Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient con...Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical ＇σ/σx＇-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.展开更多
Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessar...Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessary and sufficient conditions are obtained展开更多
We study both classical and quantum relation between two Hamiltonian systems which are mutuallyconnected by time-dependent canonical transformation.One is ordinary conservative system and the other is time-dependentHa...We study both classical and quantum relation between two Hamiltonian systems which are mutuallyconnected by time-dependent canonical transformation.One is ordinary conservative system and the other is time-dependentHamiltonian system.The quantum unitary operator relevant to classieai canonical transformation betweenthe two systems are obtained through rigorous evaluation.With the aid of the unitary operator,we have derivedquantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservativesystem.The invariant operators of the two systems are presented and the relation between them are addressed.Weshowed that there exist numerous Hamiltonians,which gives the same classical equation of motion.Though it isimpossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics,they canbe distinguishable quantum mechanically.展开更多
The dressed Hamiltonian describing the multiphoton process is presented by means of a canonical transformation.The exact eigenvalues and eigenstates are given for any set of parameters ω_(0),ω and ε.The general evo...The dressed Hamiltonian describing the multiphoton process is presented by means of a canonical transformation.The exact eigenvalues and eigenstates are given for any set of parameters ω_(0),ω and ε.The general evolution of this system is obtained.展开更多
In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two_fluid system,which consists of two layers of constant_density incompressible inviscid fluid with a horizontal bottom,a...In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two_fluid system,which consists of two layers of constant_density incompressible inviscid fluid with a horizontal bottom,an interface and a free surface. The velocity potentials are expanded in power series of the vertical coordinate. By taking the kinetic thickness of lower fluid_layer and the reduced kinetic thickness of upper fluid_layer as the generalized displacements, choosing the velocity potentials at the interface and free surface as the generalized momenta and using Hamilton's principle, the Hamiltonian canonical equations for the system are derived with the Legendre transformation under the shallow water assumption. Hence the results for single_layer fluid are extended to the case of stratified fluid.展开更多
This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressi...This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressible, inviscid fluid have been constructed in cartesian coordinates and also in cylindrical coordinates. Then Lagrangian function within a certain flow region is expanded under the assumption that the dispersion μ and the nonlinearity ε satisfied . Using Hamilton’s principle for water wave evolution Hamiltonian formulation is derived. It is obvious that the motion of the system is conservative. Then Hamilton’s canonical equation of motion is also derived.展开更多
In this work we use the Hamilton-Jacobi theory to show that locally all the Hamiltonian systems with n degrees of freedom are equivalent. That is, there is a canonical transformation connecting two arbitrary Hamiltoni...In this work we use the Hamilton-Jacobi theory to show that locally all the Hamiltonian systems with n degrees of freedom are equivalent. That is, there is a canonical transformation connecting two arbitrary Hamiltonian systems with the same number of degrees of freedom. This result in particular implies that locally all the Hamiltonian systems are equivalent to that of a free particle. We illustrate our result with two particular examples;first we show that the one-dimensional free particle is locally equivalent to the one-dimensional harmonic oscillator and second that the two-dimensional free particle is locally equivalent to the two-dimensional Kepler problem.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No 10562002） and the Natural Science Foundation of Nei Mongol, China （Grant No 200508010103）.
文摘Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical ＇σ/σx＇-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.
基金Supported in part by the National Natural Science Foundation of China (1 0 0 71 0 2 1 ) the Foundationfor University Key Teacher by MEC and Shanghai Priority Academic Discipline Foundation
文摘Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessary and sufficient conditions are obtained
基金Supported by the Korea Science and Engineering Foundation (KOSEF) Grant Funded by the Korea Government (MOST) under Grant No.F01-2007-000-10075-0
文摘We study both classical and quantum relation between two Hamiltonian systems which are mutuallyconnected by time-dependent canonical transformation.One is ordinary conservative system and the other is time-dependentHamiltonian system.The quantum unitary operator relevant to classieai canonical transformation betweenthe two systems are obtained through rigorous evaluation.With the aid of the unitary operator,we have derivedquantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservativesystem.The invariant operators of the two systems are presented and the relation between them are addressed.Weshowed that there exist numerous Hamiltonians,which gives the same classical equation of motion.Though it isimpossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics,they canbe distinguishable quantum mechanically.
基金Supported by the National Natural Science Foundation of China.
文摘The dressed Hamiltonian describing the multiphoton process is presented by means of a canonical transformation.The exact eigenvalues and eigenstates are given for any set of parameters ω_(0),ω and ε.The general evolution of this system is obtained.
文摘In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two_fluid system,which consists of two layers of constant_density incompressible inviscid fluid with a horizontal bottom,an interface and a free surface. The velocity potentials are expanded in power series of the vertical coordinate. By taking the kinetic thickness of lower fluid_layer and the reduced kinetic thickness of upper fluid_layer as the generalized displacements, choosing the velocity potentials at the interface and free surface as the generalized momenta and using Hamilton's principle, the Hamiltonian canonical equations for the system are derived with the Legendre transformation under the shallow water assumption. Hence the results for single_layer fluid are extended to the case of stratified fluid.
文摘This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressible, inviscid fluid have been constructed in cartesian coordinates and also in cylindrical coordinates. Then Lagrangian function within a certain flow region is expanded under the assumption that the dispersion μ and the nonlinearity ε satisfied . Using Hamilton’s principle for water wave evolution Hamiltonian formulation is derived. It is obvious that the motion of the system is conservative. Then Hamilton’s canonical equation of motion is also derived.
文摘In this work we use the Hamilton-Jacobi theory to show that locally all the Hamiltonian systems with n degrees of freedom are equivalent. That is, there is a canonical transformation connecting two arbitrary Hamiltonian systems with the same number of degrees of freedom. This result in particular implies that locally all the Hamiltonian systems are equivalent to that of a free particle. We illustrate our result with two particular examples;first we show that the one-dimensional free particle is locally equivalent to the one-dimensional harmonic oscillator and second that the two-dimensional free particle is locally equivalent to the two-dimensional Kepler problem.
