Liang Lifu

On the inverse problem in calculus of variations

The inverse problem in calculus of variations is studied. By introducing a new concept called Variational Integral, a new method to systematically study the inverse problem in calculus of variations is given. Using this new method to the elastodynamics and hydrodynamics of viscous fluids, some kinds of variational principles and generalized variational principles are obtained respectively.

Lagrangian theoretical framework of dynamics of nonholonomic systems

By the generalized variational principle of two kinds of variables in general mechanics, it was demonstrated that two Lagrangian classical relationships can be applied to both holonomic systems and nonholonomic systems. And the restriction that two Lagrangian classical relationships cannot be applied to nonholonomic systems for a long time was overcome. Then, one important formula of similar Lagrangian classical relationship called the popularized Lagrangian classical relationship was derived. From Vakonomic model, by two Lagrangian classical relationships and the popularized Lagrangian classical relationship, the result is the same with Chetaev’s model, and thus Chetaev’s model and Vakonomic model were unified. Simultaneously, the Lagrangian theoretical framework of dynamics of nonholonomic system was established. By some representative examples, it was validated that the Lagrangian theoretical framework of dynamics of nonholonomic systems is right.

On the unification of the Hamilton principles in non-holonomic system and in holonomic system

In this paper by means of typical engineering examples and deep theoretical analysis, we prove that: under the effect of conservative force, the Hamilton principles in holonomic and non-holonomic systems have the same formula δ∫Ldt=0. The formula ∫δLdt=0 is an evolved form of the formula δ∫Ldt=0. Therefore, the two formulas are unified.

Investigation of structural stability ofrigid-thermo-elastic coupling

For rigid-thermo-elastic coupling dynamics, a subject crossing research, there are not available governing equations until now. Dealing with the problem from entirety angle, variational principles can apply the transformation theory between work and energy and the law of conservation of energy to solve the problem conveniently. As a result, the variational principle can be used not only to establish the governing equation but also to provide convenience for finite element models of rigid-thermo-elastic coupling dynamics. In the paper, according to the transformation theory between work and energy and the law of conservation of energy, the variational principle of rigid-thermo-elastic coupling dynamics was established. Stationary value conditions of variational principle and the governing equations of rigid-elastic coupled dynamics were obtained. And then, conclusions presented above were transformed to statics expression in non-inertia coordinate system. As an application of variational principle of rigid-thermo-elastic coupling dynamics, investigation of structural stability of rigid-thermo-elastic coupling was carried out. Two-oriented critical stresses in case of rigid-elastic coupling state and rigid-thermo-elastic coupling state were presented. Finally, this paper discusses some problems.

Investigation of a theoretical problem in spacecraft dynamics

Recent study to solve the problems of flexible multi-body dynamics mainly depends on the numerical, quantitative methods and almost no one involves in the analytical discussion. It is unfavorable to understand profoundly the essence of nonlinear mechanics of the system and to predict the feature of overall dynamics of system. Therefore, it is the need to study theoretical analysis of flexible multi-body system. The research of this paper carries out to adapt this need. The theoretical problem is discussed which is coming with the projection of the vector equations in a suitable moving coordinate system. Then some important results are obtained. The important results show that not only the theoretical analysis of flexible multi-body system has been necessary, but also it has provided reference for the modeling of flexible multi-body system.

Integrating Factors and Conservation Theorems for the Nonholonomic Singular Lagrange System

We present a general approach to the construction of conservation laws for the nonholonomic singular Lagrange system. Firstly, the differential equations of motion of the system are written, the definition of integrating factors is given for the system. Next, the necessary conditions for the existence of the conserved quantity are studied in detail. Finally, the conservation theorem and its inverse are established for the system, an example is given to illustrate the application of the result.

On Form Invariance, Lie Symmetry and Three Kinds of Conserved Quantities of Generalized Lagrange's Equations

In this paper, the form invariance and the Lie symmetry of Lagranges equations for nonconservative system in generalized classical mechanics under the infinitesimal transformations of group are studied, and the Noethers conserved quantity, the new form conserved quantity, and the Hojmans conserved quantity of system are derived from them. Finally, an example is given to illustrate the application of the result.

The stationary value property of Hamilton's principle in non-holonomic systems

This paper proves that Hamiltons principle of both using the Appell-Chetaev condition and not using the Appell-Chetaev condition is the variational principle of stationary action. The relevant problems are discussed.

On the Inverse Problem in Variational Calculus

Calculus of variations is a very important tool to study mechanics, physics and other else technical science. The importance of variational method has been tested by a lot of facts in many fields. But how to find a general method to convert the boundary value problem into stationary value problem, has been attended to by many mathematicians and mechanicians, and this problem is called the inverse problem in calculus of variations.

On mechanical property of constraint

In this paper, the mechanical properties of holonomic systems and non-holonomic systems are studied. This is an important and urgent problem.