System holonomic

All "conservative holonomic" systems must satisfy (at nonrelativistic speeds) the time-dependent Schrodinger equation ... 

In the first chapter no attempt will be made to give any parts of classical dynamics but those which are useful in the treatment of atomic and molecular problems. With this restriction, we have felt justified in omitting discussion of the dynamics of rigid bodies, non-conservative systems, non-holonomic systems, systems involving impact, etc. Moreover, no use is made of Hamilton s principle or of the Hamilton-Jacobi partial differential equation. By thus limiting the subjects to be discussed, it is possible to give in a short chapter a thorough treatment of Newtonian systems of point particles. 

From a mathematical point of view, the natural concept of a numerical method for the holonomic system (4.7)-(4.9) is that of a mapping from T M to itself which approximates the dynamical evolution on a step of length h. 

Is (4.I4)-(4.I6) symplectic (in the sense outlined above for holonomic systems) ... 

In mathematics and physics, anonholonomic system is a system in which a return to the original internal configuration does not guarantee return to the original system position (Bloch, 2003). In other words, unlike with a holonomic system, the outcome of a nonholonomic system is path-dependent, and the number of generalized coor-... 

An initial and desired final configuration of a system can be used by the targeted molecular dynamics (TMD) method (Schlitter et al., 1993) to establish a suitable pathway between the given configurations. The resulting pathway, can then be employed during further SMD simulations for choosing the direction of the applied force. TMD imposes time-dependent holonomic constraints which drive the system from one known state to another. This method is also discussed in the chapter by Helms and McCammon in this volume. 

The concept of a symplectic method is easily extended to systems subject to holonomic constraints [22]. For example the RATTLE discretization is found to be a symplectic discretization. Since SHAKE is algebraically equiva lent to RATTLE, it, too, has the long-term stability of a symplectic method. 

CE uses holonomic constraints. In a constrained system the coordinates of the particles 5t independent and the equations of motion in each of the coordinate directions are cted. A second difficulty is that the magnitude of the constraint forces is unknown, in the case of the box on the slope, the gravitational force acting on the box is in the ction whereas the motion is down the slope. The motion is thus not in the same direc-s the gravitational force. As such, the total force on the box can be considered to arise wo sources one due to gravity and the other a constraint force that is perpendicular to otion of the box (Figure 7.8). As there is no motion perpendicular to the surface of the the constraint force does no work. 

The first step in the DG calculations is the generation of the holonomic distance matrix for aU pairwise atom distances of a molecule [121]. Holonomic constraints are expressed in terms of equations which restrict the atom coordinates of a molecule. For example, hydrogen atoms bound to neighboring carbon atoms have a maximum distance of 3.1 A. As a result, parts of the coordinates become interdependent and the degrees of freedom of the molecular system are confined. The acquisition of these distance restraints is based on the topology of a model structure with an arbitrary, but energetically optimized conformation. 

This section presents the notation for generalized coordinates, constraints, basis vectors, and tensors that is used throughout the paper. We consider a system consisting of N pointlike particles (beads) with positions R, ..., R with masses mi,..., mj. The positions of the beads are subject to K holonomic constraints, of the form... 

This system is called "conservative," or "holonomic," if and only if it satisfies the following two conditions ... 

We consider an A -particle mechanical system. A set of K constraints applied to it is holonomic whenever all the relationships connecting the natural coordinates Qi, i = 1,2. .. 3 N of the particles in the system plus the time t — and which are a mathematical counterpart to the existence of constraints inside the system — are of the form ... 

Any conservative mechanical system which is either free or subject to holonomic constraints and whose potential does not depend on the generalized velocities is described by standard equations of motion (either Lagrangian or Hamiltonian). The kinetic energy of the iV-particIe system is ... 

We finally note that is possible to use Gauss principle to obtain equations of motion when the system is subject to holonomic constraints such as bond length or bond angle constraints. In this case one obtains the same equations of motion as one would obtain by applying the Lagrange equation. 

Lastly it is noted that an effective potential of a form similar to Eq. (2.19) appiears when use is made of holonomic constraints in Brownian dynamics simulations, that is, in the use of fixed bond lengths and bond angles in studies of dihedral angle conformational relaxation in polymer systems. In fact the two potentials represent different physical phenomena, but each acts so as to yield the correct equilibrium distribution in the coordinates of interest. The two approaches, of thermalization and constraint, have been compared for the case of the four-particle chain. ... 

Although conventional Verlet-type molecular dynamics places restraints on bond lengths and bond angles, one could conceivably want to implement these restrictions as holo-nomic constraints. This is supported by the observation that the deviations from ideal bond lengths and bond angles are usually small in X-ray crystal structures. There are essentially two possible approaches to solve Newton s equations (Eq. 12) with holonomic constraints. The first involves a switch from Cartesian coordinates f) to generalized internal ones ft. Having thus redefined the system, one would solve equations of motion for the... 

The coding procedure involves a rather complex unitary transformation in the Hilbert space of the compound system performing such coding operations is a non-trivial quantum control problem in itself. However, one can achieve this objective by employing the idea of non-holonomic control which we have previously presented [Akulin 2001] by applying two different Hamiltonians in sequence, one can build any desired unitary transformation. 

The paper is organized as follows. In Sec. 1, we introduce the main features of quantum error-correction, and, particularly, we present the already well-developed theory of quantum error-correcting codes. In Sec. 2, we present a multidimensional generalization of the quantum Zeno effect and its application to the protection of the information contained in compound systems. Moreover, we suggest a universal physical implementation of the coding and decoding steps through the non-holonomic control. Finally, in Sec. 3, we focus on the application of our method to a rubidium isotope. 

The section starts by the presentation of a multidimensional generalization of the quantum Zeno effect, which we then employ to protect information in compound systems. Finally, we suggest the non-holonomic control technique as a physical way to implement the coding / decoding steps of our scheme. 

In this section, we have presented a method which allows one to protect information from a given set of unitary errors. Moreover, we have suggested a physical way to implement the coding and decoding steps, employing the non-holonomic control. In the last section, we propose an application of our method to a real physical system, namely a Rubidium isotope. 

Figure 4. Coding step through the non-holonomic control technique. The two Hamiltonians Ha and Hb are alternately applied to the system during pulses whose timings are reproduced in Tab. 1. The pulsations of the laser fields involved in the coding step are represented on the spectrum of the rubidium atom.

Attempts to base the Ml theory on a kinetic energy term associated with the separation parameter generally fail. Only in the non-holonomic constraint method has any success been achieved with a dynamic interpretation of the separation parameter, and it falls apart with the one-electron atomic system. 

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