Discrete Variational Methods On Lie Groups And Their Application

Small bodies in the solar system,including asteroids and comets,are important targets for future deep space exploration.Asteroid detection missions can help study the composition and motion of small bodies and find answers to fundamental astronomical questions such as the origin and evolution of the solar system.There have been many asteroid impacts on Earth in the history of the planet,and these impacts have sometimes caused changes in the Earth’s climate and even posed a potentially significant threat to human survival and development.Therefore,there is a very urgent need and far-reaching strategic significance to conduct research on the detection,development and exploitation of near-Earth asteroids and the response to asteroid impact risks.Since the first discovery of double asteroids by human asteroid probes in the 1990 s,a considerable number of NEO double asteroids have been discovered one after another.In this paper,orbital dynamics in asteroid defence is addressed,and a rigid dumbbell body is used as a simplified model for the synchronous binary system.The numerical calculation method for the rigid dumbbell system is constructed through the Li group modelling and discrete variational algorithm,and the dynamical properties of the system are studied.Firstly,a rigid dumbbell body is studied as a simplified model of a synchronous binary star.Although this system has energy conservation properties,the usual iterative algorithm cannot maintain the energy conservation of the system for a long time when studying this system numerically.Therefore,we adopt the Lie group modelling method in the modelling process,using a special orthogonal group to describe the object attitude to avoid the singularity problem brought about by the parametric representation,so that the calculation process always maintains the inherent Lie group properties of the object attitude.Secondly,in the numerical calculation process,a discrete variational method is used to discretize the action quantities to obtain the discrete action quantities for each time interval,and then use the discrete Hamiltonian variational principle to obtain the discrete kinetic equations for each time interval,which is known as the Lie group variational integration method.This method is known as the Lie group variational integration method.It is to discretize the time to obtain a discrete set of equations for each time interval and then solve for the Lie group increments,so that the errors in the previous step do not enter into the solution of the next step,thus avoiding the accumulation of errors that exist in other numerical methods,having the property of energy conservation and naturally maintaining the Lie group structure of the system.Finally,based on the geometrical mechanics theory,the Lie group variational integral formulation for the Hamiltonian system of a double dumbbell in the central force field without gravitational uptake is constructed based on the discrete Hamiltonian variational principle and the discrete Legendre transform,and the Lie group variational integral formulation and Hamel variational integral formulation for the double dumbbell system in general form without gravitational uptake are given.Numerical simulations are carried out for the double dumbbell system.The numerical results show that the numerical results obtained by using the general form of the Lie group variational integration formula and the Hamel variational integration formula based on the discrete variational principle are more accurate than those obtained by directly discrete Euler-Lagrange equations,and can maintain the inherent conserved quantities of the system better.Therefore,the results of this paper will provide an efficient and accurate numerical simulation method for solving the practical problems of asteroid orbit prediction,defence,detection and modelling of asteroid probe orbit dynamics.