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Master-slave Robots, Optimal Control, Quantum Mechanics and Informationlds
Published in Harish Parthasarathy, Electromagnetics, Control and Robotics, 2023
Let the solution to the Hamiltonian equations be Tt(q(0),p(0)) Let (q(0), p(0)) be a random vector with probability density ρ0 (q, p). Let rt (q, p) be the probability density of (q(t), p(t). Then we have by the standard change of variable formula. ρt(Tt(q,p)Jt(q,p)=ρ0(q,p) where Jt is the Jacobian determinant of Tt. The Jacobian of the infinitesimal transformation (q(t), p(t)) → (q(t +dt), p(t + dt) is one since the divergence of the Hamiltonian vector field (∇pH,−∇qH) vanishes: ∇q.∇pH−∇p.∇qH=0
Chaos in Systems with Time Variable Parameters
Published in L. Cveticanin, Dynamics of Machines with Variable Mass, 2022
where (x, τ, θ) ∈ R2n × T1 × T1, τ = μt is slow time, μ ≪ 1 is the small parameter, θ is the periodical function, f(x, τ) may be the Hamiltonian or non-Hamiltonian vector field, ε ∈ Rp is the vector of parameters, Ω is the constant value and μg is the small perturbation function. The system obtained by setting μ = 0 is referred to as the unperturbed system () x·=f(x,0),τ·=0,θ·=Ω.
Hamiltonian Mechanics and Hamilton-Jacobi Theory
Published in M.D.S. Aliyu, Nonlinear H∞-Control, Hamiltonian Systems and Hamilton-Jacobi Equations, 2017
Definition 4.1.3 Let (T⋆M, ω2) be a symplectic-manifold and letH:T⋆M→ℜ,be a Hamiltonian function. Then, the vector-field XHdetermined by the condition () ω2(XH,Y)=dH(Y)for all vector-fields Y, is called the Hamiltonian vector-field with energy function H. We call the tuple (T⋆M, ω2, XH) a Hamiltonian system.
Quasi-periodic solutions for a class of wave equation system
Published in Applicable Analysis, 2023
Let with only finitely many non-zero components of positive integers. Denote and let where is the parameter set, belongs to in parameters σ and µ. and Denote the weighted norm of F by To function F, we associate a Hamiltonian vector field defined by Its weighted norm is defined by
Exergetic port-Hamiltonian systems: modelling basics
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Markus Lohmayer, Paul Kotyczka, Sigrid Leyendecker
The Leibniz rule also implies that for some , their bracket depends only on the differentials . It follows that the Poisson bracket can be defined in terms of an antisymmetric4 contravariant -tensor field like so: . In terms of the Poisson bivector (field) , the Jacobi identity can be expressed as . Definition 2.2 (Hamiltonian vector field). Let be a state manifold and let {} be a Poisson structure on which is defined by a Poisson bivector . Let . Then, is called the Hamiltonian vector field corresponding to the Hamiltonian (function) .
Stabilization of small solutions of discrete NLS with potential having two eigenvalues
Published in Applicable Analysis, 2021
Let . Then the Hamiltonian vector field with respect to the symplectic form is defined by the relation . Comparing and we have where is defined by . In the following, we set .