Discrete Hamiltonian Systems Research Articles

Discrete mechanics, “time machines” and hybrid systems

Modifying the discrete mechanics proposed by T.D. Lee, we construct a class of discrete classical Hamiltonian systems, in which time is one of the dynamical variables. This includes a toy model of time machines which can travel forward and backward in time and which differ from models based on closed timelike curves (CTCs). In the continuum limit, we explore the interaction between such time reversing machines and quantum mechanical objects, employing a recent description of quantum-classical hybrids.

Subharmonics with Minimal Periods for Convex Discrete Hamiltonian Systems

We consider the subharmonics with minimal periods for convex discrete Hamiltonian systems. By using variational methods and dual functional, we obtain that the system has a <svg style="vertical-align:-3.50804pt;width:21.525px;" id="M1" height="15.9875" version="1.1" viewBox="0 0 21.525 15.9875" width="21.525" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g><g transform="matrix(.017,-0,0,-.017,10.143,11.55)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z" /></g> </svg>-periodic solution for each positive integer <svg style="vertical-align:-3.50804pt;width:10.2px;" id="M2" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,8.863)"><use xlink:href="#x1D45D"/></g> </svg>, and solution of system has minimal period <svg style="vertical-align:-3.50804pt;width:21.525px;" id="M3" height="15.9875" version="1.1" viewBox="0 0 21.525 15.9875" width="21.525" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.55)"><use xlink:href="#x1D45D"/></g><g transform="matrix(.017,-0,0,-.017,10.143,11.55)"><use xlink:href="#x1D447"/></g> </svg> as <svg style="vertical-align:-0.0pt;width:15.0375px;" id="M4" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g> </svg> subquadratic growth both at 0 and infinity.

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov-type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.

Group Theory in Finite Hamiltonian Systems

We give a short review ol the methods and aims ol the Óptica Matemática projects in studying discrete Hamiltonian systems with group-theoretical methods.

Discrete linear Hamiltonian systems: Lyapunov type inequalities, stability and disconjugacy criteria

In this paper, we first establish new Lyapunov type inequalities for discrete planar linear Hamiltonian systems. Next, by making use of the inequalities, we derive stability and disconjugacy criteria. Stability criteria are obtained with the help of the Floquet theory, so the system is assumed to be periodic in that case.

Infinitely many periodic solutions for discrete second order Hamiltonian systems with oscillating potential

In this article we obtain two sequence of infinitely many periodic solutions for discrete second order Hamiltonian systems with an oscillating potential. One sequence of solutions are local minimizers of the functional corresponding to the system, the other sequence are minimax type critical points of the functional.

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.

Homoclinic solutions for a class of second order discrete Hamiltonian systems

Consider the second order discrete Hamiltonian systems $$\Delta ^2 u(n - 1) - L(n)u(n) + \nabla W(n,u(n)) = f(n),$$ where n ∈ ℤ, u ∈ ℝN and W: ℤ × ℝN → ℝ and f: ℤ → ℝN are not necessarily periodic in n. Under some comparatively general assumptions on L, W and f, we establish results on the existence of homoclinic orbits. The obtained results successfully generalize those for the scalar case.

Lie Symmetries And Non-Noether Conserved Quantities For Discrete Hamiltonian Systems

We consider Lie symmetries and non-Noether conserved quantities for discrete Hamiltonian systems. Based on the infinitesimal transformations with respect to the generalized coordinates and generalized momentum, Firstly, the Lie’s theorem and non-Noether conserved quantities are obtained in continuous Hamilton systems. Secondly, the Legendre transform is introduced in the discrete variables, and we construct discrete Hamiltonian equations on the basis of the discrete equations in the Lagrange framework. Thirdly, we obtain the discrete determining equations and the non-Noether conserved quantities. Finally, an example is discussed for applications of the results.

Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

A class of first‐order noncoercive discrete Hamiltonian systems are considered. Based on a generalized mountain pass theorem, some existence results of homoclinic orbits are obtained when the discrete Hamiltonian system is not periodical and need not satisfy the global Ambrosetti‐Rabinowitz condition.

Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential

In the present paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order self-adjoint discrete Hamiltonian system where and are neither autonomous nor periodic in n. Under the assumption that is indefinite sign and subquadratic as and p(n) and L(n) are N × N real symmetric positive definite matrices for all , we establish some existence criteria to guarantee that the above system has at least one or multiple homoclinic solutions by using Clark's theorem in critical point theory.

Homoclinic orbits of first order discrete Hamiltonian systems with super linear terms

In this paper we consider the first order discrete Hamiltonian systems $$\left\{ {\begin{array}{*{20}c} {x_1 \left( {n + 1} \right) - x_1 \left( n \right) = - H_{x_2 } \left( {n,x\left( n \right)} \right),} \\ {x_2 \left( n \right) - x_2 \left( {n - 1} \right) = H_{x_1 } \left( {n,x\left( n \right)} \right),} \\ \end{array} } \right.$$ , where \(x\left( n \right) = \left( {_{x_2 \left( n \right)}^{x_1 \left( n \right)} } \right) \in \mathbb{R}^{2N} , H\left( {n,z} \right) = \frac{1} {2}S\left( n \right)z \cdot z + R\left( {n,z} \right)\) is periodic in n and superlinear as |z| → ∞. We prove the existence and infinitely many (geometrically distinct) homoclonic orbits of the system by critical point theorems for strongly indefinite functionals.

Noether Symmetry and First Integral of Discrete Nonconservative and Nonholonimic Hamiltoinian System

The letter focuses on studying Noether symmetry and conserved quantity of discrete nonconservative and nonholonomic Hamiltonian system. Firstly, the discrete Hamiltonian canonical equations and discrete energy equations of nonconservative and nonholonomic Hamiltonian systems are derived with discrete Hamiltonian action. Secondly, based on the quasi-invariance of discrete Hamiltonian action and equation of lattice under the infinitesimal transformation with respect to time, generalized coordinates and generalized momentums, the discrete analogue of Noether’s identity and determining equation of lattice are obtained for the systems. Thirdly, the discrete analogues of Noether’s theorems and conserved quantities of the systems are presented. Finally, one example is discussed to illustrate the application of the results.

Multiplicity results for periodic solutions with prescribed minimal periods to discrete Hamiltonian systems

By making use of minimax theory and geometrical index theory, some results on the existence and multiplicity of subharmonic solutions with prescribed minimal period to discrete Hamiltonian systems are obtained.

Periodic solutions for second-order discrete Hamiltonian systems

By using the least action principle and the minimax methods, some existence theorems are obtained for periodic solutions of second-order discrete Hamiltonian systems: in the cases where the nonlinearity is growing sublinearly or linearly.

Defect indices and definiteness conditions for a class of discrete linear Hamiltonian systems

This paper is concerned with a class of discrete linear Hamiltonian systems in finite or infinite intervals. A definiteness condition and its equivalent statements are discussed and three sufficient conditions for the definiteness condition are given. A precise relationship between the defect index of the minimal subspace generated by the system and the number of linearly independent square summable solutions of the system is established. In particular, they are equal if and only if the definiteness condition is satisfied. Finally, two criteria for the limit point case and one criterion for the limit circle case are obtained.

On disconjugacy and stability criteria for discrete Hamiltonian systems

By making use of new Lyapunov type inequalities, we establish disconjugacy and stability criteria for discrete Hamiltonian systems. The stability criteria are given when the system is periodic.

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

Abstract In this paper, we establish several new Lyapunov type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.

Periodic solutions of second order discrete Hamiltonian systems with potential indefinite in sign

Abstract In this paper, an existence theorem is obtained for the periodic solutions of second-order discrete Hamiltonian systems with potential indefinite in sign by Linking theorem in the critical point theory.

Error estimate of eigenvalues of discrete linear Hamiltonian systems with small perturbation

In this paper, eigenvalues of perturbed discrete linear Hamiltonian systems are considered. A new variational formula of eigenvalues is first established. Based on it, error estimates of eigenvalues of systems with small perturbation are given under certain non-singularity conditions. Small perturbations of the coefficient functions, the weight function and the coefficients of the boundary condition are all involved. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the non-singularity conditions. In addition, two examples are presented to illustrate the necessity of the non-singularity conditions and the complexity of the problem in the singularity case.