Hamiltonian systems general equations

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

The basic issue is at a higher level of generality than that of the particular mechanical assumptions (Newtonian, quantum-theoretical, etc.) concerning the system. For simplicity of exposition, we deal with the classical model of N similar molecules in a closed vessel "K, intermolecular forces being conservative, and container forces having a force-function usually involving the time. Such a system is Hamiltonian, and we assume that the potentials are such that its Hamiltonian function is bounded below. The statistics of the system are conveyed by a probability density function 3F defined over the phase space QN of our Hamiltonian system. Its time evolution is completely determined by Liouville s equation... 

Here % specify the transformation from coordinate system j to system i. In Equation 3 only Dq q (Qdm) varies with the molecular motion. Since amphiphilic liquid crystalline systems generally are cylindrically symmetrical around the director Dq q (nDM) = 0 if qf 0. If it also is assumed that a nucleus stays within a domain of a given orientation of the director over a time that is long compared with the inverse of the quadrupole interaction, one now obtains for the static quadrupole hamiltonian... 

Briefly, the aim of Lie transformations in Hamiltonian theory is to generate a symplectic (that is, canonical) change of variables depending on a small parameter as the general solution of a Hamiltonian system of differential equations. The method was first proposed by Deprit [75] (we follow the presentation in Ref. 76) and can be stated as follows. 

Until now, we have discussed NHIMs in general dynamical systems. In this section, we limit our argument to Hamiltonian systems and show how singular perturbation theory works. In particular, we discuss NHIMs in the context of reaction dynamics. First, we explain how NHIMs appear in conventional reaction theory. Then, we will show that Lie permrbation theory applied to the Hamiltonian near a saddle with index 1 acmally transforms the equation of motion near the saddle to the Fenichel normal form. This normal form can be considered as an extension of the Birkhoff normal form from stable fixed points to saddles with index 1 [2]. Finally, we discuss the transformation near saddles with index larger than 1. 

X. Chapuisat, A. Nauts, and G. Durand, A method to obtain the Eckart Hamiltonian and the equations of motion of a highly deformable polyatomic system in terms of generalized... 

The stability that we mentioned before refers to the evolution of the deviation vector ( ) = x (t) — x(l) between the perturbed solution x (t) and the periodic orbit x(t) at the same time t. If ( ) is bounded, then the periodic orbit is stable. In this case two particles, one on the periodic orbit x t) and the other on the perturbed orbit x (t), that start close to each other at t = 0, would always stay close. A necessary condition is that all the eigenvalues of the monodromy matrix be on the unit circle in the complex plane. However, in a Hamiltonian system this condition is not enough for stability, because there is only one eigenvector corresponding to the double unit eigenvalue and consequently a secular term always appears in the general solution, as can be seen from Equation (49). We remark that this secular term appears if the vector of initial deviation (0) = a (0) — s(0) has a component along the direction /2(C)). 

Consider the system of differential equations(not Hamiltonian in general)... 

For separable systems the Schrodinger equation, represented by a partial differential equation, is mapped onto uni-dimensional differential equations. The eigenvalues (separation constants) of each of the 3 uni-dimensional (in general n uni-dimensional) differential equations can be used to label the eigenfunctions (r) and hence serve as quantum numbers. Integrability of a n-dimensional Hamiltonian system requires the existence of n commuting observables O/, 1 i in involution ... 

Some General Principles of Integration OF Hamiltonian Systems OF Differential Equations... 

The proof is reduced to a direct calculation, and we leave it to the reader. Thus, on embedding the equations of motion of a rigid body into a suitable Lie algebra we find out that they have become a Hamiltonian system on the orbits of the action of the group SO (3). This fact is a particular case of the general mechanism which makes it possible to integrate many mechanical systems similar to the one just described. 

Bearing in mind the above agreements, the spin dynamics of superfluid helium-3 can be formulated within the usual Hamiltonian mechanics, and it appears to be part of the general theory of dynamical systems on orbits of Lie groups. In what follows we ignore the physically important dissipation effects (there may, in fact, be taken into account) and examine a conservative Hamiltonian system, referred to as Leggett equations, which is given by ... 

Gardner, C. S. "Korteweg-de Vries equation and generalization. IV "The Korteweg-de Vries equation as a Hamiltonian system. J. Math. Phys. 12... 

The potential energy of the particles will depend on the positions of the particles. Hamilton determined that for a generalized coordinate system, the equations of motion could be obtained from the Hamiltonian and from the following identities ... 

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

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