TITLE
	Hamiltonian canonical formulation in Hall-MHD: generalized Clebsch variables
	
AUTHOR
	Fouad SAHRAOUI

ABSTRACT 
	The different levels of description of fluid media (e.g. MHD, Hall-MHD, bi-fluid,...) 
	are commonly known under the form of Newtonian systems of equations. Nevertheless, this 
	form proves to be in general inconvenient for applying some general mathematical treatments 
	such as deriving the so-called "weak (or wave) turbulence theories" of these 
	media. We show that the Hamiltonian formalism is the most powerful mathematical frame for 
	such studies. We therefore look for Hamiltonian formulations for the different levels of 
	the fluid description of a plasma, using the variational principle with the appropriate 
	Lagrangian invariants. Starting from the bi-fluid system, we show that such a formulation 
	can be obtained by combining the Lagrangians already used for describing: i) the motion of 
	a charged particle in an electromagnetic field; ii) the evolution of an electromagnetic field 
	in presence of sources; iii) the motion of a neutral fluid (Clebsch variables). The result is 
	obtained in terms of generalized-Clebsch variables. Reducing this bi-fluid Hamiltonian 
	formulation to lower orders of the fluid approximations is then shown to be mandatory to obtain 
 	analytical results for linear waves and non linear wave-wave couplings. We show how this goal 
 	can be reached in two steps, leading to: i) a "Reduced Bi-Fluid" system, with the 
 	displacement current neglected and ii) the Hall-MHD system, with the electron mass neglected. We 
 	show that the only generalized Clebsch variables of each species are sufficient to describe the 
 	full Hall-MHD dynamics. Some future applications of such a powerful formalism are outlined.