Normal Stability Of Generalized Hamiltonian Systems

We call a nonlinear system is normally stable,if its normal form is stable.For an autonomous Hamiltonian systems with n degrees of freedom,its normal stability depends on the existence of a formal integral whose quadratic part is positive definite.The main contribution of this thesis is to extend the normal stability theory of the classical Hamiltonian system to the generalized Hamiltonian system on Poisson manifolds.More precisely,this thesis is divided into five chapters:In Chapter 1 and Chapter 2,we briefly introduce the research background and some preliminary knowledge to be used in the thesis.Chapter 3 is devoted to the linear stability of the generalized Hamiltonian system.We first transform the linear generalized Hamilton system(2.7)into a form which is easy to study by means of proper coordinate transformation.Then,we give the corresponding linear stability theorems respectly for the different cases where the equilibrium point is the regular point and the singular point of Poisson structure.As the main part of this thesis,based on the third chapter,Chapter 4 studies the normal stability of the generalized Hamiltonian system defined on Poisson manifold.It is shown that for the case of regular equilibrium point,the system can be transformed into an even-dimension Hamiltonian system on the leaf layer through the Darboux transformation and the normal stability theorem for the generalized Hamiltonian system is essentially the same as the one for the classical Hamiltonian system and the corresponding conditions for normal stability only depends on the quadratic part of the Hamiltonian function.For the case of singular equilibrium point,the generalized Hamiltonian system cannot be reduced to the one defined on a leaf of foliation.In particular,we study the case where the rank of the Poisson structural matrix at the singular equilibrium point is 2n,while the dimension of the system is m=2n+2 or 2n+3.It is found that the study on the normal stability of the generalized Hamiltonian system is much more complex than that of the regular equilibrium point.Although the normal stability of the system also only depends on the linear part of the vector field,it is different from the classical Hamiltonian system.The normal stability is not only related to the quadratic part H2,0 of the Hamiltonian function,but also to the primary part H0,1 of the Hamiltonian function and the structure matrix R(z).The purpose of Chapter 5 is the study of the nonlinear stability of the generalized Hamiltonian system.Because a normal stable system is not necessarily nonlinearly stable,in some cases,it is more convenient to discuss the nonlinear stability of the system directly.The corollary 4.11 of theorem 4.8 in the reference[2]shows that for the generalized Hamiltonian system with a group of conserved quantities,only the definiteness of the Hessian matrix of a modified Hamilton function on the intersection of the kernel spaces of the group of conserved gradient operators needs to determine that the equilibrium point of the original system is Liapunov stableSince the proof of theorem 4.8 in the reference[2]is too complex and difficult to understand,in this thesis,we present Corollary 4.11 in reference[2]as Theorem 5.1 in this thesis,and give another proof of the theorem.The proof method is simpler and easier to understand.Secondly,Theorem 2.5 in reference[3]improving Corollary 4.11 reference[2]admits advantage that when the expressions of the conserved quantities are more complicated,which is not conducive to the construction of the energy function,it can be replaced by a function F(x)satisfing certain conditions,thus makes up for the limitation of Corollary 4.11.In Chapter 5,we give and prove Theorem 5.2 which improves Theorem 2.5 in of Reference[3]and makes the conditions of the theorem simpler.At the same time,we also give an example to show that some conditions in theorem 2.5 are redundant.The conditions of theorem 5.2 in this thesis are simpler and easier to verify.