Infinite-dimensional Hamilton-Jacobi theory and L-integrability
نویسنده
چکیده
The classical Liouvile integrability means that there exist n independent first integrals in involution for 2n-dimensional phase space. However, in the infinite-dimensional case, an infinite number of independent first integrals in involution don’t indicate that the system is solvable. How many first integrals do we need in order to make the system solvable? To answer the question, we obtain an infinite dimensional Hamilton-Jacobi theory, and prove an infinite dimensional Liouville theorem. Based on the theorem, we give a modified definition of the Liouville integrability in infinite dimension. We call it the L-integrability. As examples, we prove that the string vibration equation and the KdV equation are L-integrable. In general, we show that an infinite number of integrals is complete if all action variables of a Hamilton system can reconstructed by the set of first integrals.
منابع مشابه
Hamilton-Jacobi formalism of the massive Yang-Mills theory revisited
Using Hamilton-Jacobi formalism we investigated the massive Yang-Mills theory on both extended and reduced phase-space. The integrability conditions were discussed and the actions were calculated.
متن کاملIntegrability of the Reduction Fourth-Order Eigenvalue Problem
To study the reduced fourth-order eigenvalue problem, the Bargmann constraint of this problem has been given, and the associated Lax pairs have been nonlineared. By means of the viewpoint of Hamilton mechanics, the Euler-Lagrange function and the Legendre transformations have been derived, and a reasonable Jacobi-Ostrogradsky coordinate system has been found. Then, the Hamiltonian cannonical co...
متن کاملIsospectral Deformations of Random Jacobi Operators
We show the integrability of infinite dimensional Hamiltonian systems obtained by making isospectral deformations of random Jacobi operators over an abstract dynamical system. The time 1 map of these so called random Toda flows can be expressed by a QR decomposition.
متن کاملSupersymmetry versus Integrability in two-dimensional Classical Mechanics
Supersymmetric extensions of Hamilton-Jacobi separable Liouville mechanical systems with two degrees of freedom are defined. It is shown that supersymmetry can be implemented in this type of systems in two independent ways. The structure of the constants of motion is unveiled and the entanglement between integrability and supersymmetry is explored.
متن کاملReparametrization invariance and Hamilton-Jacobi formalism
Systems invariant under the reparametrization of time were treated as constrained systems within Hamilton-Jacobi formalism. After imposing the integrability conditions the time-dependent Schrödinger equation was obtained. Three examples are investigated in details. PACS numbers: 11.10.Ef. Lagrangian and Hamiltonian approach
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009