Hamilton formalism and Noether symmetry for mechanico electrical systems with fractional derivatives

This paper presents extensions to the traditional calculus of variations for mechanico-electrical systems containing fractional derivatives. The Euler Lagrange equations and the Hamilton formalism of the mechanico-electrical systems with fractional derivatives are established. The definition and the criteria for the fractional generalized Noether quasi- symmetry are presented. Furthermore, the fractional Noether theorem and conseved quantities of the systems are obtained by virtue of the invariance of the Hamiltonian action under the infinitesimal transformations. An example is presented to illustrate the application of the results.

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

A time-dependent mild-slope equation for the extension of the classic mild-slope equation of Berkhoff is developed for the interactions of large ambient currents and waves propagating over an uneven bottom, using a Hamiltonian formulation for irrotational motions. The bottom topography consists of two components the slowly varying component which satisfies the mild-slope approximation, and the fast varying component with wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter describing the mild-slope condition. The theory is more widely applicable and contains as special cases the following famous mild-slope type equations: the classical mild-Slope equation, Kirby's extended mild-slope equation with current, and Dingemans's mild-slope equation for rippled bed. Finally, good agreement between the classic experimental data concerning Bragg reflection and the present numerical results is observed.

Solving the Optimal Control Problems of Nonlinear Duffing Oscillators By Using an Iterative Shape Functions Method

In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.

Dynamics of cantilever plates and its hybrid vibration control

Applying Lagrange-Germain＇s theory of elas- tic thin plates and Hamiltonian formulation, the dynamics of cantilever plates and the problem of its vibration control are studied, and a general solution is finally given. Based on Hamiltonian and Lagrangian density function, we can obtain the flexural wave equation of the plate and the relationship between the transverse and the longitudinal eigenvalues. Based on eigenfunction expansion, dispersion equations of propagation mode of cantilever plates are deduced. By satisfying the boundary conditions of cantilever plates, the natural frequencies of the cantilever plate structure can be given. Then, analytic solution of the problem in plate structure is obtained. An hybrid wave/mode control approach, which is based on both independent modal space control and wave control methods, is described and adopted to analyze the active vibration control of cantilever plates. The low-order （controlled by modal control） and the high-order （controlled by wave control） frequency response of plates are both improved. The control spillover is avoided and the robustness of the system is also improved. Finally, simulation results are analyzed and discussed.

AN INTEGRAL EQUATION DESCRIBING RIDING WAVES IN SHALLOW WATER OF FINITE DEPTH

An integral equation describing riding waves, i. e. , small-scaleperturbation waves superposed on unperturbed surface waves, in shallow water of finite depth wasstudied via explicit Hamiltonian formulation, and the water was regarded as ideal incompressiblefluid of uniform density. The kinetic energy, density of the perturbed fluid motion was formulatedwith Hamiltonian canonical variables, elevation of the free surface and the velocity potential atthe free surface. Then the variables were expanded to the first order at the free surface ofunperturbed waves. An integal equation for velocity potential of perturbed waves on the unperturbedfree surface was derived by conformal mapping and the Fourier transformation. The integral equationcould replace the Hamiltonian canonical equations which are difficult to solve. An explicitexpression of Lagrangian density function could be obtained by solving the integral equation. Themethod used in this paper provides a new path to study the Hamiltonian formulation of riding wavesand wave interaction problems.

Resonance Clustering in Wave Turbulent Regimes:Integrable Dynamics

Two fundamental facts of the modern wave turbulence theory are 1)existence of power energy spectra in k-space,and 2)existence of“gaps”in this spectra corresponding to the resonance clustering.Accordingly,three wave turbulent regimes are singled out:kinetic,described by wave kinetic equations and power energy spectra;discrete,characterized by resonance clustering;and mesoscopic,where both types of wave field time evolution coexist.In this review paper we present the results on integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes.Using a novel method based on the notion of dynamical invariant we show that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others-only for particular initial conditions.We also identify chaotic behaviour in some cases.Physical implications of the results obtained are discussed.

A Constructive Method for Computing Generalized Manley-Rowe Constants of Motion

The Manley-Rowe constants of motion(MRC)are conservation laws written out for a dynamical systemdescribing the time evolution of the amplitudes in resonant triad.In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants(gMRC)and give a constructive method how to compute them.We also give details of a Mathematica implementation of this method.WhileMRC provide integrability of the underlying dynamical system,gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.