Persistence of Lower Dimensional Tori of General Types in Hamiltonian Systems

2000 Mathematics Subject Classification. 37J40.The work is a generalization to [40] in which we study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.The first author was partially supported by NSFC grant 19971042, National 973 Project of China: Nonlinearity, the outstanding young's project of Ministry of Education of China, and National outstanding young's award of China. The second author was partially supported by NSF grants DMS9803581 and DMS-0204119

Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes

In this paper, we derive generalized Bern-Carrasco-Johansson relations for color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and dimensional reduction appropriately on the new discovered graphic expansion of Einstein-Yang-Mills amplitudes. These relations are also satisfied by color-ordered amplitudes in other theories such as color-scalar theory, bi-scalar theory and nonlinear sigma model (NLSM). As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in NLSM , which are derived from Feynman rules, Abelian Z-theory and Cachazo-He- Yuan formula respectively, produce the same total amplitudes. In other words, the three distinct approaches to NLSM amplitudes are equivalent to each other.Comment: 40pages, 2 figure