scholarly journals The second Birkhoff theorem for optical Hamiltonian systems

1991 ◽  
Vol 113 (2) ◽  
pp. 513-513 ◽  
Author(s):  
Leonid Polterovich
Keyword(s):  
Hamiltonian Systems ◽  
Birkhoff Theorem

scholarly journals Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

2017 ◽  
Vol 8 (1) ◽  
pp. 583-602 ◽  
Author(s):  
Alessandro Fonda ◽  
Rodica Toader
Keyword(s):  
Hamiltonian Systems ◽  
Zero Eigenvalue ◽  
Birkhoff Theorem ◽  
Subharmonic Solutions ◽  
Existence And Multiplicity ◽  
Volterra Systems

Abstract We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shuang Wang ◽  
Dingbian Qian
Keyword(s):  
Hamiltonian Systems ◽  
Rotation Number ◽  
Phase Plane Analysis ◽  
Cauchy Problems ◽  
Multiplicity Results ◽  
Birkhoff Theorem ◽  
Subharmonic Solutions ◽  
Large Solutions ◽  
Plane Analysis ◽  
Rotation Numbers

Abstract We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J ⁢ z ′ = ∇ ⁡ H ⁢ ( t , z ) {Jz^{\prime}=\nabla H(t,z)} from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂ ⁡ H ∂ ⁡ x ⁢ ( t , x , y ) {\frac{\partial H}{\partial x}(t,x,y)} , uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

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