Motion space projection

By definition, the projection of fi onto the motion space of the single joint is t. The unknown constraint forces and moments at joint 1 have been eliminated. It is also true that ai = Ji iji, since the joint velocities are all assumed to be zefo. Thus, Equation 3.6 becomes ... 

The term f2 represents the projection of onto the basis vectors of the motion space, which are expressed with respect to the cotndinate system of link 2. By Equation 3.4, we may write the two-link Jacobian as ... 

The t-link Jacobian matrix is used to project the force terms onto the motion space of the augmented manipulator... 

The component of a spatial vector which lies on or along a specific axis of a joint may be determined by performing a simple dot product between that spatial vector and the vector which represents the joint axis. This may be refen to as projecting the spatial vector onto the joint axis . Thus, we may project the terms of Equation 4.32 onto the motion space of joint 2 via the following spatial dot product ... 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

For the Berry phase, we shall quote a definition given in [164] ""The phase that can be acquired by a state moving adiabatically (slowly) around a closed path in the parameter space of the system. There is a further, somewhat more general phase, that appears in any cyclic motion, not necessarily slow in the Hilbert space, which is the Aharonov-Anandan phase [10]. Other developments and applications are abundant. An interim summai was published in 1990 [78]. A further, more up-to-date summary, especially on progress in experimental developments, is much needed. (In Section IV we list some publications that report on the experimental determinations of the Berry phase.) Regarding theoretical advances, we note (in a somewhat subjective and selective mode) some clarifications regarding parallel transport, e.g., [165], This paper discusses the projective Hilbert space and its metric (the Fubini-Study metric). The projective Hilbert space arises from the Hilbert space of the electronic manifold by the removal of the overall phase and is therefore a central geometrical concept in any treatment of the component phases, such as this chapter. 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

There is a probability N 8R do for a collision specified by energy transfer T to the translational motion of the struck atom. N is the volume density of atoms, and one approximation for do is given by Equation 9. If a collision occurs, the particle has a probability F(x — 5x, E — T,ri ) of obtaining a total projected range, x. Therefore, N 5R do F(x — 5x, E — T,r ) is the contribution from this specified collision to the total probability for the projected range, x. When this term space is integrated over all collisions, the total contribution becomes ... 

To understand these distributions, one needs to consider the fission transition nucleus. Figure 11.22 shows a coordinate system for describing this nucleus in terms of its quantum numbers J, the total angular momentum M, the projection of J upon a space-fixed axis, usually taken to be the direction of motion of the fissioning system, and K, the projection of J upon the nuclear symmetry axis. 

---