Jordan and Einstein Frames Hamiltonian Analysis for FLRW Brans-Dicke Theory

We analyze the Hamiltonian equivalence between Jordan and Einstein frames considering a mini-superspace model of the flat Friedmann–Lemaître–Robertson–Walker (FLRW) Universe in the Brans–Dicke theory. Hamiltonian equations of motion are derived in the Jordan, Einstein, and anti-gravity (or anti-Newtonian) frames. We show that, when applying the Weyl (conformal) transformations to the equations of motion in the Einstein frame, we did not obtain the equations of motion in the Jordan frame. Vice-versa, we re-obtain the equations of motion in the Jordan frame by applying the anti-gravity inverse transformation to the equations of motion in the anti-gravity frame.

Nonlocal fractal calculus based analyses of electrical circuits on fractal set

Purpose The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

On a class of integrable Hamiltonian equations in 2+1 dimensions

We classify integrable Hamiltonian equations of the form u t = ∂ x ( δ H δ u ) , H = ∫ h ( u , w ) d x d y , where the Hamiltonian density h ( u , w ) is a function of two variables: dependent variable u and the non-locality w = ∂ x − 1 ∂ y u . Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h ). We show that the generic integrable density is expressed in terms of the Weierstrass σ -function: h ( u , w ) = σ ( u ) e w . Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

Conformal Hamiltonian Mechanical Equations on Contact 9- Manifolds

In this study, we concluded the Hamiltonian equations on, being a model. Finally, introduce, some geometrical and physical results on the related mechanic systems have been discussed.

The Hamiltonian equations of motion. Poisson brackets

This chapter addresses invariance of the Hamiltonian function under a given transformation and the conservation law, the Hamiltonian function for the beam of light, the motion of a charged particle in a nonuniform magnetic field, and the motion of electrons in a metal or semiconductor. The chapter also discusses the Poisson brackets and the model of the electron and nuclear paramagnetic resonances, the Poisson brackets for the components of the particle velocity, and the “hidden symmetry” of the hydrogen atom.

The Hamiltonian equations of motion. Poisson brackets

This chapter addresses invariance of the Hamiltonian function under a given transformation and the conservation law, the Hamiltonian function for the beam of light, the motion of a charged particle in a nonuniform magnetic field, and the motion of electrons in a metal or semiconductor. The chapter also discusses the Poisson brackets and the model of the electron and nuclear paramagnetic resonances, the Poisson brackets for the components of the particle velocity, and the “hidden symmetry” of the hydrogen atom.