High-Order Hamilton's Principle and the Hamilton's Principle of High-Order Lagrangian Function

In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equations and four-order Lagrangian equations are obtained from the high-order Hamilton＇s principle. Finally, the Hamilton＇s principle of high-order Lagrangian function is given.

Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given.The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given.The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail,and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton＇s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and mo- mentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.

Noether Symmetry of Three-Order Lagrangian Equations

Based on the three-order Lagrangian equation, pseudo-Hamilton actoon I^＊ is defined and the three-order Hamilton＇s principle and the conditions are obtained in the paper. Then, the Noether symmetry about three-order Lagrangian equations is deduced. Finally, an example is given to illustrate the application of the result.

Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams

The dynamic stability of axially moving viscoelastic Rayleigh beams is pre- sented. The governing equation and simple support boundary condition are derived with the extended Hamilton＇s principle. The viscoelastic material of the beams is described as the Kelvin constitutive relationship involving the total time derivative. The axial tension is considered to vary longitudinally. The natural frequencies and solvability condition are obtained in the multi-scale process. It is of interest to investigate the summation parametric resonance and principal parametric resonance by using the Routh-Hurwitz criterion to obtain the stability condition. Numerical examples show the effects of viscos- ity coefficients, mean speed, beam stiffness, and rotary inertia factor on the summation parametric resonance and principle parametric resonance. The differential quadrature method （DQM） is used to validate the value of the stability boundary in the principle parametric resonance for the first two modes.

LAGRANGIAN MECHANICS ON KAHLER MANIFOLDS

Lagrangian mechanics on Kahler manifolds were discussed, and the complex mathematical aspects of Lagrangian operator, Lagrange＇s equation, the action functional, Hamilton＇ s principle, Hamilton＇ s equation and so on were given.

Intuitive Concept or Physical Meaning of Lagrangian

The authors have been studying on the principle of motion generation behind animals, mainly human, and have reached a certain milestone with it in [1]. Because [1] ended up being very interdisciplinary, the author has been looking for an opportunity to close in on the part where we have grasped the conceptual idea of a Lagrangian. This paper proposes the physical meaning or its intuitive concept of a Lagrangian. This is a daring attempt because the topic is over 240 years of enigma, whereby so many have neglected of its absence, and physics has gone further towards its frontiers of their time, and has successfully flourished. Meanwhile, Lagrangian is not getting enough of teachers’ attention on students getting stuck on this function, despite the fact that it is a strong foundation as is only the beginning towards Hamiltonian formalism, general relativity, and modern physics of today. This paper’s sole motive is to answer what the title says in detail, helping each and everyone who faces Lagrangian for their first time. The paper is positioned to be a supplement for [1]. This literature had three topics bound into one. Out of the three, this document focuses in the part of the intuitive meaning of Lagrangian, since the paper had contents related to multiple disciplines. The author finds it worthy to discuss this topic in an independent, more detailed manner.

Modeling and analysis of cantilever beam with active-passive hybrid piezoelectric network

The active-passive hybrid piezoelectric network (APPN) is investigated to reduce the vibration of cantilever beam. Hamilton's principle with the Rayleigh-Ritz method is used to derive the equations of motion of the beam with the APPN. Only one piezoelectric actuator is bonded on the cantilever beam, so in the segment of the beam where the piezoelectric actuator is attached, the neutral axis is not the geometric center of the beam. This change on the neutral axis is considered in the process of deriving equations. Selecting RL circuit as passive shunt circuit, open-loop analysis is performed to gain insight into the passive damping features. Velocity feedback control is then employed to analyze the characteristics of the closed-loop system. Numerical results show that the APPN has a significant effect on vibration suppression, especially at narrow frequency bands. On this basis, variable RL circuit is proposed and analyzed for broadband vibration attenuation. Numerical simulations illustrate that this scheme is effective and feasible.

Free Vibration Analysis of Nanocomposite Plates Reinforced by Graded Carbon Nanotubes Based on First-Order Shear Deformation Plate Theory

Based on the Mindlin’s first-order shear deformation plate theory this paper focuses on the free vibration behavior of functionally graded nanocomposite plates reinforced by aligned and straight single-walled carbon nanotubes(SWCNTs).The material properties of simply supported functionally graded carbon nanotube-reinforced(FGCNTR)plates are assumed to be graded in the thickness direction.The effective material properties at a point are estimated by either the Eshelby-Mori-Tanaka approach or the extended rule of mixture.Two types of symmetric carbon nanotubes(CNTs)volume fraction profiles are presented in this paper.The equations of motion and related boundary conditions are derived using the Hamilton’s principle.A semianalytical solution composed of generalized differential quadrature(GDQ)method,as an efficient and accurate numerical method,and series solution is adopted to solve the equations of motions.The primary contribution of the present work is to provide a comparative study of the natural frequencies obtained by extended rule of mixture and Eshelby-Mori-Tanaka method.The detailed parametric studies are carried out to study the influences various types of the CNTs volume fraction profiles,geometrical parameters and CNTs volume fraction on the free vibration characteristics of FGCNTR plates.The results reveal that the prediction methods of effective material properties have an insignificant influence of the variation of the frequency parameters with the plate aspect ratio and the CNTs volume fraction.

Magnetoelastic Instability of a Long Graphene Nano-Ribbon Carrying Electric Current

This paper aims at investigating the resonance frequencies and stability of a long Graphene Nano-Ribbon(GNR)carrying electric current.The governing equation of motion is obtained based on the Euler-Bernoulli beam model along with Hamilton’s principle.The transverse force distribution on the GNR due to the interaction of the electric current with its own magnetic field is determined by the Biot-Savart and Lorentz force laws.Using Galerkin’s method,the governing equation is solved and the effect of current strength and dimensions of the GNR on the stability and resonance frequencies are investigated.