Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

The Mei symmetry and conservation laws for time-scale nonshifted Hamilton equations are explored, and the Mei symmetry theorem is presented and proved. Firstly, the time-scale Hamilton principle is established and extended to the nonconservative case. Based on the Hamilton principles, the dynamic equations of time-scale nonshifted constrained mechanical systems are derived. Secondly, for the time-scale nonshifted Hamilton equations, the definitions of Mei symmetry and their criterion equations are given. Thirdly, Mei symmetry theorems are proved, and the Mei-type conservation laws in time-scale phase space are driven. Two examples show the validity of the results.

INVESTIGATION OF THE PLANE VIBRATION OF AN ELASTIC PLATE WITH A KINEMATIC VIBRATION PROTECTION SYSTEM

Creation of vibro-protective devices on rolling contact bearings is widely spread in transportation technology and seismic protection. In this work, mathematical modeling of the oscillation movements of the elastic plate will be considered. The equations of motion for elastic plate on vibration supports bounded by high-order rotation surfaces by the Ostrogradsky-Hamilton principle are obtained. The natural frequencies of elastic plate are determined. It is established that the value of the natural frequencies of elastic plate decreases with increasing height and increases with the width of the bases. The ratio of the natural frequency of the second form to the natural frequency of the first form does not depend on the geometrical parameters of the plate.

The Generalized Hamilton Principle and Non-Hermitian Quantum Theory

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the Hermitian quantum theory, i.e., the standard Schrodinger equation. In this paper, we have generalized the Hamilton principle to the generalized Hamilton principle, which can describe the open system (mass or energy exchange systems) and nonconservative force systems or dissipative systems, and given the generalized Lagrange function, it has to do with the kinetic energy, potential energy and the work of nonconservative forces to do. With the Feynman path integration, we have given the non-Hermitian quantum theory of the nonconservative force systems. Otherwise, with the generalized Hamilton principle, we have given the generalized Hamiltonian for the particle exchanging heat with the outside world, which is the sum of kinetic energy, potential energy and thermal energy, and further given the equation of quantum thermodynamics. PACS: 03.65.-w, 05.70.Ce, 05.30.Rt

An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution

An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.

Noether’s Symmetries and Its Inverse for Fractional Logarithmic Lagrangian Systems

In this paper, Noether’s theorem and its inverse theorem are proved for the fractional variational problems based on logarithmic Lagrangian systems. The Hamilton principle of the systems is derived. And the definitions and the criterions of Noether’s symmetry and Noether’s quasi-symmetry of the systems based on logarithmic Lagrangians are given. The intrinsic relation between Noether’s symmetry and the conserved quantity is established. At last an example is given to illustrate the application of the results.

Stability of a hydraulic telescopic cylinder subjected to generalised load by a force directed towards the positive pole

The paper presents the boundary problem of the stability of a telescopic hydraulic cylinder subjected to a generalized load with a force directed to the positive pole. The boundary problem was formulated on the basis of the Hamilton principle. Numerical calculations were carried out, taking into account the influence of the parameters of the load heads (radii of loading and receiving head, length of bolt). On the basis of the numerical calculations, regions of load heads parameters were presented, at which the load bearing capacity of the analysed telescopic hydraulic cylinder is the largest from the buckling standpoint.