Lie transform perturbation theory for Hamiltonian systems

Cary, John R.

doi:10.1016/0370-1573(81)90175-7

Cited by 266 publications

(202 citation statements)

References 10 publications

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“…The KAM theorem provides information about which tori survive a specific perturbation. [25] Van Vleck's formulation of quantum-mechanical perturbation theory, [26,27] which is akin to the Lie transform method, [28] seems to be suitable for the implementation of the diagonalizing unitary transformation U for a one-spin system. Here the strategy is to systematically remove the (off-diagonal) creation and annihilation operators (or spin ladder operators) from the Hamiltonian in increasing orders.…”

Section: Discussionmentioning

confidence: 99%

Quantum integrability and action operators in spin dynamics

Weigert¹,

Müller²

1995

Chaos, Solitons & Fractals

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A new formulation of the quantum integrability condition for spin systems is proposed. It eliminates the ambiguities inherent in formulations derived from a direct transcription of the classical integrability criterion. In the new formulation, quantum integrability of an N -spin system depends on the existence of a unitary transformation which expresses the Hamiltonian as a function of N action operators. All operators are understood to be algebraic expressions of the spin-components with no restriction to any finite-dimensional matrix representation. The consequences of quantum (non-)integrability on the structure of quantum invariants are discussed in comparison with the consequences of classical (non-)integrability on the corresponding classical invariants. Our results indicate that quantum integrability is universal for systems with N = 1 and contingent for systems with N ≥ 2.

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Section: Discussionmentioning

confidence: 99%

Quantum integrability and action operators in spin dynamics

Weigert¹,

Müller²

1995

Chaos, Solitons & Fractals

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“…Our approach is based on a Hamiltonian action-angle formulation with the canonical perturbation method [9] and Lie transform techniques [17] being utilized for the calculation of angle averaged variations of the actions corresponding to the particle parallel and angular momentum as well as its guiding center position. The method naturally couples analytical information on single particle dynamics to the collective particle behavior as described by the aforementioned averaged action variations.…”

Section: Introductionmentioning

confidence: 99%

Interaction of charged particles with localized electrostatic waves in a magnetized plasma

2012

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Charged particle interaction with localized wave packets in a magnetic field is formulated using the canonical perturbation theory and the Lie transform theory. An electrostatic wave packet characterized by a wide range of group and phase velocities as well as spatial extent along and across the magnetic field is considered. The averaged changes in the momentum along the magnetic field, the angular momentum, and the guiding center position for an ensemble of particles due to their interaction with the wave packet are determined analytically. Both resonant and ponderomotive effects are included. For the case of a Gaussian wave packet, closed-form expressions include the dependency of the ensemble averaged particle momenta and gc position variations on wave packet parameters and particle initial conditions. These expressions elucidate the physics of the interaction which is markedly different from the well known case of particle interaction with plane waves and are relevant to a variety of applications ranging from space and astrophysical plasmas to laboratory and fusion plasmas as well as accelerators and microwave devices.

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“…According to the method of Deprit [5], the old Hamiltonian H, the new Hamiltonian K and the transformation T along with the Lie generator w are expanded in power series of ε. The equations providing the Lie generator function at order n can be written in the general form…”

mentioning

confidence: 99%

Nonlinear theory of cyclotron resonant wave-particle interactions: Analytical results beyond the quasilinear approximation

Kominis

2008

Phys. Rev. E

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Lie transform perturbation theory for Hamiltonian systems

Cited by 266 publications

References 10 publications

Quantum integrability and action operators in spin dynamics

Quantum integrability and action operators in spin dynamics

Interaction of charged particles with localized electrostatic waves in a magnetized plasma

Nonlinear theory of cyclotron resonant wave-particle interactions: Analytical results beyond the quasilinear approximation

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