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.
In this paper, a recursive O(n) method to obtain a set of Hamiltonian equations for open-loop and constrained multibody system is briefly discussed. The method is then used to perform a numerical comparison of acceleration based and canonical momenta based equations of motion. A relatively simple example consisting of a biped during double support phase is used for that purpose. While no significant difference in efficiency is found when using a fixed step numerical integration method, the Hamiltonian equations perform considerably better when using an adaptive method. This is at least the case when the error control is applied straightforwardly. Both methods can be made equally efficient by removing the error control on the velocities for the acceleration based equations.
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.
First-order perturbative Hamiltonian equations of motion for a point particle orbiting a Schwarzschild black hole
