时间尺度上Hamilton系统的Noether理论

提出并研究时间尺度上Hamilton系统的Noether对称性与守恒量问题．建立了时间尺度上Hamilton原理，导出了相应的Hamilton正则方程．基于时间尺度上Hamilton作用量在群的无限小变换下的不变性，建立了时间尺度上Hamilton系统的Noether定理．定理的证明分成两步：第一步，在时间不变的无限小变换群下给出证明；第二步，利用时间重新参数化技术得到了一般无限小变换群下的定理．给出了经典和离散两种情况下Hamilton系统的Noether守恒量．文末举例说明结果的应用．

Noether Theory for Hamiltonian System on Time Scales

The Noether symmetry and the conserved quantity for a Hamiltonian system on time scales are proposed and studied in this paper. The Hamilton principle on time scales is established, and corresponding Hamilton canonical equations are deduced. Based upon the invariance of the Hamilton action on time scales under the infinitesimal transformations of a group, the Noether theorem for the Hamiltonian system on time scales is established. The proof of the theorem is composed of two steps. First, we prove the Noether theorem under the infinitesimal transformations of a special one-parameter group without varying the time. Second using the technique of time-re-parameterization, we obtain the Noether theorem in its general form. The Noether-type conserved quantities for Hamiltonian system in both the classical and the discrete cases are given. At the end of the paper, two examples are given to illustrate the application of the theorem.