Gauge invariance,charge conservation,and variational principles

We present new results on the correspondence between symmetries, conservation laws and variational principles for field equations in general non-abelian gauge theories. Our main result states that second order field equations possessing translational and gauge symmetries and the corresponding conservation laws are always derivable from a variational principle. We also show by the way of examples that the above result fails in general for third order field equations.     

Water wave theories and variational principles

Summary A unified account of the most outstanding models for water wave propagation is given. A first scheme assembles the shallow-water equations, the Boussinesq equations and the Korteweg-de Vries equation as particular cases of the Green and Naghdi equations. Instead, Benjamin-Bona-Mahony’s and Jeffrey’s equations cannot be set in such a scheme: this is ascribed to their non-Galileian invariance. A second scheme emphasizes the different peculiarities of the various models through the corresponding variational formulations. In this context a variational principle for Green and Naghdi’s equations is set up.

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Riassunto Si presenta una trattazione compatta delle più note equazioni che descrivono la propagazione delle onde d’acqua. Un primo schema riunisce le equazioni dell’acqua bassa, di Boussinesq e di Korteweg-de Vries come casi particolari delle equazioni di Green e Naghdi. Per contro, l’equazione di Benjamin-Bona-Mahony e quella di Jeffrey non rientrano in tale schema: ciò è attribuito al fatto che esse non sono invarianti per trasformazioni di Galileo. Un secondo schema mette in rilievo le diverse caratteristiche dei vari modelli tramite le corrispondenti formulazioni variazionali. In questo contesto si fornisce un principio variazionale per le equazioni di Green e Naghdi.

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Резюме Предлагается единое описание моделей для распространения водяных волн. Первая схема описывает уравнения мелкой воды, уравнения Буссинэ и уравнение Кортевега-де Вриса, как частные случаи уравнений Грина и Нагди. Уравнения Бенджамина-Бона-Махони и Джефри не могут Быть получены в такой схеме: это приписывается негалилеевой инвариантнности этих уравнений. Вторая схема придает особое значение особенностям различных моделей посредством соответствующих вариационных формулировок. В этом контексте устанавливается вариационный принцип для уравнений Грина и Нагди.

     

不同积分变分原理的统一

依据定量因果原理的数学表示，统一地导出了Lagrange量中含坐标关于时间一阶、二阶导数 的积分型的Hamilton原理、Voss原理、Hlder原理和Maupertuis-Lagrange原理等，给出了 这些原理的本质联系和统一描述.得出f₀=0并不是通常的保持Euler-Lagrange方 程不 变的结果，而是满足定量因果原理的结果.还得出Lagrange量的所有的积分型变分原理等价 地对应于两类满足定量因果原理的不变形式.同时发现所有积分型变分原理的运动方程都是E uler-Lagrange 方程，但不同条件的变分原理所对应的不同群Ｇ作用下的守恒量是不同 的.从而可对过去众多零散的积分型变分原理有一个系统和深入的理解，并使这些变分原理 自然地成为定量因果原理的推论. 关键词： 变分原理 因果原理 运动方程 对称性     

Entropy,limit theorems,and variational principles for disordered lattice systems

We study infinite volume limits and Gibbs states of disordered lattice systems with bounded and continuous potentials. Our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes. We find that many familiar results of invariant potentials, such as large deviation theorems, variational principles, and equivalence of ensembles, continue to hold for disordered models, with suitably modified statements.     

General Zakharov-Shabat equations,multi-time Hamiltonian formalism,and constants of motion

We construct a Hamiltonian formalism for general Zakharov-Shabat equations (zero curvature equations with rational dependence on a parameter) as well as their constants of motion, and prove that the latter are in involution. The field-theoretical (multi-time) Hamiltonian formalism is used.Supported by National Science Foundation, Research Grant DMS 8703407.     

Non-invariance symmetries and constants of the motion

It is shown that conserved quantities for lagrangian systems can be associated with symmetry groups which do not leave the action integral invariant. An analogy with a recently described invariant for hamiltonian systems is pointed out.     

Stability of general relativistic gaseous masses and variational principles

The Einstein field equations for a self-gravitating fluid that obeys an equation of state of the formp=p(w),p the pressure andw the energy density may be derived from a variational principle. The perturbations of the metric tensor and the fluid dynamic variables satisfy equations which may be derived from a related variational principle, namely the principle associated with the second variation problem. It is shown that the variational principle given by Chandrasekhar from which a sufficient criterion may be obtained for deciding when a self gravitating spherical gaseous mass is unstable against spherically symmetric perturbations is that given by the second variation problem. It is further shown that this criterion is equivalent to requiring that the integral entering into the second variation be negative. The latter form of the criterion may be used in general situations.This work was supported in part by the United States Atomic Energy Commission under contract number AT (11-1)-34, Project Agreement Number 125.     

On variational principles of relativistic theories involving torsion

We consider variational principles that are distinct from the formulation of Hehl et al. which reproduce their field equations in reductions of U₄. The results are at odds with conclusions which appeared in a recent letter by Safko et al.     

On the interior scattering of waves,defined by hyperbolic variational principles

The Transformation of waves of different kinds at some interior points of nonhomogeneous media is studied for waves defined by linear hyperbolic variational systems.A formal normal form is given for the light hypersurface (i.e. for the dispersion relation) at its generic singular points in terms of the contact geometry. This normal form describes the behaviour of the rays and of the wavefronts at the singular points corresponding to the multiple eigenvalues of the principal symbol of a generic hyperbolic variational problem.     

Formally exact quantum variational principles for collective motion based on the invariance principle of the Schrödinger equation

Utilizing the concept of invariant collective subspace of a many-body system (or invariance principle of the time-dependent Schrödinger equation), we derive a number of formally exact variational principles to characterize the subspace. Previous studies based on time-dependent or adiabatic time-dependent Hartree-Fock theory are, in principle, contained as approximations.     

Arcs of discrete dynamics and constants of motion

A theorem is proved which may relate the attractors of families of dissipative discrete-time dynamical systems to certain closed orbits of conservative systems. The result is illustrated by an example taken from dynamics defined by mappings.     

On the motion of particles and strings,presymplectic mechanics,and the variational bicomplex

Examples of equations of motion in classical relativistic mechanics are studied: the equations of motion of a charged spinning particle moving in a space-time (with or without torsion) in the presence of an electromagnetic field are derived via Souriau presymplectic reduction. Then, the extension of Souriaus ideas to Lagrangian field theory due to Witten, Crnkovi, Zuckerman is reviewed using the variational bicomplex, the basic properties of the Lund–Regge equations describing the motion of a string interacting with a scalar field and moving in Minkowski spacetime are recalled, and a symplectic structure for their space of solutions is found.This revised version was published online in April 2005. The publishing date was inserted.     

Variational principles for collective motion: Relation between invariance principle of the Schrödinger equation and the trace variational principle

The invariance principle of the Schrödinger equation provides a basis for theories of collective motion with the help of the time-dependent variational principle. It is formulated here with maximum generality, requiring only the motion of intrinsic state in the collective space. Special cases arise when the trial vector is generalized coherent state and when it is a uniform superposition of collective eigenstates. The latter example yields variational principles uncovered previously only within the framework of the equations of motion method.     

Practical use of variational principles for modeling water waves

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a ‘relaxed’ variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving a variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact traveling wave solutions are constructed when possible.     

三类大气基本波动的广义变分原理

半反推法是He为了寻求数学物理问题的变分原理而提出的, 可避免由拉氏乘子法引起的临界变分现象. 应用半反推法分别获得了动力气象中Rossby波、大气声波和重力外波等三类基本大气波动的广义变分原理, 并验证了它们的正确性.