On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions
Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the invariance structure of given nonlinear dynamical systems on the infinite-dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrow...
Збережено в:
| Дата: | 2005 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2005
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| Назва видання: | Нелінійні коливання |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/178006 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions / Y.A. Prykarpatsky, A.M. Samoilenko // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 360-387. — Бібліогр.: 41 назв. — англ. |
Репозитарії
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irk-123456789-1780062021-02-18T01:28:29Z On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions Prykarpatsky, Y.A. Samoilenko, A.M. Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the invariance structure of given nonlinear dynamical systems on the infinite-dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrowed from the Cartan’s theory of differential systems on the associated jet-manifolds. The symmetry structure reduced on the invariant submanifolds of critical points of some nonlocal Euler – Lagrange functional is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integraldifferential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. The connection of this hierarchy with integrable by Lax spatially two-dimensional systems is studied. Наведено деякi аспекти опису лагранжевого та гамiльтонового формалiзму, який природно виникає iз структури iнварiантностi заданих нелiнiйних динамiчних систем на нескiнченновимiрному функцiональному многовидi. Основнi iдеї, якi використовуються для формування канонiчної симплектичної структури, взято з теорiї Картана диференцiальних систем на вiдповiдних многовидах струмiв. Для диференцiальних та диференцiальних дискретних динамiчних систем наведено детальний опис структури симетрiй, якi редукованi на iнварiантнi пiдмноговиди критичних точок деяких нелокальних ейлерово-лагранжевих функцiоналiв. За допомогою деякого перетворення Беклунда отримано гамiльтонове зображення для iєрархiї рiвнянь лаксового типу на двоїстому до алгебри Лi просторi iнтегрально-диференцiальних операторiв з матричними коефiцiєнтами, яке продовжено еволюцiями власних функцiй та спряжених власних функцiй вiдповiдних спектральних задач. Вивчено зв’язок мiж цiєю iєрархiєю та iнтегровними за Лаксом просторово-двовимiрними системами. 2005 Article On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions / Y.A. Prykarpatsky, A.M. Samoilenko // Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 360-387. — Бібліогр.: 41 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178006 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Some aspects of the description of Lagrangian and Hamiltonian formalisms naturally arising from the
invariance structure of given nonlinear dynamical systems on the infinite-dimensional functional manifold is presented. The basic ideas used to formulate the canonical symplectic structure are borrowed
from the Cartan’s theory of differential systems on the associated jet-manifolds. The symmetry structure
reduced on the invariant submanifolds of critical points of some nonlocal Euler – Lagrange functional
is described thoroughly for both differential and differential discrete dynamical systems. The Hamiltonian representation for a hierarchy of Lax-type equations on a dual space to the Lie algebra of integraldifferential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. The connection of this hierarchy with integrable by Lax spatially two-dimensional systems is studied. |
| format |
Article |
| author |
Prykarpatsky, Y.A. Samoilenko, A.M. |
| spellingShingle |
Prykarpatsky, Y.A. Samoilenko, A.M. On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions Нелінійні коливання |
| author_facet |
Prykarpatsky, Y.A. Samoilenko, A.M. |
| author_sort |
Prykarpatsky, Y.A. |
| title |
On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions |
| title_short |
On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions |
| title_full |
On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions |
| title_fullStr |
On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions |
| title_full_unstemmed |
On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions |
| title_sort |
on the lagrangian and hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/178006 |
| citation_txt |
On the Lagrangian and Hamiltonian aspects of infinite-dimensional dynamical systems and their finite-dimensional reductions / Y.A. Prykarpatsky, A.M. Samoilenko
// Нелінійні коливання. — 2005. — Т. 8, № 3. — С. 360-387. — Бібліогр.: 41 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT prykarpatskyya onthelagrangianandhamiltonianaspectsofinfinitedimensionaldynamicalsystemsandtheirfinitedimensionalreductions AT samoilenkoam onthelagrangianandhamiltonianaspectsofinfinitedimensionaldynamicalsystemsandtheirfinitedimensionalreductions |
| first_indexed |
2023-10-18T22:44:59Z |
| last_indexed |
2023-10-18T22:44:59Z |
| _version_ |
1796156333158825984 |