The law of the inverse cube P = μu3 is interesting by way of contrast. The orbits may be divided into two classes according as h2 ≷ μ, i.e. according as the transverse velocity (hu) is greater or less than the velocity √μ·u appropriate to a circular orbit at the same distance. In the former case the equation (17) takes the form
| d 2u | + m2u = 0, |
| dθ2 |
the solution of which is
The orbit has therefore two asymptotes, inclined at an angle π/m. In the latter case the differential equation is of the form
| d 2u | = m2u, |
| dθ2 |
so that
If A, B have the same sign, this is equivalent to
if the origin of θ be suitably adjusted; hence r has a maximum value α, and the particle ultimately approaches the pole asymptotically by an infinite number of convolutions. If A, B have opposite signs the form is
this has an asymptote parallel to θ = 0, but the path near the origin has the same general form as in the case of (23). If A or B vanish we have an equiangular spiral, and the velocity at infinity is zero. In the critical case of h2 = μ, we have d 2u/dθ2 = 0, and
the orbit is therefore a “reciprocal spiral,” except in the special case of A = 0, when it is a circle. It will be seen that unless the conditions be exactly adjusted for a circular orbit the particle will either recede to infinity or approach the pole asymptotically. This problem was investigated by R. Cotes (1682–1716), and the various curves obtained are known as Coles’s spirals.
A point on a central orbit where the radial velocity (dr/dt) vanishes is called an apse, and the corresponding radius is called an apse-line. If the force is always the same at the same distance any apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be reversed. It follows that the angle between successive apse-lines is constant; it is called the apsidal angle of the orbit.
If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5),
| h2 | = 2 Pdr ; |
| p2 |
this determines the form of the critical orbit, as it is called. If P = μ/r n, its polar equation is
where m = 12(3 − n), except in the case n = 3, when the orbit is an equiangular spiral. The case n = 2 gives the parabola as before.
If we eliminate dθ/dt between (15) and (16) we obtain
| d 2r | − | h2 | = −P = −ƒ(r), |
| dt 2 | r 3 |
say. We may apply this to the investigation of the stability of a circular orbit. Assuming that r = a + x, where x is small, we have, approximately,
| d 2x | − | h2 | ( 1 − | 3x | ) = −ƒ(a) − xƒ′(a). |
| dt 2 | a3 | a |
Hence if h and a be connected by the relation h2 = a3ƒ(a) proper to a circular orbit, we have
| d 2x | + { ƒ′(a) + | 3 | ƒ(a) } x = 0 |
| dt 2 | a |
If the coefficient of x be positive the variations of x are simple-harmonic, and x can remain permanently small; the circular orbit is then said to be stable. The condition for this may be written
| d | { a3ƒ(a) } > 0, |
| da |
i.e. the intensity of the force in the region for which r = a, nearly, must diminish with increasing distance less rapidly than according to the law of the inverse cube. Again, the half-period of x is π/√{ƒ′(a) + 3−1ƒ(a)}, and since the angular velocity in the orbit is h/a2, approximately, the apsidal angle is, ultimately,
| π √ | ƒ(a) | , |
| aƒ′(a) + 3ƒ(a) |
or, in the case of ƒ(a) = μ/r n, π/√(3 − n). This is in agreement with
the known results for n = 2, n = −1.
We have seen that under the law of the inverse square all finite orbits are elliptical. The question presents itself whether there then is any other law of force, giving a finite velocity from infinity, under which all finite orbits are necessarily closed curves. If this is the case, the apsidal angle must evidently be commensurable with π, and since it cannot vary discontinuously the apsidal angle in a nearly circular orbit must be constant. Equating the expression (30) to π/m, we find that ƒ(a) = C/an, where n = 3 − m2. The force must therefore vary as a power of the distance, and n must be less than 3. Moreover, the case n = 2 is the only one in which the critical orbit (27) can be regarded as the limiting form of a closed curve. Hence the only law of force which satisfies the conditions is that of the inverse square.
At the beginning of § 13 the velocity of a moving point P was represented by a vector OV→ drawn from a fixed origin O. The locus of the point V is called the hodograph (q.v.); and it appears that the velocity of the point V along the hodograph represents in magnitude and in direction the acceleration in the original orbit. Thus in the case of a plane orbit, if v be the velocity of P, ψ the inclination of the direction of motion to some fixed direction, the polar co-ordinates of V may be taken to be v, ψ; hence the velocities of V along and perpendicular to OV will be dv/dt and vdψ/dt. These expressions therefore give the tangential and normal accelerations of P; cf. § 13 (12).
 |
| Fig. 69. |
In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity. In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit. In the case of a central orbit described under the law of the inverse square we have v = h/SY = h. SZ/b2, where S is the centre of force, SY is the perpendicular to the tangent at P, and Z is the point where YS meets the auxiliary circle again. Hence the hodograph is similar and similarly situated to the locus of Z (the auxiliary circle) turned about S through a right angle. This applies to an elliptic or hyperbolic orbit; the case of the parabolic orbit may be examined separately or treated as a limiting case. The annexed fig. 70 exhibits the various cases, with the hodograph in its proper orientation. The pole O of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola. In any case of a central orbit the hodograph (when turned through a right angle) is similar and similarly situated to the “reciprocal polar” of the orbit with respect to the centre of force. Thus for a circular orbit with the centre of force at an excentric point, the hodograph is a conic with the pole as focus. In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hodograph is a circle through the pole, described with constant velocity.
§ 15. Kinetics of a System of Discrete Particles.—The momenta of the several particles constitute a system of localized vectors which, for purposes of resolving and taking moments, may be reduced like a system of forces in statics (§ 8). Thus taking any point O as base, we have first a linear momentum whose components referred to rectangular axes through O are
its representative vector is the same whatever point O be chosen. Secondly, we have an angular momentum whose components are
these being the sums of the moments of the momenta of the several particles about the respective axes. This is subject to the same relations as a couple in statics; it may be represented by a vector which will, however, in general vary with the position of O.
The linear momentum is the same as if the whole mass were concentrated at the centre of mass G, and endowed with the velocity of this point. This follows at once from equation (8) of § 11, if we imagine the two configurations of the system there referred to to be those corresponding to the instants t, t + δt. Thus
| Σ ( m· | PP→ | ) = Σ(m)· | GG′→ | . |
| δt | δt |
Analytically we have
| Σ(mẋ) = | d | Σ(mx) = Σ(m)· | dx̄ | , |
| dt | dt |
with two similar formulae.
