(1)
The momentum of a particle is the vector obtained by multiplying the velocity by the mass m. Theimpulseof a force in any infinitely small interval of time δt is the product of the force into δt; it is to be regarded as a vector. The total impulse in any finite interval of time is the integral of the impulses corresponding to the infinitesimal elements δt into which the interval may be subdivided; the summation of which the integral is the limit is of course to be understood in the vectorial sense.
Newton’s Second Law asserts that change of momentum is equal to the impulse; this is a statement as to equality of vectors and so implies identity of direction as well as of magnitude. If X, Y, Z are the components of force, then considering the changes in an infinitely short time δt we have, by projection on the co-ordinate axes, δ(mu) = Xδt, and so on, or
(2)
For example, the path of a particle projected anyhow under gravity will obviously be confined to the vertical plane through the initial direction of motion. Taking this as the plane xy, with the axis of x drawn horizontally, and that of y vertically upwards, we have X = 0, Y = −mg; so that
(3)
The solution is
x = At + B, y = −1⁄2gt2+ Ct + D.
(4)
If the initial values of x, y, ẋ, ẏ are given, we have four conditions to determine the four arbitrary constants A, B, C, D. Thus if the particle start at time t = 0 from the origin, with the component velocities u0, v0, we have
x = u0t, y = v0t −1⁄2gt2.
(5)
Eliminating t we have the equation of the path, viz.
(6)
This is a parabola with vertical axis, of latus-rectum 2u02/g. The range on a horizontal plane through O is got by putting y = 0, viz. it is 2u0v0/g. we denote the resultant velocity at any instant by ṡ we have
ṡ2= ẋ2+ ẏ2= ṡ02− 2gy.
(7)
Another important example is that of a particle subject to an acceleration which is directed always towards a fixed point O and is proportional to the distance from O. The motion will evidently be in one plane, which we take as the plane z = 0. If μ be the acceleration at unit distance, the component accelerations parallel to axes of x and y through O as origin will be −μx, −μy, whence
(8)
The solution is
x = A cos nt + B sin nt, y = C cos nt + D sin nt,
(9)
where n = √μ. If P be the initial position of the particle, we may conveniently take OP as axis of x, and draw Oy parallel to the direction of motion at P. If OP = a, and ṡ0be the velocity at P, we have, initially, x = a, y = 0, ẋ = 0, ẏ = ṡ0whence
x = a cos nt, y = b sin nt,
(10)
if b = ṡ0/n. The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is described in the period 2π/√μ; moreover, the velocity at any point P is equal to √μ·OD, where OD is the semi-diameter conjugate to OP. This type of motion is calledelliptic harmonic. If the co-ordinate axes are the principal axes of the ellipse, the angle nt in (10) is identicalwith the “excentric angle.” The motion of the bob of a “spherical pendulum,”i.e.a simple pendulum whose oscillations are not confined to one vertical plane, is of this character, provided the extreme inclination of the string to the vertical be small. The acceleration is towards the vertical through the point of suspension, and is equal to gr/l, approximately, if r denote distance from this vertical. Hence the path is approximately an ellipse, and the period is 2π √(l/g).
The above problem is identical with that of the oscillation of a particle in a smooth spherical bowl, in the neighbourhood of the lowest point. If the bowl has any other shape, the axes Ox, Oy may be taken tangential to the lines of curvature at the lowest point O; the equations of small motion then ared2x= −gx,d2y= −gy,dt2ρ1dt2ρ2(11)where ρ1, ρ2, are the principal radii of curvature at O. The motion is therefore the resultant of two simple vibrations in perpendicular directions, of periods 2π √(ρ1/g), 2π √(ρ2/g). The circumstances are realized in “Blackburn’s pendulum,” which consists of a weight P hanging from a point C of a string ACB whose ends A, B are fixed. If E be the point in which the line of the string meets AB, we have ρ1= CP, ρ2= EP. Many contrivances for actually drawing the resulting curves have been devised.
The above problem is identical with that of the oscillation of a particle in a smooth spherical bowl, in the neighbourhood of the lowest point. If the bowl has any other shape, the axes Ox, Oy may be taken tangential to the lines of curvature at the lowest point O; the equations of small motion then are
(11)
where ρ1, ρ2, are the principal radii of curvature at O. The motion is therefore the resultant of two simple vibrations in perpendicular directions, of periods 2π √(ρ1/g), 2π √(ρ2/g). The circumstances are realized in “Blackburn’s pendulum,” which consists of a weight P hanging from a point C of a string ACB whose ends A, B are fixed. If E be the point in which the line of the string meets AB, we have ρ1= CP, ρ2= EP. Many contrivances for actually drawing the resulting curves have been devised.
It is sometimes convenient to resolve the accelerations in directions having a more intrinsic relation to the path. Thus, in a plane path, let P, Q be two consecutive positions, corresponding to the times t, t + δt; and let the normals at P, Q meet in C, making an angle δψ. Let v (= ṡ) be the velocity at P, v + δv that at Q. In the time δt the velocity parallel to the tangent at P changes from v to v + δv, ultimately, and the tangential acceleration at P is therefore dv/dt or s̈. Again, the velocity parallel to the normal at P changes from 0 to vδψ, ultimately, so that the normal acceleration is v dψ/dt. Since
(12)
where ρ is the radius of curvature of the path at P, the tangential and normal accelerations are also expressed by v dv/ds and v2/ρ, respectively. Take, for example, the case of a particle moving on a smooth curve in a vertical plane, under the action of gravity and the pressure R of the curve. If the axes of x and y be drawn horizontal and vertical (upwards), and if ψ be the inclination of the tangent to the horizontal, we have
(13)
The former equation gives
v2= C − 2gy,
(14)
and the latter then determines R.
In the case of the pendulum the tension of the string takes the place of the pressure of the curve. If l be the length of the string, ψ its inclination to the downward vertical, we have δs = lδψ, so that v = ldψ/dt. The tangential resolution then givesld2ψ= − g sin ψ.dt2(15)If we multiply by 2dψ/dt and integrate, we obtain(dψ)2=2gcos ψ + const.,dtl(16)which is seen to be equivalent to (14). If the pendulum oscillate between the limits ψ = ±α, we have(δψ)2=2g(cos ψ − cos α) =4g(sin21⁄2α − sin21⁄2ψ);dtll(17)and, putting sin1⁄2ψ = sin1⁄2α. sin φ, we find for the period (τ) of a complete oscillationτ = 4∫1⁄2π0dtdφ = 4√l·∫1⁄2π0dφdφg√(1 − sin21⁄2α · sin2φ)= 4√l· F1(sin1⁄2α),g(18)in the notation of elliptic integrals. The function F1(sin β) was tabulated by A. M. Legendre for values of β ranging from 0° to 90°. The following table gives the period, for various amplitudes α, in terms of that of oscillation in an infinitely small arc [viz. 2π√(l/g)] as unit.α/πτα/πτ.11.0062.61.2817.21.0253.71.4283.31.0585.81.6551.41.1087.92.0724.51.18041.0∞The value of τ can also be obtained as an infinite series, by expanding the integrand in (18) by the binomial theorem, and integrating term by term. Thusτ = 2π√l·{1 +12sin21⁄2α +12· 32sin41⁄2α + ...}.g2222· 42(19)If α be small, an approximation (usually sufficient) isτ = 2π √(l/g) · (1 +1⁄16α2).In the extreme case of α = π, the equation (17) is immediately integrable; thus the time from the lowest position ist = √(l/g) · log tan (1⁄4π +1⁄4ψ).(20)This becomes infinite for ψ = π, showing that the pendulum only tends asymptotically to the highest position.The variation of period with amplitude was at one time a hindrance to the accurate performance of pendulum clocks, since the errors produced are cumulative. It was therefore sought to replace the circular pendulum by some other contrivance free from this defect. The equation of motion of a particle in any smooth path isd2s= −g sin ψ,dt2(21)where ψ is the inclination of the tangent to the horizontal. If sin ψ were accurately and not merely approximately proportional to the arc s, says = k sin ψ,(22)Fig.67.the equation (21) would assume the same form as § 12 (5). The motion along the arc would then be accurately simple-harmonic, and the period 2π √(k/g) would be the same for all amplitudes. Now equation (22) is the intrinsic equation of a cycloid; viz. the curve is that traced by a point on the circumference of a circle of radius1⁄4k which rolls on the under side of a horizontal straight line. Since the evolute of a cycloid is an equal cycloid the object is attained by means of two metal cheeks, having the form of the evolute near the cusp, on which the string wraps itself alternately as the pendulum swings. The device has long been abandoned, the difficulty being met in other ways, but the problem, originally investigated by C. Huygens, is important in the history of mathematics.
In the case of the pendulum the tension of the string takes the place of the pressure of the curve. If l be the length of the string, ψ its inclination to the downward vertical, we have δs = lδψ, so that v = ldψ/dt. The tangential resolution then gives
(15)
If we multiply by 2dψ/dt and integrate, we obtain
(16)
which is seen to be equivalent to (14). If the pendulum oscillate between the limits ψ = ±α, we have
(17)
and, putting sin1⁄2ψ = sin1⁄2α. sin φ, we find for the period (τ) of a complete oscillation
(18)
in the notation of elliptic integrals. The function F1(sin β) was tabulated by A. M. Legendre for values of β ranging from 0° to 90°. The following table gives the period, for various amplitudes α, in terms of that of oscillation in an infinitely small arc [viz. 2π√(l/g)] as unit.
The value of τ can also be obtained as an infinite series, by expanding the integrand in (18) by the binomial theorem, and integrating term by term. Thus
(19)
If α be small, an approximation (usually sufficient) is
τ = 2π √(l/g) · (1 +1⁄16α2).
In the extreme case of α = π, the equation (17) is immediately integrable; thus the time from the lowest position is
t = √(l/g) · log tan (1⁄4π +1⁄4ψ).
(20)
This becomes infinite for ψ = π, showing that the pendulum only tends asymptotically to the highest position.
The variation of period with amplitude was at one time a hindrance to the accurate performance of pendulum clocks, since the errors produced are cumulative. It was therefore sought to replace the circular pendulum by some other contrivance free from this defect. The equation of motion of a particle in any smooth path is
(21)
where ψ is the inclination of the tangent to the horizontal. If sin ψ were accurately and not merely approximately proportional to the arc s, say
s = k sin ψ,
(22)
the equation (21) would assume the same form as § 12 (5). The motion along the arc would then be accurately simple-harmonic, and the period 2π √(k/g) would be the same for all amplitudes. Now equation (22) is the intrinsic equation of a cycloid; viz. the curve is that traced by a point on the circumference of a circle of radius1⁄4k which rolls on the under side of a horizontal straight line. Since the evolute of a cycloid is an equal cycloid the object is attained by means of two metal cheeks, having the form of the evolute near the cusp, on which the string wraps itself alternately as the pendulum swings. The device has long been abandoned, the difficulty being met in other ways, but the problem, originally investigated by C. Huygens, is important in the history of mathematics.
The component accelerations of a point describing a tortuous curve, in the directions of the tangent, the principal normal, and the binormal, respectively, are found as follows. IfOV>,OV′>be vectors representing the velocities at two consecutive points P, P′ of the path, the plane VOV′ is ultimately parallel to the osculating plane of the path at P; the resultant acceleration is therefore in the osculating plane. Also, the projections ofVV′>on OV and on a perpendicular to OV in the plane VOV′ are δv and vδε, where δε is the angle between the directions of the tangents at P, P′. Since δε = δs/ρ, where δs = PP′ = vδt and ρ is the radius of principal curvature at P, the component accelerations along the tangent and principal normal are dv/dt and vdε/dt, respectively, or vdv/ds and v2/ρ. For example, if a particle moves on a smooth surface, under no forces except the reaction of the surface, v is constant, and the principal normal to the path will coincide with the normal to the surface. Hence the path is a “geodesic” on the surface.
If we resolve along the tangent to the path (whether plane or tortuous), the equation of motion of a particle may be written
(23)
whereTis the tangential component of the force. Integrating with respect to s we find
1⁄2mv12−1⁄2mv02=∫s1s0Tds;
(24)
i.e.the increase of kinetic energy between any two positions is equal to the work done by the forces. The result follows also from the Cartesian equations (2); viz. we have
m (ẋẍ + ẏÿ + żz̈) = Xẋ + Yẏ + Zż,
(25)
whence, on integration with respect to t,
(26)
If the axes be rectangular, this has the same interpretation as (24).
Suppose now that we have a constant field of force;i.e.the force acting on the particle is always the same at the same place. The work which must be done by forces extraneous to the field in order to bring the particle from rest in some standard position A to rest in any other position P will not necessarily be the same for all paths between A and P. If it is different for different paths, then by bringing the particle from A to P by one path, and back again from P to A by another, we might secure a gain of work, and the process could be repeated indefinitely. If the work required is the same for all paths between A and P, and therefore zero for a closed circuit, the field is said to beconservative. In this case the work required to bring the particle from rest at A to rest at P is called thepotential energyof the particle in the position P; we denote it by V. If PP′ be a linear element δs drawn in any direction from P, and S be the force due to the field, resolved in the direction PP′, we have δV = −Sδs or
(27)
In particular, by taking PP′ parallel to each of the (rectangular) co-ordinate axes in succession, we find
(28)
The equation (24) or (26) now gives
1⁄2mv12+ V1=1⁄2mv02+ V0;
(29)
i.e.the sum of the kinetic and potential energies is constant when no work is done by extraneous forces. For example, if the field be that due to gravity we have V = ƒmg dy = mgy + const., if the axis of y be drawn vertically upwards; hence
1⁄2mv2+ mgy = const.
(30)
This applies to motion on a smooth curve, as well as to the free motion of a projectile; cf. (7), (14). Again, in the case of a force Kr towards O, where r denotes distance from O we have V = ∫ Kr dr =1⁄2Kr2+ const., whence
1⁄2mv2+1⁄2Kr2= const.
(31)
It has been seen that the orbit is in this case an ellipse; also that if we put μ = K/m the velocity at any point P is v = √μ. OD, where OD is the semi-diameter conjugate to OP. Hence (31) is consistent with the known property of the ellipse that OP2+ OD2is constant.
The forms assumed by the dynamical equations when the axes of reference are themselves in motion will be considered in § 21. At present we take only the case where the rectangular axes Ox, Oy rotate in their own plane, with angular velocity ω about Oz, which is fixed. In the interval δt the projections of the line joining the origin to any point (x, y, z) on the directions of the co-ordinate axes at time t are changed from x, y, z to (x + δx) cos ω δt − (y + δy) sin ωδt, (x + δx) sin ω δt + (y + δy) cos ω δt, z respectively. Hence the component velocities parallel to the instantaneous positions of the co-ordinate axes at time t areu = ẋ − ωy, v = ẏ + ωz, ω = ż.(32)In the same way we find that the component accelerations areu̇ − ωv, v̇ + ωu, ω̇(33)Hence if ω be constant the equations of motion take the formsm (ẍ − 2ωẏ − ω2ẋ) = X, m (ÿ + 2ωẋ − ω2y) = Y, mz̈ = Z.(34)These become identical with the equations of motion relative to fixed axes provided we introduce a fictitious force mω2r acting outwards from the axis of z, where r = √(x2+ y2), and a second fictitious force 2mωv at right angles to the path, where v is the component of the relative velocity parallel to the plane xy. The former force is called by French writers theforce centrifuge ordinaire, and the latter theforce centrifuge composée, orforce de Coriolis. As an application of (34) we may take the case of a symmetrical Blackburn’s pendulum hanging from a horizontal bar which is made to rotateabout a vertical axis half-way between the points of attachment of the upper string. The equations of small motion are then of the typeẍ − 2ωẏ − ω2x = −p2x, ÿ + 2ωẋ − ω2y = −q2y.(35)This is satisfied byẍ = A cos (σt + ε), y = B sin (σt + ε),(36)provided(σ2+ ω2− p2) A + 2σωB = 0,2σωA + (σ2+ ω2− q2) B = 0.(37)Eliminating the ratio A : B we have(σ2+ ω2− p2) (σ2+ ω2− q2) − 4σ2ω2= 0.(38)It is easily proved that the roots of this quadratic in σ2are always real, and that they are moreover both positive unless ω2lies between p2and q2. The ratio B/A is determined in each case by either of the equations (37); hence each root of the quadratic gives a solution of the type (36), with two arbitrary constants A, ε. Since the equations (35) are linear, these two solutions are to be superposed. If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x = 0, y = 0 is unstable. This occurs only when the period (2π/ω) of revolution of the arm lies between the two periods (2π/p, 2π/q) of oscillation when the arm is fixed.
The forms assumed by the dynamical equations when the axes of reference are themselves in motion will be considered in § 21. At present we take only the case where the rectangular axes Ox, Oy rotate in their own plane, with angular velocity ω about Oz, which is fixed. In the interval δt the projections of the line joining the origin to any point (x, y, z) on the directions of the co-ordinate axes at time t are changed from x, y, z to (x + δx) cos ω δt − (y + δy) sin ωδt, (x + δx) sin ω δt + (y + δy) cos ω δt, z respectively. Hence the component velocities parallel to the instantaneous positions of the co-ordinate axes at time t are
u = ẋ − ωy, v = ẏ + ωz, ω = ż.
(32)
In the same way we find that the component accelerations are
u̇ − ωv, v̇ + ωu, ω̇
(33)
Hence if ω be constant the equations of motion take the forms
m (ẍ − 2ωẏ − ω2ẋ) = X, m (ÿ + 2ωẋ − ω2y) = Y, mz̈ = Z.
(34)
These become identical with the equations of motion relative to fixed axes provided we introduce a fictitious force mω2r acting outwards from the axis of z, where r = √(x2+ y2), and a second fictitious force 2mωv at right angles to the path, where v is the component of the relative velocity parallel to the plane xy. The former force is called by French writers theforce centrifuge ordinaire, and the latter theforce centrifuge composée, orforce de Coriolis. As an application of (34) we may take the case of a symmetrical Blackburn’s pendulum hanging from a horizontal bar which is made to rotateabout a vertical axis half-way between the points of attachment of the upper string. The equations of small motion are then of the type
ẍ − 2ωẏ − ω2x = −p2x, ÿ + 2ωẋ − ω2y = −q2y.
(35)
This is satisfied by
ẍ = A cos (σt + ε), y = B sin (σt + ε),
(36)
provided
(37)
Eliminating the ratio A : B we have
(σ2+ ω2− p2) (σ2+ ω2− q2) − 4σ2ω2= 0.
(38)
It is easily proved that the roots of this quadratic in σ2are always real, and that they are moreover both positive unless ω2lies between p2and q2. The ratio B/A is determined in each case by either of the equations (37); hence each root of the quadratic gives a solution of the type (36), with two arbitrary constants A, ε. Since the equations (35) are linear, these two solutions are to be superposed. If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x = 0, y = 0 is unstable. This occurs only when the period (2π/ω) of revolution of the arm lies between the two periods (2π/p, 2π/q) of oscillation when the arm is fixed.
§ 14.Central Forces. Hodograph.—The motion of a particle subject to a force which passes always through a fixed point O is necessarily in a plane orbit. For its investigation we require two equations; these may be obtained in a variety of forms.
Since the impulse of the force in any element of time δt has zero moment about O, the same will be true of the additional momentum generated. Hence the moment of the momentum (considered as a localized vector) about O will be constant. In symbols, if v be the velocity and p the perpendicular from O to the tangent to the path,
pv = h,
(1)
where h is a constant. If δs be an element of the path, pδs is twice the area enclosed by δs and the radii drawn to its extremities from O. Hence if δA be this area, we have δA =1⁄2pδs =1⁄2hδt, or
(2)
Hence equal areas are swept over by the radius vector in equal times.
If P be the acceleration towards O, we have
(3)
since dr/ds is the cosine of the angle between the directions of r and δs. We will suppose that P is a function of r only; then integrating (3) we find
1⁄2v2= −∫P dr + const.,
(4)
which is recognized as the equation of energy. Combining this with (1) we have
(5)
which completely determines the path except as to its orientation with respect to O.
If the law of attraction be that of the inverse square of the distance, we have P = μ/r2, and
(6)
Now in a conic whose focus is at O we have
(7)
where l is half the latus-rectum, a is half the major axis, and the upper or lower sign is to be taken according as the conic is an ellipse or hyperbola. In the intermediate case of the parabola we have a = ∞ and the last term disappears. The equations (6) and (7) are identified by putting
l = h2/μ, a = ± μ/C.
(8)
Since
(9)
it appears that the orbit is an ellipse, parabola or hyperbola, according as v2is less than, equal to, or greater than 2μ/r. Now it appears from (6) that 2μ/r is the square of the velocity which would be acquired by a particle falling from rest at infinity to the distance r. Hence the character of the orbit depends on whether the velocity at any point is less than, equal to, or greater than thevelocity from infinity, as it is called. In an elliptic orbit the area πab is swept over in the time
(10)
since h = μ1/2l1/2= μ1/2ba−1/2by (8).
The converse problem, to determine the law of force under which a given orbit can be described about a given pole, is solved by differentiating (5) with respect to r; thusP =h2dp.p3dr(11)In the case of an ellipse described about the centre as pole we havea2b2= a2+ b2− r2;p2(12)hence P = μr, if μ = h2/a2b2. This merely shows that a particular ellipse may be described under the law of the direct distance provided the circumstances of projection be suitably adjusted. But since an ellipse can always be constructed with a given centre so as to touch a given line at a given point, and to have a given value of ab (= h/√μ) we infer that the orbit will be elliptic whatever the initial circumstances. Also the period is 2πab/h = 2π/√μ, as previously found.Fig.68.Again, in the equiangular spiral we have p = r sinα, and therefore P = μ/r3, if μ = h2/sin2α. But since an equiangular spiral having a given pole is completely determined by a given point and a given tangent, this type of orbit is not a general one for the law of the inverse cube. In order that the spiral may be described it is necessary that the velocity of projection should be adjusted to make h = √μ·sinα. Similarly, in the case of a circle with the pole on the circumference we have p2= r2/2a, P = μ/r5, if μ = 8h2a2; but this orbit is not a general one for the law of the inverse fifth power.
The converse problem, to determine the law of force under which a given orbit can be described about a given pole, is solved by differentiating (5) with respect to r; thus
(11)
In the case of an ellipse described about the centre as pole we have
(12)
hence P = μr, if μ = h2/a2b2. This merely shows that a particular ellipse may be described under the law of the direct distance provided the circumstances of projection be suitably adjusted. But since an ellipse can always be constructed with a given centre so as to touch a given line at a given point, and to have a given value of ab (= h/√μ) we infer that the orbit will be elliptic whatever the initial circumstances. Also the period is 2πab/h = 2π/√μ, as previously found.
Again, in the equiangular spiral we have p = r sinα, and therefore P = μ/r3, if μ = h2/sin2α. But since an equiangular spiral having a given pole is completely determined by a given point and a given tangent, this type of orbit is not a general one for the law of the inverse cube. In order that the spiral may be described it is necessary that the velocity of projection should be adjusted to make h = √μ·sinα. Similarly, in the case of a circle with the pole on the circumference we have p2= r2/2a, P = μ/r5, if μ = 8h2a2; but this orbit is not a general one for the law of the inverse fifth power.
In astronomical and other investigations relating to central forces it is often convenient to use polar co-ordinates with the centre of force as pole. Let P, Q be the positions of a moving point at times t, t + δt, and write OP = r, OQ = r + δr, ∠POQ = δθ, O being any fixed origin. If u, v be the component velocities at P along and perpendicular to OP (in the direction of θ increasing), we have
(13)
Again, the velocities parallel and perpendicular to OP change in the time δt from u, v to u − v δθ, v + u δθ, ultimately. The component accelerations at P in these directions are therefore
(14)
respectively.
In the case of a central force, with O as pole, the transverse acceleration vanishes, so that
r2dθ / dt = h,
(15)
where h is constant; this shows (again) that the radius vector sweeps over equal areas in equal times. The radial resolution gives
(16)
where P, as before, denotes the acceleration towards O. If in this we put r = 1/u, and eliminate t by means of (15), we obtain the general differential equation of central orbits, viz.
(17)
If, for example, the law be that of the inverse square, we have P = μu2, and the solution is of the formu =μ{1 + e cos (θ − α)},h2(18)where e, α are arbitrary constants. This is recognized as the polar equation of a conic referred to the focus, the half latus-rectum being h2/μ.The law of the inverse cube P = μu3is 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 formd2u+ m2u = 0,dθ2(19)the solution of which isau = sin m (θ − α).(20)The orbit has therefore two asymptotes, inclined at an angle π/m. In the latter case the differential equation is of the formd2u= m2u,dθ2(21)so thatu = Aemθ+ Be−mθ(22)If A, B have the same sign, this is equivalent toau = cosh mθ,(23)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 isau = sinh mθ,(24)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 d2u/dθ2= 0, andu = Aθ + B;(25)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 arc known asColes’s spirals.
If, for example, the law be that of the inverse square, we have P = μu2, and the solution is of the form
(18)
where e, α are arbitrary constants. This is recognized as the polar equation of a conic referred to the focus, the half latus-rectum being h2/μ.
The law of the inverse cube P = μu3is 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
(19)
the solution of which is
au = sin m (θ − α).
(20)
The orbit has therefore two asymptotes, inclined at an angle π/m. In the latter case the differential equation is of the form
(21)
so that
u = Aemθ+ Be−mθ
(22)
If A, B have the same sign, this is equivalent to
au = cosh mθ,
(23)
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
au = sinh mθ,
(24)
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 d2u/dθ2= 0, and
u = Aθ + B;
(25)
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 arc known asColes’s spirals.
A point on a central orbit where the radial velocity (dr/dt) vanishes is called anapse, and the corresponding radius is called anapse-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 theapsidal angleof the orbit.
If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5),
(26)
this determines the form of the critical orbit, as it is called. If P = μ/rn, its polar equation is
rmcos mθ = am,
(27)
where m =1⁄2(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 obtaind2r−h2= −P = −ƒ(r),dt2r3say. 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,d2x−h2(1 −3x)= −ƒ(a) − xƒ′(a).dt2a3aHence if h and a be connected by the relation h2= a3ƒ(a) proper to a circular orbit, we haved2x+{ƒ′(a) +3ƒ(a)}x = 0dt2a(28)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 writtend{ a3ƒ(a) } > 0,da(29)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)(30)or, in the case of ƒ(a) = μ/rn, 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.
If we eliminate dθ/dt between (15) and (16) we obtain
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,
Hence if h and a be connected by the relation h2= a3ƒ(a) proper to a circular orbit, we have
(28)
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
(29)
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,
(30)
or, in the case of ƒ(a) = μ/rn, 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 vectorOV>drawn from a fixed origin O. The locus of the point V is called thehodograph(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 v dψ/dt. These expressions therefore give the tangential and normal accelerations of P; cf. § 13 (12).