Now let the liquid given at rest in such a space be set in motion by any arbitrarily specified variation of position of the envelope. The liquid within will be set in motion in a manner depending entirely on the motion of the envelope. It is possible to conceive of other motions of the liquid than that taken, which all agree in having the specified motion of the surface. Thomson's theorem asserts that the motion actually taken has less kinetic energy than that of any of the other motions which have the same motion of the bounding surface.
The motion produced has the property described by the word "irrotational," that is, the elements of the fluid have no spinning motion—they move without rotation. A small portion of a fluid may describe any path—may go round in a circle, for example—and yet have no rotation. The reader may imagine a ball carried round in a circle, but in such a way that no line in the body ever changes its direction. The body has translation, but no spin.
Irrotationality of a fluid is secured, as stated above, when the velocity of each element in any direction is the rate of variation per unit distance in that direction of a certain function of the coordinates, the distances, taken parallel to three lines perpendicular to one another and drawn from a point, which specify the position of the particle. In fact, what is called a velocity-potential exists, similar to the potential described in Chapter IV above, for an electric field. This condition, together with the specified motion of the surface, suffices to determine the motion of the fluid.
Two important particular consequences were pointed out by Thomson: (1) that the motion of the fluid at any instant depends solely on the form and motion of the bounding surface, and is therefore independent of the previous motion; and (2) that if the bounding surface be instantaneously brought to rest, the liquid throughout the vessel will also be instantly brought to rest.
This theorem was afterwards generalised by Thomson (Proc. R.S.E., 1863), and applied to any material system of connected particles set into motion by specified velocities simultaneously and suddenly imposed at selected points of the system. It was already known that the kinetic energy of a system of bodies connected in any manner, and set in motion by impulses applied at specified points, was either a maximum or a minimum, as compared with that for any other motion compatible with these impulses, and with the connections of the system. This was proved by Lagrange in the Mécanique Analytique as a generalisation of a theorem given by Euler for a rigid body set into rotation by an impulse.
Bertrand proved in 1842 that when the impulses applied are given in amount, and are applied at specified points, the system starts off with kinetic energy greater than that of any other motion which is consistent with the given impulses and the connections of the system. This other motion must be such as could be produced in the system by the given impulses, together with any other set of impulses capable of doing no work on the whole.
Thomson's theorem is curiously complementary to Bertrand's. Let the system be acted on by impulses applied at certain specified points, and by no other impulses of any kind; and let the impulses be such as to start those selected points with any prescribed velocities. The system will start off with kinetic energy which is less than that of any other motion which the system could have consistently with the prescribed velocities, and which it could be constrained to take by impulses which do no work on the whole. In each case the difference of energies is the energy of the motion which must be compounded with one motion to give the other which is compared with it.
A simple example, such as might be taken of the particular case considered by Euler, may help to make these theorems clear. Imagine a straight uniform rod to lie on a horizontal table, between which and the rod there is no friction. Let the rod be struck a blow at one end in a horizontal direction at right angles to the length of the rod. If no other impulse acts, the end of the rod will move off with a certain definite velocity, and the other parts of the rod (which is supposed perfectly unbending) will be started by the connections of the system. It is obvious that any number of other motions of the rod can be imagined, all of which give the same motion of the extremity struck. But the actual motion taken is one of turning about that point of the rod which is two-thirds of the length from the end struck. If the reader will consider the kinetic energy for any other horizontal turning motion consistent with the same motion of the end, he will find that the kinetic energy is greater than that of the motion just specified. This motion could be produced by applying at the point about which the rod turns the impulse required to keep that point at rest. The impulse so applied would do no work. The actual value is 1⁄8mv2, where m denotes the mass of the rod and v the velocity of the end. If the motion taken were one of rotation about a point of the rod at distance x from the end struck, the kinetic energy would be m (4l2 − 6lx + 3x2) v2 ⁄ 6x2, where 2l is the length of the rod, and this has its least value 1⁄8mv2 for x = 4l ⁄ 3. For example, x = 2l gives 1⁄6mv2, which is greater than the value just found.
Bertrand's theorem applied to this case of motion is not quite so easy, perhaps, to understand. The motion which is said to have maximum energy is one given by a specified impulse at the end struck, and this, in the absence of any other impulses, would be a motion of minimum energy. But let the alternative motion, which is to be compared with that actually taken, be one constrained by additional impulses such as can together effect no work, and the existence of the maximum is accounted for. The kinetic energy produced is one-half the product of the impulse into the velocity of the point struck, that is ½Iv, and it has just been seen that this is the product of 1⁄6mv2 by the factor (4l2 − 6lx + 3x2) ⁄ x2. This factor is 3I ⁄ mv, and is a minimum when x = 4l ⁄ 3. Thus for a given I, v will have its maximum value when the factor referred to is least, and ½Iv will then be a maximum.
The bar can be constrained to turn about another point by a fixed pivot there situated. An impulse will be applied to the rod by the pivot, simultaneously with the blow; and it is obvious that this impulse does no work, since there is no displacement of the point to which it is applied.