-Vlasov equations are shown
to be in Hamiltonian form relative to the Lie-Poisson bracket on the dual of the (nite dimensional) Lie algebra of innitesimal canonical transformations. Then we write Maxwell's equations in Hamiltonian form using the canonical
symplectic structure on the phase space of the electromagnetic elds, regarded as a gauge theory. In the last step we couple these two systems via the reduction
procedure for interacting systems. We also show that two other standard models in plasma physics, ideal MHD and two-
uid electrodynamics, can be written in Hamiltonian form using similar group theoretic techniques.https://authors.library.caltech.edu/records/rpq2q-xr717Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids
https://resolver.caltech.edu/CaltechAUTHORS:20100910-121621517
Authors: {'items': [{'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'Jerrold E.'}}, {'id': 'Weinstein-Alan-J-Math', 'name': {'family': 'Weinstein', 'given': 'Alan J.'}}]}
Year: 1983
DOI: 10.1016/0167-2789(83)90134-3
This paper is a study of incompressible fluids, especially their Clebsch variables and vortices, using symplectic geometry and the Lie-Poisson structure on the dual of a Lie algebra. Following ideas of Arnold and others it is shown that Euler's equations are Lie-Poisson equations associated to the group of volume-preserving diffeomorphisms. The dual of the Lie algebra is seen to be the space of vortices, and Kelvin's circulation theorem is interpreted as preservation of coadjoint orbits. In this context, Clebsch variables can be understood as momentum maps. The motion of N point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit, and the standard canonical variables for them are a special kind of Clebsch variables. Point vortices with cores, vortex patches, and vortex filaments can be understood in a similar way. This leads to an explanation of the geometry behind the Hald-Beale-Majda convergence theorems for vorticity algorithms. Symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the KdV and the SchrÃ¶dinger equations respectively.https://authors.library.caltech.edu/records/bgbfy-j3x09Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics
https://resolver.caltech.edu/CaltechAUTHORS:20100812-143646961
Authors: {'items': [{'id': 'Holm-D-D', 'name': {'family': 'Holm', 'given': 'Darryl D.'}}, {'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'Jerrold E.'}}, {'id': 'Ratiu-T-S', 'name': {'family': 'Ratiu', 'given': 'T.'}, 'orcid': '0000-0003-1972-5768'}, {'id': 'Weinstein-Alan-J-Math', 'name': {'family': 'Weinstein', 'given': 'Alan J.'}}]}
Year: 1983
DOI: 10.1016/0375-9601(83)90534-0
A method developed by Arnold to prove nonlinear stability of certain steady states for ideal incompressible flow in two dimensions is extended to the barotropic compressible case. The results are applied to plane shear flow.https://authors.library.caltech.edu/records/xyvc0-9fe38Semidirect products and reduction in mechanics
https://resolver.caltech.edu/CaltechAUTHORS:20100907-154555674
Authors: {'items': [{'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'Jerrold E.'}}, {'id': 'Ratiu-T-S', 'name': {'family': 'Ratiu', 'given': 'Tudor'}, 'orcid': '0000-0003-1972-5768'}, {'id': 'Weinstein-Alan-J-Math', 'name': {'family': 'Weinstein', 'given': 'Alan J.'}}]}
Year: 1984
This paper shows how to reduce a Hamiltonian system on the cotangent bundle of a Lie group to a Hamiltonian system in the dual of the Lie algebra of a semidirect product. The procedure simplifies, unifies, and extends work of Greene, Guillemin, Holm, Holmes, Kupershmidt, Marsden, Morrison, Ratiu, Sternberg and others. The heavy top, compressible fluids, magnetohydrodynamics, elasticity, the Maxwell-Vlasov equations and multifluid plasmas are presented as examples. Starting with Lagrangian variables, our method explains in a direct way why semidirect products occur so frequently in examples. It also provides a framework for the systematic introduction of Clebsch, or canonical, variableshttps://authors.library.caltech.edu/records/erjct-dyv39Nonlinear stability of fluid and plasma equilibria
https://resolver.caltech.edu/CaltechAUTHORS:20100817-084719378
Authors: {'items': [{'id': 'Holm-D-D', 'name': {'family': 'Holm', 'given': 'Darryl D.'}}, {'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'Jerrold E.'}}, {'id': 'Ratiu-T-S', 'name': {'family': 'Ratiu', 'given': 'T.'}, 'orcid': '0000-0003-1972-5768'}, {'id': 'Weinstein-Alan-J-Math', 'name': {'family': 'Weinstein', 'given': 'Alan J.'}}]}
Year: 1985
DOI: 10.1016/0370-1573(85)90028-6
The Liapunov method for establishing stability has been used in a variety of fluid and plasma problems. For nondissipative systems, this stability method is related to well-known energy principles. A development of the Liapunov method for Hamiltonian systems due to Arnold uses the energy plus other conserved quantities, together with second variations and convexity estimates, to establish stability. For Hamiltonian systems, a useful class of these conserved quantities consists of the Casimir functionals, which Poisson-commute with all functionals of the given dynamical variables. Such conserved quantities, when added to the energy, help to provide convexity estimates bounding the growth of perturbations. These estimates enable one to prove nonlinear stability, whereas the commonly used second variation or spectral arguments only prove linearized stability. When combined with recent advances in the Hamiltonian structure of fluid and plasma systems, this convexity method proves to be widely and easily applicable. This paper obtains new nonlinear stability criteria for equilibria for MHD, multifluid plasmas and the Maxwell-Vlasov equations in two and three dimensions. Related systems, such as multilayer quasigeostrophic flow, adiabatic flow and the Poisson-Vlasov equation are also treated. Other related systems, such as stratified flow and reduced magnetohydrodynamic equilibria are mentioned where appropriate, but are treated in detail in other publications.https://authors.library.caltech.edu/records/hvs1a-fax31