Lissajous motion

Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a hamionic excited state is used because that is the way the first calculations were perfomied.) (a) The classical trajectory that originates from rest on the ground-state surface makes a vertical transition to the excited state, and subsequently undergoes Lissajous motion, which is shown superimposed, (b) Assuming a vertical transition down at time (position and momentum conserved) the trajectory continues to evolve on the ground-state surface and exits from chaimel 1. (c) If the transition down is at time 2 the classical trajectory exits from chaimel 2 (reprinted from [52]).

Even with this simple model it is clear that if one of the nuclei is given a sudden displacement it is very likely that the whole molecule will undergo a very complicated motion, a Lissajous motion, consisting of a mixture of angle-bending and bond-stretching. The Lissajous motion can always be broken down into a combination of the so-called normal vibrations of the system which, in the Lissajous motion, are superimposed in varying proportions. 

In the non-periodic case the motion is analogous to that which in two dimensions is called a Lissajous-motion, the path being closed only in the event of a rational relation between the vks. We consider the path in the te-space, confined to a standard unit cell of the period lattice (see 13) by replacing every point on the actual path by the equivalent point in the standard cell. If there are no linear integral relations between the efc s, this path in the w-spacc approaches indefinitely near to each point in the standard cell (as proved in Appendix I). The representation of the 5-space in the w-space is continuous so in this case the path in the 5-space approaches indefinitely close to every point of an/-dimensional region. 

The Martindale abrader is usually seen as a four station machine which uses cloth as the abradant, but coarser and faster acting materials can be substituted. The principal feature of this machine is that the test pieces are rubbed successively in different directions as the motion takes the form of a Lissajous figure. It is mostly used with coated fabrics62. 

Consider the case where a circular annulus of fluid is undergoing circular motion at radius r with angular speed w. The two-dimensional VEXSY spectrum has the character of a Lissajous figure ... 

Fig. 6.—Curve representing the motion of a system with two degrees of freedom, in which the two frequencies I l and I a are incommensurable (Lissajous figure).

At the lower energy, 0.05e [see Fig. 2.6(a)], in all the orders one ean see the approximate Lissajous figures, whieh implies that in the subspaee of these two nonreactive dof, the motions are eomposed of two approximately decoupled, simple harmonic oscillations. As the total energy increases to 10 times higher, 0.5e, as one may anticipate from Figure 2.5, that no approximate invariants of motion survive in the nonreactive subspace, and the nonreactive modes change from regular to fully chaotic dynamics [see Fig. 2.6(h)]. 

More precisely the two energy surfaces A and B will intersect each other along a curve. The chance of predissociation for a vibration-rotation level of B will be great—apart from the fulfilment of selection rules perhaps introduced by the symmetry properties of the molecule—if its energy is about the same as that of a point on the intersection curve. For then the vibratory motion of the molecule represented by a sort of Lissajous figure on the surface V pa) come somewhere near the line of intersection with the surface V (p, pa) of A, making it easy for the molecule to jump from the former to the latter surface. 

Lissajous figures A curve in one plane traced by a point moving under the influence of two independent harmonic motions. In the common case the harmonic motions are simple, perpendicular to each other, and have a simple frequency ratio. They can be displayed by applying sinusoidal alternating potentials to the X- and Y-inputs of a cathode-ray oscilloscope. They are named after Jules Lissajous (1822-80). 

A three-dimensional representation of an ion trajectory, shown in Figure 8, has the general appearance of a Lissajous curve composed of two fundamental frequency components, rUf O and >z,o of the secular motion. Higher-order (n) frequencies exist and the family of frequencies is described by given by... 

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