A Plethora of Integrable Bi-Hamiltonian Equations

1997 ◽  
pp. 93-101 ◽  
Author(s):  
A. S. Fokas ◽  
P. J. Olver ◽  
P. Rosenau
Keyword(s):  
Hamiltonian Equations

A HIERARCHY OF HAMILTONIAN EQUATIONS ASSOCIATED WITH THE MODIFIED JAULENT–MIODEK HIERARCHY

2010 ◽  
Vol 24 (02) ◽  
pp. 183-193
Author(s):  
HAI-YONG DING ◽  
HONG-XIANG YANG ◽  
YE-PENG SUN ◽  
LI-LI ZHU
Keyword(s):  
Eigenvalue Problem ◽  
Evolution Equations ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Matrix Eigenvalue Problem ◽  
Hamiltonian Structures ◽  
Hamiltonian Equations ◽  
Matrix Eigenvalue ◽  
Integrable Evolution

By considering a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is presented.

Ergodic properties and Weyl M-functions for random linear Hamiltonian systems

2000 ◽  
Vol 130 (5) ◽  
pp. 1045-1079 ◽  
Author(s):  
R. Johnson ◽  
S. Novo ◽  
R. Obaya
Keyword(s):  
Hamiltonian Systems ◽  
Qualitative Description ◽  
Invariant Region ◽  
Absolutely Continuous ◽  
Hamiltonian Equations ◽  
Radial Limits ◽  
Ergodic Properties ◽  
Ergodic Measures ◽  
Linear Hamiltonian Systems ◽  
Dynamical Information

This paper provides a topological and ergodic analysis of random linear Hamiltonian systems. We consider a class of Hamiltonian equations presenting absolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at ±∞ define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrödinger equation at the first point of the spectrum.

Nonlocal fractal calculus based analyses of electrical circuits on fractal set

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rawid Banchuin
Keyword(s):  
Cantor Set ◽  
Source Terms ◽  
Electrical Circuits ◽  
Hamiltonian Equations ◽  
Content Type ◽  
Fractal Calculus ◽  
Lc Circuit ◽  
Lagrangian And Hamiltonian Formalisms ◽  
Rlc Circuits ◽  
First Time

Purpose The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

Hamiltonian Dynamics and Phase Space

2019 ◽  
pp. 83-108
Author(s):  
David D. Nolte
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Hamiltonian Dynamics ◽  
Orbital Dynamics ◽  
Legendre Transform ◽  
Lagrange Equations ◽  
Conservation Of Energy ◽  
Hamiltonian Equations ◽  
Generalized Coordinates ◽  
Conservation Of Momentum

Hamiltonian dynamics are derived from the Lagrange equations through the Legendre Transform that expresses the equations of dynamics in terms of the Hamiltonian, which is a function of the generalized coordinates and of their conjugate momenta. Consequences of the Lagrangian and Hamiltonian equations of dynamics are conservation of energy and conservation of momentum, with applications to collisions and orbital dynamics. Action-angle coordinates can be defined for integrable Hamiltonian systems and reduce all dynamical motions to phase space trajectories on a hyperdimensional torus.

scholarly journals Invariance and first integrals of continuous and discrete Hamiltonian equations

2009 ◽  
Vol 66 (1-3) ◽  
pp. 253-270 ◽  
Author(s):  
Vladimir Dorodnitsyn ◽  
Roman Kozlov
Keyword(s):  
First Integrals ◽  
Hamiltonian Equations
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