Applications Of Classical Particle-field Theory In Plasma Physics

Conservation laws,especially energy and momentum conservations,are funda-mentally applied to plasma physics.In tokamak physics,for example,the exact energy conservation laws can be used to analyze the energy flux and transport property.The mean flows and radial electric field,which are crucial for studying tokamak equilibrium and stability,are mainly determined by the momentum conservation.Furthermore,the exact conservation laws have become a rigorous test on the accuracy and the control for the numerical simulation codes.However,there still doesn't have a systematical and general theory to derive conservation laws for different models of plasma physics.To get the conservation laws of a system,there are two ways to proceed.One may construct a conservation law by guessing conservation and verify it by the equations of motion of the system.However,it is very hard to guess a conservation property for a complex system,for example,the gyrokinetic system.Another approach is to find the symmetry of the action from the viewpoint of the field theory and to determine the conservation laws of the systems by using the Noether's theorem.In this thesis,the latter method is well applied to derive the conservation laws of general plasma physics.Two basic equations are need according to Noether's the-orem:the equation of motion of the system derived from the least action principle or Hamilton's principle and the equation of symmetry called infinitesimal invariant crite-rion.For a single particle system or pure field system,the equation of motion derived from Hamilton's principle is the standard Euler-Lagrange equation.The combination of this equation and the infinitesimal invariant criterion will give us the corresponding conservation law.However,in a plasma system where many particles evolve under the self-generated interacting field,the standard approach is not applicable.In the previous theory,the particles are described by the distribution function which inevitably intro-duces Liouville's equation as a constraint.The existence of the constraint makes the variational process hard to deal with and it is not easy to popularize.To avoid the constraint,the particle-field theory is employed.To study the particle-field model,fundamental mathematics is introduced.Most of the mathematic concepts are from differential geometry,which is manifold,tensor field,Lie group,Lie algebra,especially fiber bundle,cross section,jet bundle,jet space and prolonga-tion of the vector field.By these concepts of differential geometry,the action of the particle-field system can be regarded as the functional of the cross section of particles'trajectories instead of the distribution function.The particle's trajector just depends on one parameter whereas the field depends on space-time which is four parameters,or in the fiber language,the different dimensions of the base manifold of the cross section for particle's trajector and field,which make the standard Euler-Lagrange equation is not applicable for Noether's theorem.What we have discovered is that when the standard Euler-Lagrange equation breaks down,the field equations of these systems assume a more general form that can be viewed as a weak Euler-Lagrange equation.It's a pleas-ant surprise to find out that this weak Euler-Lagrange equation can also link symmetries with conservation laws as in the standard field theory.Using the general theory we developed here,the conservation laws and sym-metries for reduced models in plasma physics such as Klimontovich-Poisson(KP),Klimontovich-Darwin(KD)and the general gyrokinetic systems are systematically studied.As a benchmark of the general theory we developed here,we first calculate the energy,momentum and angular momentum conservation laws for the KP system,which are agree with the results shown in the existed reference.Similarly,for KD sys-tem,we also calculate the energy,momentum and angular momentum conservation laws.According to the KD system is gauge dependent and the gauge cannot be cho-sen arbitrarily,the momentum conservation laws given by Kaufman is then incorrect.Moreover,we obtain general conservation laws for gyrokinetic systems.With the best of our knowledge,there isn't exist the general theory to calculate the corresponding con-servation laws for previous work.Particularly,we obtained the energy and momentum conservation laws for the second-order gyrokinetic model.The theory we developed here is also suitable for relativistic particle-field sys-tems when an inertial frame of reference is chosen.However,if the action and the theory are required to be manifestly covariant,then the mass shell constraint has occurred.To deal with this constraint,the manifestly covariant weak Euer-Lagrange equation instead of the standard weak Euer-Lagrange equation is developed.The original infinitesimal invariant criterion is replaced by two manifestly covariant conditions,called manifestly covariant criterions,where the other criterion is to satisfy the mass shell constraint.Us-ing the manifestly covariant weak Euer-Lagrange equation,conservation laws can be systematically derived from the underlying manifestly covariant criterions.Based on the general theory,the link between symmetries and conservation for the relativistic particle-electromagnetic-field system is constructed.Especially,the manifestly covari-ant energy-momentum tensor is derived.