Lie Transform Perturbation Theory

1999 ◽  
pp. 125-178 ◽  
Author(s):  
Dino Boccaletti ◽  
Giuseppe Pucacco
Keyword(s):  
Perturbation Theory ◽  
Lie Transform

Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation

2001 ◽  
Author(s):  
Randolph S. Zounes ◽  
Richard H. Rand
Keyword(s):  
Perturbation Theory ◽  
Nonlinear Term ◽  
Mathieu Equation ◽  
Elliptic Functions ◽  
Action Space ◽  
Global Behavior ◽  
Trigonometric Functions ◽  
Lie Transform ◽  
Analytic Expressions ◽  
Subharmonic Resonances

Abstract We investigate the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation,(1)x..+[δ+ϵ(cos⁡ω1t+cos⁡ω2t)]x+αx3=0. We assume that ϵ ≪ 1 and that the coefficient of the nonlinear term, α, is positive but not necessarily small. We utilize Lie transform perturbation theory with elliptic functions — rather than the usual trigonometric functions — to study subharmonic resonances associated with orbits in 2m : 1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (δ, ϵ, ω1, ω2) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov’s overlap criterion to obtain an overview of the O(ϵ) global behavior of equation (1) as a function of δ and ω2 with ω1, α, and ϵ fixed.

Oscillation centres and mode coupling in non-uniform Vlasov plasma

1979 ◽  
Vol 22 (1) ◽  
pp. 105-119 ◽  
Author(s):  
Shayne Johnston ◽  
Allan N. Kaufman
Keyword(s):  
Perturbation Theory ◽  
Single Particle ◽  
Mode Coupling ◽  
Poisson Brackets ◽  
Lie Transform ◽  
Time Integral ◽  
Compact Expression ◽  
Hamiltonian Perturbation Theory ◽  
Insight Into ◽  
Vlasov Plasma

The general coupling coefficient for three electromagnetic linear modes of an inhomogeneous and relativistic plasma is derived from the oscillation-centre viewpoint. A concise and manifestly symmetric formula is obtained; it is cast in terms of Poisson brackets of the single-particle perturbation Hamiltonian and its convective time-integral along unperturbed orbits. The simplicity of the compact expression obtained is shown to lead to a new insight into the essence of three-wave coupling and of the Manley–Rowe relations governing such interactions. Thus, the interaction Hamiltonian of the three waves is identified as simply the trilinear contribution to the single-particle (new) Hamiltonian, summed over all non-resonant particles. The relation between this work and the Lie-transform approach to Hamiltonian perturbation theory is discussed.

Covariant ponderomotive Hamiltonian

1986 ◽  
Vol 35 (2) ◽  
pp. 257-266 ◽  
Author(s):  
A. Achterberg
Keyword(s):  
Perturbation Theory ◽  
Covariant Formulation ◽  
Lie Transform

A covariant formulation for the ponderomotive Hamiltonian is developed using Lie-transform perturbation theory. The case of unmagnetized as well as magnetized particles is discussed.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

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