Cycloidal motion. Conservative positional, and no motional, forces. Equation of energy in realized general solution. Artificial or ideal ac system. 343 p. A short algebraic proof of (b) could no doubt be easily given; but our somewhat elaborate discussion of the subject is important as showing in (15)...(20) the whole relation between the previous short algebraic investigation, conducted in terms involving quantities which are essentially imaginary for the case of oscillations about a configuration of stable equilibrium, and the fully realized solution, with formulas for the potential and kinetic energies realized both for oscillations and for fallings away from unstable equilibrium. We now see definitively by (15) and (17) that, in real dynamics (that is to say T essentially positive) the factors (a, a), etc., all Adding (a', a'), etc., are all negative, and V (r, r), V (r', r′), (a', a), etc. + o2T (r, r) + o' (r', r') + etc. }.....(21). It is interesting to see in this formula how the constancy of the sum of the potential and kinetic energies is attained in any solution of the form Aet+ Be- [which, with λ=σ-1, includes the form r cos (σt − e)], and to remark that for any single solution act, or solution compounded of single solutions depending on unequal values of A3 (whether real or imaginary), the sum of the potential and kinetic energies is essentially zero. 344. When the positional forces of a system violate the law cumulative of conservatism, we have seen (§ 272) that energy without limit may be drawn from it by guiding it perpetually through a returning cycle of configurations, and we have inferred that in every real system, not supplied with energy from without, the positional forces fulfil the conservative law. But it is easy to arrange a system artificially, in connexion with a source of energy, so that its positional forces shall be non-conservative; and the consideration of the kinetic effects of such an arrangement, especially of its oscillations about or motions round a configuration of equilibrium, is most instructive, by the contrasts which it presents to the phenomena of a natural system. The preceding formulas, (7)...(9) of § 343 ƒ and § 343 g, express the general solution of the problem-to find the infinitely small motion of a cycloidal system, when, without motional forces, there is deviation from conservatism by the character of the positional forces. or ideal ac system. In this case [(10) not fulfilled,] just as in the case of motional Artificial forces fulfilling the conservative law (10), the character of the cumulative equilibrium as to stability or instability is discriminated accord- Criterion of ing to the character of the roots of an algebraic equation of degree equal to the number of degrees of freedom of the system. If the roots (A) of the determinantal equation § 343 (8) are all real and negative, the equilibrium is stable in every other case it is unstable. 345. But although, when the equilibrium is stable, no possible infinitely small displacement and velocity given to the system can cause it, when left to itself, to go on moving farther and farther away till either a finite displacement is reached, or a finite velocity acquired; it is very remarkable that stability should be possible, considering that even in the case of stability an endless increase of velocity may, as is easily seen from § 272, be obtained merely by constraining the system to a particular closed course, or circuit of configurations, nowhere deviating by more than an infinitely small amount from the configuration of equilibrium, and leaving it at rest anywhere in a certain part of this circuit. This result, and the distinct peculiarities of the cases of stability and instability, will be sufficiently illustrated by the simplest possible example, that of a material particle moving in a plane. Let the mass be unity, and the components of force parallel to two rectangular axes be ax + by, and a'x+b'y, when the position of the particle is (x, y). The equations of motion will be The terms ey and ex are clearly the components of a force e (x+y), perpendicular to the radius-vector of the particle. Hence if we turn the axes of co-ordinates through any angle, the VOL. I. 25 Artificial or ideal accumulative system. corresponding terms in the transformed components are still -ey and ex. If, therefore, we choose the axes so that the equations of motion become, without loss of generality, To integrate these, assume, as in general [§ 343 (2)], (2), This shows that the equilibrium is stable if both aß + e2 and a+ẞ are positive and e2 < (a - 6) but unstable in every other case. But let the particle be constrained to remain on a circle, of radius r. Denoting by its angle-vector from OX, and transforming (§ 27) the equations of motion, we have == (3 − a) sin cos 0 + e − − 1 (ẞ − a) sin 20 + e................. (4). = = If we had e 0 (a conservative system of force) the positions of equilibrium would be at 0=0, 0, 0, and 3; and the motion would be that of the quadrantal pendulum. But when e has any finite value less than (B-a) which, for convenience, we may suppose positive, there are positions of equilibrium at third being positions of stable, and the second and fourth of unstable, equilibrium. Thus it appears that the effect of the constant tangential force is to displace the positions of stable and unstable equilibrium forwards and backwards on the circle. through angles each equal to 9. And, by multiplying (4) by 20dt and integrating, we have as the integral equation of energy 0o = C + } (ẞ − a) cos 20 + 2e0.. ............ ..(5). ideal ac From this we see that the value of C, to make the particle Artificial or just reach the position of unstable equilibrium, is cumulative system. 4 and by equating to zero the expression (5) for Ở3, with this value Π 2 -0, behind it, it will vibrate. But if placed anywhere beyond those limits and left either at rest or moving with any velocity in either direction, it will end by flying round and round forwards with a periodically increasing and diminishing velocity, but increasing every half turn by equal additions to its squares. If on the other hand e>(B-a), the positions both of stable and unstable equilibrium are imaginary; the tangential force predominating in every position. If the particle be left at rest in any part of the circle it will fly round with continually increasing velocity, but periodically increasing and diminishing acceleration. 345'. Leaving now the ideal case of positional forces violating the law of conservatism, interestingly curious as it is, and instructive in respect to the contrast it presents with the positional forces of nature which are essentially conservative, let us henceforth suppose the positional forces of our system to be conservative and let us admit infringement of conservatism only as in nature through motional forces. We shall soon see (§ 345l and) that we may have motional forces which do not violate the law of conservatism. At present we make no restriction Cycloidal upon the motional forces and no other restriction on the positional forces than that they are conservative. system with conserva tive positional forces and unrestricted forces. The differential equations of motion, taken from (1) of 343a motional above, with the relations (10), and with V to denote the potential energy, are, Multiplying the first of these by 1, the second by 4, adding and transposing, we find Dissipativity defined. where Q =11 + (12 + 21),+22, +(13+31) j,Åg + etc......(3). 345". The quadratic function of the velocities here denoted by Q has been called by Lord Rayleigh* the Dissipation Function. We prefer to call it Dissipativity. It expresses the rate at which the palpable energy of our supposed cycloidal system is lost, not, as we now know, annihilated but (§§ 278, 340, 341, 342) dissipated away into other forms of energy. It is essentially positive when the assumed motional forces are such as can exist theorem of in nature. That it is equal to a quadratic function of the velocities is an interesting and important theorem. Lord Rayleigh's Dissipa tivity. 0 where E is a constant denoting the sum of the kinetic and potential energies at the instant t=0. Now T and Q are each of them essentially positive except when the system is at rest, and then each of them is zero. Therefore .00. fedt must increase to infinity unless the system comes more and more nearly to rest * Proceedings of the London Mathematical Society, May, 1873; Theory of Sound, Vol. 1. § 81. |