Variational integrators for nonlinear electric circuits

The variational framework for linear electric circuits introduced in [1] is extended to general nonlinear circuits. Based on a constrained Lagrangian formulation that takes the basic circuit laws into account the equations of motion of a nonlinear electric circuit are derived. The resulting differential-algebraic system can be reduced by performing the variational principle on a reduced space and regularity conditions for the reduced Lagrangian are presented. A variational integrator for the structure-preserving simulation of nonlinear electric circuits is derived and demonstrated by numerical examples. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilität bei Systemen mit quasizyklischen Koordinaten

Quasicyclic or quasiignorable coordinates occur in mechanical systems with equations of motion of the Lagrangian type and can be interpreted as a generalisation of the well known cyclic or ignorable coordinates. In this paper parts of the theory of cyclic coordinates are extended to quasicyclic coordinates. Especially we present sufficient conditions for the existence of the reduced system and generalize the Routh-Salvadori theorem about the stability of stationary motions.     

On the Jacobi Last Multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé–Gambier classification

We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painlevé equations. Indeed this method yields the Lagrangians of many of the equations of the Painlevé–Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Liénard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of second-order ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian.     

Effective Lagrangians of the Randall–Sundrum Model

We construct the second variation Lagrangian for the Randall*Sundrum model with two branes, study its gauge invariance, and introduce and decouple the corresponding equations of motion. For the physical degrees of freedom in this model, we find the effective four-dimensional Lagrangians describing the massless graviton, massive gravitons, and the massless scalar radion. We show that the masses of these fields and their matter coupling constants are different on the different branes.     

Linearized gravity in the Randall-Sundrum model with stabilized distance between branes

We consider linearized gravity in the Randall-Sundrum model in which the distance between branes is stabilized by introducing the scalar Goldberger-Wise field. We construct the second variation Lagrangian for fluctuations of gravitational and scalar fields over the background solution and investigate its gauge invariance. We obtain, separate, and solve the corresponding equations of motion. For physical degrees of freedom, we obtain the effective four-dimensional Lagrangian describing the massless graviton, massive gravitons, and the set of massive scalar fields. We also find masses and coupling constants of these fields to the matter on the negative-tension brane. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 339–353, December, 2006.     

双荷子系统运动的经典解

根据力学理论和经典电磁理论研究双荷子系统的运动.列出双荷子系统的运动微分方程,导出运动积分,说明系统的对称性,包括SO(4)对称性;利用变分法逆问题方法,构造双荷子系统的Lagrange(拉格朗日)函数和Hamilton(哈密顿)函数;解出双荷子系统的运动规律.     

On the motion of a mechanical system inside a rolling ball

We consider a mechanical system inside a rolling ball and show that if the constraints have spherical symmetry, the equations of motion have Lagrangian form. Without symmetry, this is not true.     

A stochastic Lagrangian representation of the three‐dimensional incompressible Navier‐Stokes equations

In this paper we derive a probabilistic representation of the deterministic three‐dimensional Navier‐Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self‐contained proof of local existence for the nonlinear stochastic system and can be extended to formulate stochastic representations of related hydrodynamic‐type equations, including viscous Burgers equations and Lagrangian‐averaged Navier‐Stokes alpha models. © 2007 Wiley Periodicals, Inc.     

On the Existence and Uniqueness of the Solution to the Cauchy Problem for a System of Integral Equations Describing the Motion of a Rarefied Mass of a Self-Gravitating Gas

Computational Mathematics and Mathematical Physics - The Cauchy problem for a system of nonlinear Volterra-type integral equations that describes, in Lagrangian coordinates, the motion of a finite...     

Principle of stationary action in the theory of superfluid systems with spontaneously broken translational symmetry

We construct a Lagrangian describing the low-frequency dynamics of a system with spontaneously broken phase and translational symmetry (a supersolid). Using the principle of stationary action, we obtain the hydrodynamic equations for the considered system. We give a relativistic generalization of the obtained equations of motion in terms of the the Gibbs thermodynamic potential density.     

Symmetries of the Maxwell-Bloch equations with the rotating wave approximation

In this paper a symplectic realization for the Maxwell-Bloch equations with the rotating wave approximation is given, which also leads to a Lagrangian formulation. We show how Lie point symmetries generate a third constant of motion for the dynamical system considered.     

The Hamilton formalism with fractional derivatives

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.     

扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用

首先,我们给出了引入伴随方程(组)扩充原方程(组)的策略使给定偏微分方程(组)的扩充方程组具有对应泛瓯即,成为Lagrange系统的方法,以此为基础提出了作为偏微分方程(组)传统守恒律和对称概念的一种推广-偏微分方程(组)扩充守恒律和扩充对称的概念;其次,以得到的Lagrange系统为基础给定了确定原方程(组)扩充守恒律和扩充对称的方法,从而达到扩充给定偏微分方程(组)的首恒律和对称的目的;第三,提出了适用于一般形式微分方程(组)的计算固有守恒律的方法;第四,实现以上算法过程中,我们先把计算(扩充)守恒律和对称问题均归结为求解超定线性齐次偏微分方程组(确定方程组)的问题.然后,对此关键问题我们提出了用微分形式吴方法处理的有效算法;最后,作为方法的应用我们计算确定了非线性电报方程组在内的五个发展方程(组)的新守恒律和对称,同时也说明了方法的有效性.     

Pseudo-Rigid Bodies: A Geometric Lagrangian Approach

The pseudo-rigid body model is viewed within the context of continuum mechanics and elasticity theory. A Lagrangian reduction, based on variational principles, is developed for both anisotropic and isotropic pseudo-rigid bodies. For isotropic Lagrangians, the reduced equations of motion for the pseudo-rigid body are a system of two (coupled) Lax equations on so(3)×so(3) and a second-order differential equation on the set of diagonal matrices with a positive determinant. Several examples of pseudo-rigid bodies such as stretching bodies, spinning gas cloud and Riemann ellipsoids are presented.     

Reduced Variational Formulations in Free Boundary Continuum Mechanics

We present the material, spatial, and convective representations for elasticity and fluids with a free boundary from the Lagrangian reduction point of view, using the material and spatial symmetries of these systems. The associated constrained variational principles are formulated and the resulting equations of motion are deduced. In addition, we introduce general free boundary continua that contain both elasticity and free boundary hydrodynamics, extend for them various classical notions, and present the constrained variational principles and the equations of motion in the three representations.     

Group properties of the extended Green–Naghdi equations

The group analysis method is applied to the extended Green–Naghdi equations. The equations are studied in the Eulerian and Lagrangian coordinates. The complete group classification of the equations is provided. The derived Lie symmetries are used to reduce the equations to ordinary differential equations. For solving the ordinary differential equations the Runge–Kutta methods were applied. Comparisons between solutions of the Green–Naghdi equations and the extended Green–Naghdi equations are given.     

Noether, partial Noether operators and first integrals for a linear system

We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians.     

On the Lagrangian form of the variational equations of Lagrangian dynamical systems

An extremal principle for obtaining the variational equations of a Lagrangian system is reviewed and formalized. Formalization is accomplished by relating the new Lagrangian function γ needed in such scheme to a prolongation of the original Lagrangian L. This formalization may be regarded as a necessary step before using the approach for stablishing nonintegrability of dynamical systems, or before applying it to analyse chaos-producing perturbations of integrable Lagrangian systems. The configuration manifold in which γ is defined is the double tangent bundle T(TQ) of the original configuration manifold Q modulo a flip mapping in such manifold. Our main result establishes that both the Euler–Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the composition of the first prolongation of the original Lagrangian with a flip mapping. Some applications of the approach to chaos and integrability issues are discussed.     

Dirac reduction for nonholonomic mechanical systems and semidirect products

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange–Dirac and Hamilton–Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler–Poincaré–Suslov equations with advected parameters and the implicit Lie–Poisson–Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin–Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.