Hamilton系统同宿轨与双曲轨的研究

The Hamiltonian systems are usually used to describe the behaviors of the evolution for various systems in the state space, such as the motion of the planets in the universe, the motion of the charged particles under the magnetic and electric fields and so on. Since Hamiltonian systems possess the natural variational structure, the existence of periodic, homoclinic, heteroclinic, hyperbolic and parabolic orbits has drown more and more attention of mathematicians after the landmark paper by P.H.Rabinowitz in 1978.The homoclinic orbit has been proved that they could describe the asymptotic behaviors of the systems when time variable goes to infinity and the equilibrium points of the systems better. Moreover, the appearance of chaos is closely associated with the existence of homoclinic orbits. Hence the homoclinic orbit is a kind of important solutions for Hamiltonian systems, which influences the properties of the whole systems. In addition, the hyperbolic orbit is a kind of solutions for two body problem which was found by Newton. In fact, when we study the motion of the hevean bodies, we find the orbits of the rocket near the moon is hyperbolic. Therefore, the hyperbolic orbit is important in applications. We will study the existence of homoclinic and hyperbolic orbits for Hamiltonian systems under various conditions.

Hamilton系统常常用来描述各种系统在状态空间上随着时间而演化的行为，如宇宙中行星的运动，微观世界中带电粒子在电磁场下的运动等等。由于Hamilton系统有着自然的变分结构，自1978年，P.H.Rabinowitz发表了开创性的结果之后，越来越多的数学工作者将精力投入到了对Hamilton系统周期轨，同宿轨，异宿轨，双曲轨和抛物轨等不同类型轨道存在性的研究中。其中Hamilton系统的同宿轨被证明了可以用来刻画系统在时间趋于无穷时轨道的渐近行为，以及更好地描述系统的平衡点。并且混沌现象的出现跟同宿轨有着紧密的联系，所以同宿轨是一种很重要的解，它影响着整个系统的性质。另外，人们在研究天体运行时发现，当火箭在月球附近运行的时候轨道是属于双曲线的，所以双曲轨道在实际应用中有着重要的作用。本项目拟利用变分方法，在不同的条件下得到Hamilton系统同宿轨与双曲轨的存在性。

Hamilton系统在许多力学系统中出现，它可以用来描述系统中物体的运动行为，但是要求出该系统的解是复杂的。在本项目中，我们利用变分法得到了带关于时间无界的非线性项的Hamilton系统同宿轨的存在性和多重性。利用给出新的博弈条件，我们得到了新的嵌入定理。该条件在研究关于带时间无界的非线性项的Hamilton系统时有着一定的理论意义。其次，我们还得到了一类关于带混合位势的Hamilton系统双曲轨的存在性。这帮助我们更加清晰地认识Hamilton系统在不同条件下的解的情况。