Cristel Chandre

On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets

The role of projectors associated with Poisson brackets of constrained Hamiltonian systems is analyzed. Projectors act in two instances in a bracket: in the explicit dependence on the variables and in the computation of the functional derivatives. The role of these projectors is investigated by using Dirac’s theory of constrained Hamiltonian systems. Results are illustrated by three examples taken from plasma physics: magnetohydrodynamics, the Vlasov–Maxwell system, and the linear two-species Vlasov system with quasineutrality.

Abundance of Regular Orbits and Nonequilibrium Phase Transitions in the Thermodynamic Limit for Long-Range Systems

We investigate the dynamics of many-body long-range interacting systems, taking the Hamiltonian mean-field model as a case study. We show that regular trajectories, associated with invariant tori of the single-particle dynamics, prevail. The presence of such tori provides a dynamical interpretation of the emergence of long-lasting out-of-equilibrium regimes observed generically in long-range systems. This is alternative to a previous statistical mechanics approach to such phenomena which was based on a maximum entropy principle. Previously detected out-of-equilibrium phase transitions are also reinterpreted within this framework.

Control of stochasticity in magnetic field lines

We present a method of control which is able to create barriers to magnetic field line diffusion by a small modification of the magnetic perturbation. This method of control is based on a localized control of chaos in Hamiltonian systems. The aim is to modify the perturbation (of order e) locally by a small control term (of order e2) which creates invariant tori acting as barriers to diffusion for Hamiltonian systems with two degrees of freedom. The location of the invariant torus is enforced in the vicinity of the chosen target (at a distance of order e due to the angle dependence). Given the importance of confinement in magnetic fusion devices, the method is applied to two examples with a loss of magnetic confinement. In the case of locked tearing modes, an invariant torus can be restored that aims at showing the current quench and therefore the generation of runaway electrons. In the second case, the method is applied to the control of stochastic boundaries allowing one to define a transport barrier within the stochastic boundary and therefore to monitor the volume of closed field lines.

Controlling chaotic transport in a Hamiltonian model of interest to magnetized plasmas

We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a model that reproduces turbulent E × B drift and show numerically that the control is able to drastically reduce chaotic transport.

On the Hamiltonian formulation of incompressible ideal fluids and magnetohydrodynamics via Dirac's theory of constraints

Abstract The Hamiltonian structures of the incompressible ideal fluid, including entropy advection, and magnetohydrodynamics are investigated by making use of Diracʼs theory of constrained Hamiltonian systems. A Dirac bracket for these systems is constructed by assuming a primary constraint of constant density. The resulting bracket is seen to naturally project onto solenoidal velocity fields.

Dynamical bottlenecks to intramolecular energy flow.

Vibrational energy flows unevenly in molecules, repeatedly going back and forth between trapping and roaming. We identify bottlenecks between diffusive and chaotic behavior, and describe generic mechanisms of these transitions, taking the carbonyl sulfide molecule OCS as a case study. The bottlenecks are found to be lower-dimensional tori; their bifurcations and unstable manifolds govern the transition mechanisms.

Hamiltonian derivation of the Charney-Hasegawa-Mima equation

The Charney–Hasegawa–Mima equation is an infinite-dimensional Hamiltonian system with dynamics generated by a noncanonical Poisson bracket. Here a first principle Hamiltonian derivation of this system, beginning with the ion fluid dynamics and its known Hamiltonian form, is given.

Newton's descent method for the determination of invariant tori

We formulate a variational fictitious-time flow which drives an initial guess torus to a torus invariant under given dynamics. The method is general and applies in principle to continuous time flows and discrete time maps in arbitrary dimension, and to both Hamiltonian and dissipative systems.

Derivation of reduced two-dimensional fluid models via Dirac’s theory of constrained Hamiltonian systems

We present a Hamiltonian derivation of a class of reduced plasma two-dimensional fluid models, an example being the Charney–Hasegawa–Mima equation. These models are obtained from the same parent Hamiltonian model, which consists of the ion momentum equation coupled to the continuity equation, by imposing dynamical constraints. It is shown that the Poisson bracket associated with these reduced models is the Dirac bracket obtained from the Poisson bracket of the parent model.

Quantum-classical correspondence in circularly polarized high harmonic generation

Using numerical simulations, we show that atomic high order harmonic generation (HHG) with a circularly polarized laser field offers an ideal framework for quantum-classical correspondence in strong field physics. With an appropriate initialization of the system, corresponding to a superposition of ground and excited state(s), simulated HHG spectra display a narrow strip of strong harmonic radiation preceded by a gap of missing harmonics in the lower part of the spectrum. In specific regions of the spectra, HHG tends to lock to circularly polarized harmonic emission. All these properties are shown to be closely related to a set of key classical periodic orbits that organize the recollision dynamics in an intense, circularly polarized field.