We consider the self-action problem in classical electrodynamics of a point-like charge arbitrarily moving in flat space-time of four or six dimensions. A consistent regularization procedure is proposed which exploits the symmetry properties of the theory. The energy-momentum and angular momentum balance equations allow us to derive the radiation reaction forces in both 4D and 6D. It is shown that a point-like source in 6D possesses an internal angular momentum with magnitude which is proportional to the square of acceleration. 6D action functional contains, apart from usual "bare" mass, an additional renormalization constant which corresponds to the curvature of the world line (i.e. to the magnitude of internal angular momentum of "bare" particle). It is demonstrated that the {\it Poincaré-invariant} six-dimensional electrodynamics is renormalizable theory.
Conservation Laws of Multidimensional Diffusion-Convection Equations
利用直接方法和复合变分原理构造了$n$维扩散-对流方程$u_t=(A(u))_{ii}+(B^i(u))_i$的所有可能的线性无关局部守恒律.守恒律分类法在点变换群上的应用[R. O. Popovych，新墨西哥州Ivanova，J. Math. Phys.，2005，V.46，043502（math-ph/0407008）]允许我们以显式封闭形式来公式化结果。研究了对称群对扩散方程守恒律的作用，构造了守恒律的生成元集。
In the present paper we consider nonlinear multidimensional Cahn-Hilliard and Kuramoto-Sivashinsky equations that have many important applications in physics and chemistry, and a certain natural generalization of these equations. For an arbitrary number of spatial independent variables we present a complete list of cases when the generalized Cahn-Hilliard-Kuramoto-Sivashinsky equation admits nontrivial local conservation laws of any order, and for each of those cases we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under stu…
We characterize all cases when a certain natural generalization of the Infeld--Rowlands equation admits nontrivial local conservation laws of any order, and give explicit form of these conservation laws modulo trivial ones. Furthermore, we prove that the equation under study admits no nontrivial local Hamiltonian and symplectic structures and no nontrivial local Noether and inverse Noether operators; the method of establishing the said nonexistence results can be readily applied to many other PDEs.
A recent paper considered symmetries and conservation laws of the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. This paper analyses the one-dimensional magnetohydrodynamics flows with cylindrical symmetry in the mass Lagrangian coordinates. The medium is assumed inviscid and thermally non-conducting. It is modeled by a polytropic gas. Symmetries and conservation laws are found. The cases of finite and infinite electric conductivity need to be analyzed separately. For finite electric conductivity $σ(ρ,p)$ we perform Lie group classification, which iden…
We find noncommutative analogs for well-known polynomial evolution systems with higher conservation laws and symmetries. The integrability of obtained non-Abelian systems is justified by explicit zero curvature representations with spectral parameter.
We consider a new nonlocal formulation of the water-wave problem for a free surface with an irrotational flow based on the work of Ablowitz, Fokas, and Musslimani and presented in the recent work of Oliveras. The main focus of the short paper is to show how one can systematically derive Olver's eight conservation laws not only for an irrotational fluid, but also for constant vorticity (linear shear flow) without explicitly relying on the underlying Lie symmetries. This allows us to make draw new conclusions about conservation laws and posit the existence of additional, nonlocal, conservation l…
For systems of partial differential equations in three spatial dimensions, dynamical conservation laws holding on volumes, surfaces, and curves, as well as topological conservation laws holding on surfaces and curves, are studied in a unified framework. Both global and local formulations of these different conservation laws are discussed, including the forms of global constants of motion. The main results consist of providing an explicit characterization for when two conservation laws are locally or globally equivalent, and for when a conservation law is locally or globally trivial, as well as…
We study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are "flux--gradient flows". We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow and Burgers' equation, are presented. Incorporating both primal-dual algorithms and monotone schemes, we successfully com…
