Lie Symmetry and Conserved Quantity of Three-Order Lagrangian Equations for Non-conserved Mechanical System

Based on the infinitesimal and one parameter transformation, the problem of Lie symmetry of three-order Lagrangian equations has been studied. Under Lie transformation, the sufficient and necessary condition which keeps three-order Lagrangian equations to be unchanged and the invariant are obtained in this paper.

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.

Integrating Factors and Conservation Theorems of Lagrangian Equations for Nonconservative Mechanical System in Generalized Classical Mechanics

The conservation theorems of the generalized Lagrangian equations for nonconservative mechanical system are studied by using method of integrating factors. Firstly, the differential equations of motion of system are given, and the definition of integrating factors is given. Next, the necessary conditions for the existence of the conserved quantity are studied in detail. Finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the result.

Higher-Order Lagrangian Equations of Higher-Order Motive Mechanical System

In this paper, if the condition of variation δt = 0 is satisfied, the higher-order Lagrangian equations and higher-order Hamilton＇s equations, which show the consistency with the results of traditional analytical mechanics, are obtained from the higher-order Lagrangian equations and higher-order Hamilton＇s equations. The results can enrich the theory of analytical mechanics.

The Neumann Problem for Special Lagrangian Equations with Critical Phase

In this paper,we consider the Neumann problem for special Lagrangian equations with critical phase.The global gradient and Hessian estimates are obtained.Using the method of continuity,we prove the existence of solutions to this problem.

Three-order pseudo-Hamilton canonical equationsReceived

Based on the three-order Lagrangian equations, Hamilton＇s function of acceleration H^＊ and generalized acceleration momentum P^＊α are defined, and pseudo-Hamilton canonical equations corresponding to three-order Lagrangian equations are obtained. The equations are similar to Hamilton＇s canonical equations of analytical mechanics in form.

A robust shell element in meshfree SPH method

With the incorporation of total Lagrangian smoothed particle hydrodynamics（SPH） method equation and moving least square（MLS） function,the traditional SPH method is improved regarding the stability and consistency.Based on Mindlin-Ressiner plate theory,the SPH method simulating dynamic behavior via one layer of particles is applied to plate＇s mid-plane,i.e.,a SPH shell model is constructed.Finally,through comparative analyses on the dynamic response of square,stiffened shells and cylindrical shells under various strong impact loads with common finite element software,the feasibility,validity and numerical accuracy of the SPH shell method are verified.Consequently,further researches on SPH shell may well pave the way towards solving problems involving dynamic plastic damage,tearing or even crushing.

Research on Planar Double Compound Pendulum Based on RK-8 Algorithm

Establishing the Lagrangian equation of double complex pendulum system and obtaining the dynamic differential equation,we can analyze the motion law of double compound pendulum with application of the numerical simulation of RK-8 algorithm.When the double compound pendulum swings at a small angle,the Lagrangian equation can be simplified and the normal solution of the system can be solved.And we can walk further on the relationship between normal frequency and swing frequency of double pendulum.When the external force of normal frequency is applied to the double compound pendulum,the forced vibration of the double compound pendulum will show the characteristics of beats.

Gravitational Field as a Pressure Force from Logarithmic Lagrangians and Non-Standard Hamiltonians：The Case of Stellar Halo of Milky Way

Recently,the notion of non-standard Lagrangians was discussed widely in literature in an attempt to explore the inverse variational problem of nonlinear differential equations.Different forms of non-standard Lagrangians were introduced in literature and have revealed nice mathematical and physical properties.One interesting form related to the inverse variational problem is the logarithmic Lagrangian,which has a number of motivating features related to the Li′enard-type and Emden nonlinear differential equations.Such types of Lagrangians lead to nonlinear dynamics based on non-standard Hamiltonians.In this communication,we show that some new dynamical properties are obtained in stellar dynamics if standard Lagrangians are replaced by Logarithmic Lagrangians and their corresponding non-standard Hamiltonians.One interesting consequence concerns the emergence of an extra pressure term,which is related to the gravitational field suggesting that gravitation may act as a pressure in a strong gravitational field.The case of the stellar halo of the Milky Way is considered.

MPM simulations of dam-break floods

Owing to its ability of modelling large deformations and the ease of dealing with moving boundary conditions,the material point method is gaining popularity in geotechnical engineering applications.In this paper,this promising Lagrangian method is applied to hydrodynamic problems to further explore its potential.The collapse of water columns with different initial aspect ratios is simulated by the material point method.In order to test the accuracy and stability of the material point method,simulations are first validated using experimental data and results of mature numerical models.Then,the model is used to ascertain the critical aspect ratio for the widely-used shallow water equations to give satisfactory approximation.From the comparisons between the simulations based on the material point method and the shallow water equations,the critical aspect ratio for the suitable use of the shallow water equations is found to be 1.