A Study on Time Scale Non-Shifted Hamiltonian Dynamics and Noether's Theorems

The time-scale non-shifted Hamiltonian dynamics are investigated,including both general holonomic systems and nonholonomic systems.The time-scale non-shifted Hamilton principle is presented and extended to nonconservative system,and the dynamic equations in Hamiltonian framework are deduced.The Noether symmetry,including its definition and criteria,for time-scale non-shifted Hamiltonian dynamics is put forward,and Noether's theorems for both holonomic and nonholonomic systems are presented and proved.The nonshifted Noether conservation laws are given.The validity of the theorems is verified by two examples.

NONLINEAR INTERNAL RESONANCE OF FUNCTIONALLY GRADED CYLINDRICAL SHELLS USING THE HAMILTONIAN DYNAMICS

Internal resonance in nonlinear vibration of functionally graded （FG） circular cylin- drical shells in thermal environment is studied using the Hamiltonian dynamics formulation. The material properties are considered to be temperature-dependent. Based on the Karman-Donnell＇s nonlinear shell theory, the kinetic and potential energy of FG cylindrical thin shells are formu- lated. The primary target is to investigate the two-mode internal resonance, which is triggered by geometric and material parameters of shells. Following a secular perturbation procedure, the underlying dynamic characteristics of the two-mode interactions in both exact and near resonance cases are fully discussed. It is revealed that the system will undergo a bifurcation in near resonance case, which induces the dynamic response at high energy level being distinct from the motion at low energy level. The effects of temperature and volume fractions of composition on the exact resonance condition and bifurcation characteristics of FG cylindrical shells are also investigated.

Dynamics of Quantum State and Effective Hamiltonian with Vector Differential Form of Motion Method

Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective Hamiltonian using various methods has gained great interest. We consider a vector differential equation of motion to derive the effective Hamiltonian for any periodically driven two-level system, and the dynamics of the spin vector are an evolution under the Bloch sphere. Here, we investigate the properties of this equation and show that a sudden change of the effective Hamiltonian is expected. Furthermore, we present several exact relations, whose expressions are independent of the different starting points. Moreover, we deduce the effective Hamiltonian from the high-frequency limit, which approximately equals the results in previous studies. Our results show that the vector differential equation of motion is not affected by a convergence problem, and thus, can be used to numerically investigate the effective models in any periodic modulating system. Finally, we anticipate that the proposed method can be applied to experimental platforms that require time-periodic modulation, such as ultracold atoms and optical lattices.

Motion stability dynamics for spacecraft coupled with partially filled liquid container

This paper presents a canonical Hamiltonian model of liquid sloshing for the container coupled with spacecraft. Elliptical shape of rigid body is considered as spacecraft structure. Hamiltonian system is an important form of mechanical system. It mostly used to stabilize the potential shaping of dynamical system. Free surface movement of liquid inside the container is called sloshing. If there is uncontrolled resonance between the motion of tank and liquid-frequency inside the tank then such sloshing can be a reason of attitude disturbance or structural damage of spacecraft. Equivalent mechanical model of simple pendulum or mass attached with spring for sloshing is used by many researchers. Mass attached with spring is used as an equivalent model of sloshing to derive the mathematical equations in terms of Hamiltonian model. Analytical method of Lyapunov function with Casimir energy function is used to find the stability for spacecraft dynamics. Vertical axial rotation is taken as the major axial steady rotation for the moving rigid body.

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.

Symplectic orbit propagation based on Deprit’s radial intermediary

The constantly challenging requirements for orbit prediction have opened the need for better onboard propagation tools.Runge-Kutta(RK)integrators have been widely used for this purpose;however RK integrators are not symplectic,which means that RK integrators may lead to incorrect global behavior and degraded accuracy.Emanating from Deprit’s radial intermediary,obtained by the elimination of the parallax transformation,we present the development of symplectic integrators of different orders for spacecraft orbit propagation.Through a set of numerical simulations,it is shown that these integrators are more accurate and substantially faster than Runge-Kutta-based methods.Moreover,it is also shown that the proposed integrators are more accurate than analytic propagation algorithms based on Deprit’s radial intermediary solution,and even other previously-developed symplectic integrators.

Modelling of dynamic contact length in rail grinding process

Rails endure frequent dynamic loads from the passing trains for supporting trains and guiding wheels. The accumulated stress concentrations will cause the plastic deformation of rail towards generating corruga- tions, contact fatigue cracks and also other defects, resulting in more dangerous status even the derailment risks. So the rail grinding technology has been invented with rotating grinding stones pressed on the rail with defects removal. Such rail grinding works are directed by experiences rather than scientifically guidance, lacking of flexible and scientific operating methods. With grinding control unit holding the grinding stones, the rail grinding process has the characteristics not only the surface grinding but also the running railway vehicles. First of all, it＇s important to analyze the contact length between the grinding stone and the rail, because the contact length is a critical parameter to measure the grinding capabilities of stones. Moreover, it＇s needed to build up models of railway vehicle unit bonded with the grinding stone to represent the rail grinding car. Therefore the theoretical model for contact length is developed based on the geometrical analysis. And the calculating models are improved considering the grinding car＇s dynamic behaviors during the grinding process. Eventually, results are obtained based on the models by taking both the operation parameters and the structure parameters into the calculation, which are suitable for revealing the process of rail grinding by combining the grinding mechanism and the railway vehicle systems.