from a general mechanical point of view you can say that the Hamiltonian reduces the order of your differential equations of motion which makes the calculation considerably easier (or sometimes which makes the calculations possible at all).

but the integrability of the Hamiltonian equation is strongly affected by the degree of freedom of the physical system and many of physical systems have infinite number of degree of freedom and confined plasma system is one of them.???

True, that is another point which rapidly increases your calculation time (however no real physical system has infinite number of degrees of freedom, although the number is usually very high).

But consider this: if you are dealing with the Lagrangian equations of motion you end up with n coupled differential equations of second order in time (for n degrees of freedom). However, if you are using the Hamiltonian you have 2n coupled differential equations, but of order one - so you get much simpler diff-equations, but a larger number of them.

If you are dealing with a large number of particles than I would rather apply fluid models instead of treating each particle separately - you will loose some accuracy due to the averaging process that comes with fluid models, but the calculations will be simplified (the big disadvantage is that you may loose some information about the motions of individual particles which might just be the information you need for describing your turbulences...)

weak turbulence theory is based on Hamiltonian formulation, involving three or four wave resonant interactions. Sagdeev and Galeev (1969) give an introduction in their book on nonlinear plasma physics. Zakharov also developed much of the theory.

Applications: weak turbulence in fluid mechanics (e.g. water waves) and in plasma

turbulence.

So, what about anomalous diffusion???

Dear Hamdi Abd-El-Hamid, I would like to give you an extra answer to your question.

As a matter of fact, an appear of hamiltonian approaches to modeling nonlinear plasma phenomena was conditioned by traditions of plasma theory, and the latter were natural to the bed of development of theoretical physics in the lapsed XX-th century. Meanwhile, it is just the plasma theoretical studies that have helped to uncover that the very above traditions do insufficiently well comply with basic goal of physical theoretical researches, namely, with a desire to formulate possibly more informative conclusions regarding physical phenomena in surrounding world. Let me expose you the basic argumentation in support of this statement, and also unroll a bit more its content.

In the nuclear fusion research, the most natural basis of plasma consideration constitute the full plasma description. It is given by simultaneous Maxwell equations and Klimontovich-Dupree equation: They contain in cumulative form the classical motion equations of all individual charged plasma particles. (I comment that respective plasmas have high temperatures, therefore the notion of particle trajectory is well consistent.) However, the full plasma description is not constructive: its equations cannot be integrated to full extent. (The reason is that, on the one side, the data on initial positions and momentums of individual plasma particles are never known to full extent. On the other, even had it been known, there were not possible to integrate the equations from the technical viewpoint, because of immensely large numbers of charged particles.) In view of this, plasma theorists should reduce the full plasma description to more easy models of plasma kinetics. Just the simplified models of plasma kinetic were underlain, particularly, both the modeling of the nonlinear plasma phenomena by a Hamiltonian and the machinery of theoretical consideration of the anomalous (say, neoclassical) transport phenomena in ionized plasmas. Meanwhile, traditional practice of reducing full plasma description to simplified plasma kinetic models was formed without attention to the most important aspects of the problem. They can be clarified as follows.

Note that none of constructive plasma descriptions (i.e., the inevitably simplified ones as compared to the full plasma description) can give a scenario of the plasma macrophysical evolution that does not diverge over time from real physical picture of the plasma evolution. (Recall above mentioned incompleteness of initial data of plasma particles and the technical impossibility of its full-scale account.) Meanwhile, the plasma macrophysical evolution constitute the key interest in plasma research. Therefore, an extremely important aspect of a theory constitutes the degree of adequateness of its prognoses on behavior of evolving plasmas to real pictures of their macrophysical evolutions. I introduce there the category INFORMATIVENESS: The longer the theoretical scenario of the plasma evolution depicts the real picture of the macrophysical evolution of the plasma, the higher an estimate the researcher should suggest for the scenario informativeness. Respectively, the heightening of informativeness of plasma scenario to real plasma physical phenomena in surrounding world should constitute the leading motif of the theory development. It supposes an extremely careful separation of the theory informational basis from the full (unknown) plasma initial data. In the latter respect, traditional methods of developing simplified plasma kinetic models were formulated without proper understanding of two key restrictions which resulted in that respective plasma kinetic models happen to possess by an inappropriately low informativeness. To enlighten the latter to a greater degree, it is sufficient to note that the recipes of traditional theory yield equally rigorous justifications to incompatible versions of the same physical phenomenon. For instance, it helps to substantiate both the thesis of conservation of Langmuir wave quanta in a weakly turbulent plasma and the thesis of an intense wave quanta decay with thermalization of Langmuir wave energy via the stochastic plasma electron acceleration.

Now - the very shortcomings of traditional approaches. First of them consists in culturally conditioned practice of substituting original plasmas by probabilistic plasma ensembles. It was accepted after the success of the gibbsian statistical thermodynamics: Deductions on reciprocal influence of plasma ensemble statistics were commonly believed to comprise objective laws of the plasma macrophysical evolution. In reality, the ensemble substitution generally obscures the picture of the plasma physical evolution. Following to the most elementary common sense, laws of evolution of plasma ensemble statistics essentially depend on the ensemble content and therefore cannot be regarded as objective laws of the plasma physical evolution.

Second reason of non-informativeness of traditionally developed plasma physical scenarios stems from an absence of adequate understanding of essence and significance of THE ASYMPTOTIC NATURE OF THEORY CONVERGENCE. The researcher inevitably generates some nonlinear successive perturbations in the process of reducing full plasma description to simpler models of plasma macrophysical behavior. Regardless their essence, they may converge at best ASYMPTOTICALLY only: The heightening of the order of consideration entails the factorial growth of number of new summands in the perturbation theory, and after some order the factorial growth of this number outweighs the possible power-law decrease of individual summands. Respectively, one may rigorously deduce diverse pictures of the plasma physical evolution even using a single perturbation theory, having merely varied its lowest order approximation. The point is that the choice of the theory leading order specifies respective conditional limit of converging (first coming) sequential orders of the theory, whereas the latter comprises corresponding version of the plasma physical scenario.

Two above reasons of theory non-informativeness cannot be separated (without unrolling of the thesis in these my explanations).

Having formulated the problem of heightening the informativeness of plasma theoretical deductions, I have clarified basic principles for its solving. First, the researcher should refrain from the traditional plasma ensemble substitution.

Such a refrain forces to modify the concept of basic objects of the theory, THE DISTRIBUTION FUNCTIONS OF PLASMA PARTICLES. Mathematically, it represents some STATISTIC of distribution of discrete charged particles in phase space of particle positions and momentums. Usual approaches implied the

developing of such a statistic, the distribution function of a Vlasovian type, just via the ensemble averaging of its counterpart from full plasma description (i.e., from the Klimontovich's distribution function of charged plasma particles). The only constructive way to refrain from the ensemble averaging consists in its substitution by a contextually oriented averaging in phase space of positions and momentums of plasma particles. An appropriate arrangement of the averaging depends essentially on the physical problem under consideration. In the case of a homogeneous plasma with weakly turbulent wave fields one can average over 6-dimensional parallelepipeds with extended spatial dimensions: at the expense of these dimensions one can reach appropriate small momentum gradations of a statistically reliable particle distribution. The consideration of plasma leakage from a tokamak dictates toroidal geometry of spatial projection of a 6-dimensional averaging volume. Naturally, other physical situations shall dictate their own geometrical aspects of the phase space averaging.

Second principle of developing informative plasma kinetic scenarios is that the researcher should develop successive iterations of plasma scenario using A DIRECT TIME INTEGRATION of necessary evolutional equations: It is on this path that he can properly account for an available information on current plasma state and its recent history and simultaneously discriminate an indeterminate information on time remote plasma states.

Finally, a CRUCIAL position in developing informative models of plasma physical evolution is that its first sequential iterations may result in heightening the accuracy of plasma scenario only when the characteristic expansion parameter of the perturbation theory is small (here I also will not unroll the thesis to a greater extent.)

The practice of plasma studies consists of many activities whereat the heightening of informativeness of plasma scenarios is strongly desirable. Particularly, just in theoretical studies of plasmas in tokamaks and stellarators that seems to interest you first of all.

From my above narration understand please that the very question "What is the advantage of Hamiltonian mechanics in describing transport and turbulence in magnetized plasma" is not an important one. Functionally, the hamiltonian approaches substantially simplify considerations of nonlinear plasma phenomena. However, respective simplified considerations have no scientific value, since they cannot lead to appropriately informative pictures of plasma physical evolution. For instance, the hamiltonian formulation of weak plasma turbulence theory does not complies with real macrophysical behavior of weakly turbulent plasmas. Respective truth you may learn only from my papers, ones that were published after 2000. Basically, I have considered a number of phenomena in a homogeneous plasma with weak Langmuir turbulence and have shown that many traditional understandings of respective physics are no more than myths. I have developed for my consideration a new approach that can be well categorized as A HIGH-INFORMATIVE CORRELATION ANALYSIS OF PLASMA KINETICS. My respective studies, and the very technique of plasma kinetic modelling, are exposed in a systematic fashion in my monograph "High-Informative Plasma Theory" (LAP, Saarbrucken, 2011). Naturally, I am performing further on my studies on informative versions of nonlinear phenomena in weakly turbulent plasmas, and above my monograph is a bit outdated. The further extension of my high-informative correlation analysis is reported in new paper, "A High-Informative Version of Nonlinear Transformation of Langmuir Waves to Electromagnetic Waves" (DOI: 10.1017/S002237781300127X, URL = http://journals.cambridge.org/article_S002237781300127X). I invite you and other young plasma theorists to learn my ideas and approaches and to adopt them to plasma contexts differing from that of weakly turbulent plasmas. Particularly, this can be said about the problems of transport phenomena. I suspect that consideration of plasma transport phenomena on a basis of new ideas may substantially modify final conclusions.