Central Forces and Statics

Derives Proposition 27.4. Let \(O'\) be displaced from \(O\) by \(\vect{a}\), so that the position of the \(m\)-th point of application measured from \(O'\) is \(\vect{x}_{m}-\vect{a}\). By Equation (19.15) the total torque about \(O'\) is

the last step by Equation (27.2). The displacement \(\vect{a}\) was arbitrary.

Derivation. Derives Phenomenon 27.5. Apply forces \(\vect{F}\) and \(-\vect{F}\) at the points \(\vect{x}_1\) and \(\vect{x}_2\) of a rigid body. Equation (27.2) is satisfied, since the two cancel, so by the second law of Newtonian Dynamics the centre of mass keeps whatever velocity it had. The total torque about the origin, however, is

which is independent of the choice of origin — a couple has the same moment about every point, which is Proposition 27.4 for the special case of a vanishing resultant — and which vanishes only if \(\vect{F}\) is parallel to the line joining the two points of application. Whenever it does not vanish, Equation (27.3) fails, the angular momentum grows, and the body turns although no net force acts upon it. For the converse case, a single force applied at the centre of mass has \(\vect{x}=\vect{0}\) measured from there, hence zero moment about it, and the body accelerates without rotating. The two conditions are therefore logically independent, and neither implies the other.

Derives Proposition 27.7. The weight of the \(m\)-th element is \(m_{m}\vect{g}\) with one and the same \(\vect{g}\), so by Equation (19.15)

using Equation (19.38) in the form \(\sum_{m}m_{m}\vect{x}_{m}=M\vect{X}\). The right-hand side is the torque of a single force \(M\vect{g}\) applied at \(\vect{X}\), which is Definition 27.6.

Derivation. Derives Phenomenon 27.8. Take torques about the point of support \(S\), so that the unknown reaction there contributes nothing. The only other force is the weight, whose torque is \(\left(\vect{X}-\vect{x}_{S}\right)\times M\vect{g}\) by Equation (27.6). Rotational equilibrium Equation (27.3) therefore demands

that is, that \(\vect{X}-\vect{x}_{S}\) be parallel to \(\vect{g}\): the centre of gravity lies on the vertical through the support. Two positions satisfy this, one with \(\vect{X}\) below \(S\) and one with it above; only the first is stable, by Theorem 27.10, since rotating the body about the pin from that position raises the centre of gravity and so raises the potential energy \(MgZ\), \(Z\) being its height, whereas rotating it from the second position lowers \(Z\). Repeating the suspension from a second point gives a second vertical through the same material point, and two non-parallel lines meet in one point.

Derives Theorem 27.9. By Theorem 21.18 the system is in equilibrium if and only if the virtual work of the active forces vanishes for every virtual displacement compatible with the constraints. Because the constraints are holonomic and time-independent, a compatible virtual displacement is exactly one of the form \(\delta\vect{x}^{s} =\left(\pp\vect{x}^{s}/\pp q^{a}\right)\delta q^{a}\) with the \(\delta q^{a}\) arbitrary and independent, so that by Equation (21.31) the virtual work is

For forces derived from a potential, \(\vect{F}^{s}=-\nabla_{s}U\) and the chain rule (Proposition 7.72) gives \(Q_{a}=-\pp U/\pp q^{a}\). Vanishing of \(Q_{a}\delta q^{a}\) for arbitrary independent \(\delta q^{a}\) is vanishing of every \(Q_{a}\), which is Equation (27.9).

Derives Theorem 27.10. Set \(U(q_{0})=0\) without loss of generality. The kinetic energy \(T\) of a system with time-independent constraints is a positive definite quadratic form in the generalized velocities (Definition 21.66), so \(T\geq0\) with equality only at rest, and the conserved energy \(E=T+U\) of Theorem 21.43 is a positive definite function of \(\left(q-q_{0},\dot q\right)\) near the equilibrium, because \(U\) has a strict minimum there. Given a neighbourhood \(N\), choose \(\varepsilon>0\) smaller than the least value of \(U\) on the boundary of a ball inside \(N\); then any motion starting with \(E<\varepsilon\) can never reach that boundary, since doing so would require \(U\geq\varepsilon\) and hence \(E\geq\varepsilon\). The set \(\set{E<\varepsilon}\) is the required smaller neighbourhood.

Derives Lemma 27.15. Differentiating the two unit vectors with respect to time by the chain rule (Proposition 7.31), \(\dot{\hat{\vect{r}}} =\dot{\phi}\left(-\sin\phi,\cos\phi\right) =\dot{\phi}\,\hat{\vect{\phi}}\) and \(\dot{\hat{\vect{\phi}}} =\dot{\phi}\left(-\cos\phi,-\sin\phi\right) =-\dot{\phi}\,\hat{\vect{r}}\), which is Equation (27.14). Then \(\vect{v}=\dv{}{t}\left(r\hat{\vect{r}}\right) =\dot{r}\hat{\vect{r}}+r\dot{\hat{\vect{r}}}\), giving Equation (27.15), and differentiating once more,

which on substituting \(\dot{\hat{\vect{\phi}}}\) and collecting terms is Equation (27.16).

Derives Proposition 27.17. Between the times \(t\) and \(t+\dd t\) the radius turns through \(\dd\phi=\dot{\phi}\,\dd t\) and lengthens by \(\dd r=\dot{r}\,\dd t\). The region swept out contains the circular sector of radius \(r\) and angle \(\dd\phi\), and is contained in the sector of the same angle and radius \(r+\abs{\dd r}\); by the sector area recorded in Remark 27.1 these bound it as

so the two differ by \(O\!\left(\dd t^{2}\right)\) and \(\dd A=\tfrac{1}{2}r^{2}\dd\phi+O\!\left(\dd t^{2}\right)\). Dividing by \(\dd t\) and letting \(\dd t\to0\) gives Equation (27.18). Differentiating that result,

by Equation (27.16), which rearranged is Equation (27.19).

Derives Proposition 27.18. By Equation (19.14) the rate of change of the angular momentum is the torque, and by Equation (19.15) the torque is \(\vect{x}\times\vect{F}\). For a central force Equation (27.13) the force is parallel to \(\vect{x}=r\hat{\vect{r}}\), so

and \(\vect{L}\) is a fixed vector. Since \(\vect{L}=\vect{x}\times m\vect{v}\) is orthogonal to \(\vect{x}\) by construction, the position stays in the fixed plane through the centre perpendicular to \(\vect{L}\); choosing Cartesian axes with \(\vect{L}\) along \(z\) makes that the \(xy\)-plane, so \(\vect{L}=L\hat{\vect{z}}\) with no other component. Finally, by Equation (27.15) the radial part of \(\vect{v}\) is parallel to \(\vect{x}\) and contributes nothing to the cross product, leaving \(\vect{L}=r\hat{\vect{r}}\times m r\dot{\phi}\hat{\vect{\phi}} =m r^{2}\dot{\phi}\,\hat{\vect{z}}\), which is Equation (27.20). The degenerate case \(\vect{L}=\vect{0}\) is the rectilinear motion through the centre; the plane is then not unique, and any plane containing the line will serve.

Derivation. Derives Phenomenon 27.19. By Proposition 27.18 the motion is plane and \(L=m r^{2}\dot{\phi}\) is constant. Comparing with the areal velocity Equation (27.18),

a constant — the equal-area law, and the relation Equation (27.27) read from left to right. Mark what the argument did not use: the form of \(f(r)\) never entered. Kepler's second law therefore tests the centrality of the force and nothing else, and would hold equally for an inverse-cube or a linear attraction.

The converse also holds, and is what makes the law evidence for anything. If \(v_{a}\) is constant then by Equation (27.19) the transverse acceleration \(a_{\phi}\) vanishes identically, so by Equation (19.4) the force has no transverse component and is central. Equal areas in equal times and centrality of the force are therefore the same statement.

Derives Proposition 27.20. The gradient of the radius is \(\vect{\nabla}r=\hat{\vect{r}}\), since \(r=\left(x^{2}+y^{2}+z^{2}\right)^{1/2}\) gives \(\pp r/\pp x^{i}=x^{i}/r\). Hence, by the chain rule (Proposition 7.72) and the fundamental theorem of calculus (Theorem 7.42) applied to Equation (27.23),

which is Equation (27.24); Equation (27.25) then follows because the curl of a gradient vanishes, Equation (19.20). Note that no topological hypothesis was needed: the general criterion Proposition 7.104 requires a simply connected domain because it must reconstruct the potential from line integrals, whereas here Equation (27.23) writes it down.

For the conservation, differentiate \(E=\tfrac{1}{2}m\abs{\vect{v}}^{2} +V\) along the motion and use Equation (19.4) together with Equation (27.24):

From Equations (19.4) and (27.24) (the chain rule on \(V(\vect{x}(t))\) and the second law on the kinetic term). Equivalently, the work Equation (19.25) done between two points depends only on the radii of the endpoints.

Derives Proposition 27.21. A central force Equation (27.13) has no \(\hat{\vect{\phi}}\)-component, so by Equation (19.4) neither does the acceleration: \(a_{\phi}=0\). Equation (27.19) then gives \(\dd v_{a}/\dd t=\tfrac{1}{2}r\,a_{\phi}=0\). This is the same fact as Equation (27.22), obtained without passing through the angular momentum, and the equality of the two routes is Proposition 27.22.

Derives Proposition 27.22. The first equality is Equation (27.20) divided by \(m\), and the second is twice Equation (27.18). The content of the chain is that the two conserved quantities of the previous two propositions are not independent: the areal speed is the angular momentum per unit mass, halved. Kepler's second law and the conservation of angular momentum are one law counted twice, and the constant \(L/m\) is the one that appears in every formula below.

Derives Proposition 27.23. The device is to use \(\phi\) rather than \(t\) as the independent variable, which is legitimate whenever \(L\neq0\), since then \(\dot{\phi}=L/\left(mr^{2}\right)\) never vanishes and \(\phi\) is a strictly monotone function of \(t\). Write \(u:=1/r\), so that

Then, by the chain rule (Proposition 7.31),

a first simplification worth noticing on its own: the radial velocity is \(-L/m\) times \(\dd u/\dd\phi\), with a constant factor. Differentiating once more in the same way,

while the centrifugal term of Equation (27.16) is \(r\dot{\phi}^{2}=u^{-1}L^{2}u^{4}/m^{2}=L^{2}u^{3}/m^{2}\). Subtracting,

From Equations (27.16) and (27.29) (eliminating \(t\) in favour of \(\phi\) through \(\dot\phi=Lu^{2}/m\) and writing \(u=1/r\)). which is Equation (27.28) with \(u\) restored to \(1/r\). Combined with \(m a_{r}=f(r)\) from Equation (19.4), it turns the dynamical problem into a single second-order equation for the shape of the orbit, with the time eliminated altogether.

Derives Proposition 27.24. The motion is plane by Proposition 27.18, and the two terms of Equation (27.15) are orthogonal, so

and the kinetic energy Equation (19.18) splits into a radial and a transverse part, \(T=\tfrac{1}{2}m\dot{r}^{2}+\tfrac{1}{2}m r^{2}\dot{\phi}^{2}\). Eliminating \(\dot{\phi}\) with Equation (27.20),

which added to the radial part and to the potential Equation (27.24) gives Equation (27.31). The constancy of \(E\) is Proposition 27.20, and the constancy of \(L\) is what makes the middle term a function of \(r\) alone.

Derivation. Derives Phenomenon 27.26. By Equation (27.33) the radial coordinate obeys a one-dimensional energy equation, \(E=\tfrac{1}{2}m\dot{r}^2+V_{e}\), with the effective potential Equation (27.32). Since \(\tfrac{1}{2}m\dot{r}^2\geq0\), the motion is confined to the set where \(E\geq V_{e}(r)\), which is Equation (27.35), and the boundary points of that set, where \(E=V_{e}(R)\), are the turning radii at which \(\dot{r}\) vanishes and changes sign. Now suppose \(L\neq0\). The centrifugal term \(L^2/(2mr^2)\) grows like \(r^{-2}\) as \(r\to0\) and so dominates every attractive potential less singular than \(r^{-2}\) — the inverse-square attraction among them, whose potential goes only as \(r^{-1}\). Hence \(V_{e}\to+\infty\) at the centre, and the condition \(E\geq V_{e}\) excludes a whole neighbourhood of \(r=0\): the body cannot reach the centre at any finite energy. If instead \(L=0\) the centrifugal term is absent, \(V_{e}=V\), and nothing forbids the collision — which shows that the barrier is angular momentum and not force. For an attraction that vanishes at infinity, a state with \(E<0\) is trapped between an inner and an outer turning radius and the orbit is bounded; a state with \(E\geq0\) has only an inner one and the body escapes. That the bounded orbit should also close is a further and much stronger property, discussed in Phenomenon 27.32 and Theorem 27.41.

Derives Proposition 27.28. Equation (27.34) is a separable first-order equation for \(r(t)\): dividing by the root and integrating gives Equation (27.37), which is legitimate on any interval where \(E>V_{e}\), that is strictly between turning points, since there the integrand is continuous. For the orbit, divide \(\dd\phi=\dot{\phi}\,\dd t\) by \(\dd r=\dot{r}\,\dd t\) and use Equation (27.20) and Equation (27.34):

where the last step multiplied numerator and denominator by \(m\) and used Equation (27.32). Integrating gives Equation (27.38).

Derives Theorem 27.30. Adding the two equations of motion and using Equation (27.40) gives \(m_{1}\ddot{\vect{x}}_{1}+m_{2}\ddot{\vect{x}}_{2}=\vect{0}\), that is \(\ddot{\vect{X}}=\vect{0}\) by Equation (19.40): the centre of mass moves uniformly, as Phenomenon 19.54 states in general. Subtracting instead, after dividing each equation by its own mass,

From Equations (19.4) and (27.40) (dividing each equation of motion by its mass and subtracting, with \(\mu^{-1}=m_{1}^{-1}+m_{2}^{-1}\)). which is Equation (27.41). For the splittings, invert the change of variables: with \(M=m_{1}+m_{2}\),

Substituting into \(T=\tfrac{1}{2}m_{1}\abs{\dot{\vect{x}}_{1}}^{2} +\tfrac{1}{2}m_{2}\abs{\dot{\vect{x}}_{2}}^{2}\), the cross terms carry the factor \(m_{1}m_{2}/M-m_{2}m_{1}/M=0\) and cancel, while the remaining coefficients of \(\abs{\dot{\vect{x}}}^{2}\) combine to \(\tfrac{1}{2}\left(m_{1}m_{2}^{2}+m_{2}m_{1}^{2}\right)/M^{2} =\tfrac{1}{2}\mu\), giving Equation (27.42). The same substitution in \(\vect{L}=m_{1}\vect{x}_{1}\times\dot{\vect{x}}_{1} +m_{2}\vect{x}_{2}\times\dot{\vect{x}}_{2}\) cancels the cross terms for the same reason and gives Equation (27.43).

Derivation. Derives Phenomenon 27.32. Take the Newtonian attraction between the two bodies,

whose potential Equation (27.23) is \(V(r)=-k/r\) once the additive constant is fixed by requiring \(V\to0\) as \(r\to\infty\), and which by Theorem 27.30 makes the separation obey the one-body central equation with mass \(\mu\). Insert Equation (27.45) into the Binet equation Equation (27.30), written with \(\mu\) in place of \(m\) and \(u=1/r\), using \(\mu a_{r}=f=-ku^{2}\):

and the factor \(u^{2}\) cancels on both sides — which happens for the inverse square and for no other power. What is left is linear with constant coefficients,

From Equations (27.30) and (27.45) (substituting \(f=-ku^{2}\) and cancelling the common factor \(u^{2}\)). By Theorem 9.13 its general solution is any one solution plus the general solution of the homogeneous equation; the constant \(u=\mu k/L^{2}\) is a solution, and the homogeneous equation is Equation (9.106) with \(\lambda=1\), whose real general solution is Equation (9.108). Choosing the origin of \(\phi\) at the maximum of \(u\) — the point of closest approach — absorbs the phase, and writing the remaining constant as \(e\,\mu k/L^{2}\) with \(e\geq0\),

This is the polar equation of a conic of eccentricity \(e\) with one focus at the centre of force and semi-latus rectum \(p\) (Remark 27.33); for \(0\leq e<1\) it is an ellipse, and the centre of force is at a focus and not at the centre, since \(r\) takes its extreme values \(p/(1+e)\) and \(p/(1-e)\) at \(\phi=0\) and \(\phi=\pi\) and these are unequal unless \(e=0\). The solution is single-valued and \(2\pi\)-periodic in \(\phi\), so the orbit closes exactly, and the identification of \(e\) with the energy is Proposition 27.34.

Derives Proposition 27.34. At a turning point \(\dot{r}=0\), so by Equation (27.31) with \(V=-k/r=-ku\) and \(\mu\) for \(m\),

The turning values of \(u\) are the extremes of Equation (27.47), namely \(u_{\pm}=\left(\mu k/L^{2}\right)\left(1\pm e\right)\). Substituting \(u_{+}\) into Equation (27.50) and using \(p=L^{2}/\left(\mu k\right)\),

and solving for \(e\) gives Equation (27.48); the same substitution with \(u_{-}\) gives the same relation, as it must, since both turning points belong to one orbit. For Equation (27.49), \(r_{\pm}=p/\left(1\mp e\right)\) so

and inserting \(1-e^{2}=-2EL^{2}/\left(\mu k^{2}\right)\) from Equation (27.48) together with \(p=L^{2}/\left(\mu k\right)\) gives \(a=-k/(2E)\), positive precisely when \(E<0\).

Derivation. Derives Phenomenon 27.35. For a circular orbit the result is immediate. A body of mass \(m\) carried around a circle of radius \(r\) requires, by the kinematics of Kinematics, the centripetal acceleration \(\omega^2 r\), and the attraction of a central mass \(M\) supplies it, so that \(GM/r^{2}=\omega^{2}r\) and, with \(\omega=2\pi/T\),

The orbiting mass has cancelled — it appears once as the gravitational charge and once as the inertia — which is why a single constant serves every planet.

For the general bounded orbit, integrate the equal-area law. By Equation (27.22) the areal speed is \(L/\left(2\mu\right)\), and by Equation (27.20) the period is

using Equation (27.47) in the last step. The integral is elementary by residues. For \(0<e<\lambda\) the substitution \(z=\ee^{\ii\phi}\), under which \(\cos\phi=\left(z+z^{-1}\right)/2\) and \(\dd\phi=\dd z/\left(\ii z\right)\), turns \(I(\lambda)=\int_{0}^{2\pi}\left(\lambda+e\cos\phi\right)^{-1}\dd\phi\) into a contour integral on the unit circle,

whose two poles have product \(1\), so that exactly one lies inside, namely \(z_{+}=\left(-\lambda+\sqrt{\lambda^{2}-e^{2}}\right)/e\); by the residue theorem Theorem 8.24 and Proposition 8.23, \(I(\lambda)=2\pi/\sqrt{\lambda^{2}-e^{2}}\), which is also the value at \(e=0\), where the integral is immediate. Differentiating under the integral sign in \(\lambda\) and then setting \(\lambda=1\),

Hence, with \(a=p/\left(1-e^{2}\right)\) from Equation (27.49) and \(p=L^{2}/\left(\mu k\right)\),

Squaring, and putting \(k=Gm_{1}m_{2}\) and \(\mu=m_{1}m_{2}/\left(m_{1}+m_{2}\right)\) from Theorem 27.30,

From Equations (27.41), (27.49) and (27.52) (integrating the equal-area law over one revolution and eliminating \(p\) and \(L\)). This is the exact law. It contains Equation (27.51) as the limit \(m_{2}\ll m_{1}\), it replaces the radius by the semi-major axis, and it shows precisely how the “constant” fails to be one: the proportionality factor depends on the total mass, so two planets of different masses about the same star obey slightly different laws. The check of Equation (33.19) in the Cavendish chapter is this same relation applied to the Moon.

Derives Theorem 27.38. Differentiate Equation (27.54). Since \(\vect{L}\) is constant by Proposition 27.18, \(\dd\left(\vect{p}\times\vect{L}\right)/\dd t =\dot{\vect{p}}\times\vect{L}=\vect{F}\times\vect{L}\) by Equation (27.41). With \(\vect{F}=-k\hat{\vect{r}}/r^{2}\) and \(\vect{L}=\mu\,\vect{x}\times\vect{v}\), the expansion Equation (27.1) of the double cross product gives

using \(\hat{\vect{r}}\cdot\vect{v}=\dot{r}\) and \(\hat{\vect{r}}\cdot\vect{x}=r\). What multiplies \(k\mu\) on the right is exactly the derivative of the unit radial vector,

so \(\dd\left(\vect{p}\times\vect{L}\right)/\dd t =\mu k\,\dd\hat{\vect{r}}/\dd t\) and the two terms of Equation (27.54) cancel in the derivative. Finally \(\vect{A}\cdot\vect{L} =\left(\vect{p}\times\vect{L}\right)\cdot\vect{L} -\mu k\,\hat{\vect{r}}\cdot\vect{L}=0\), the first term because \(\vect{p}\times\vect{L}\) is orthogonal to \(\vect{L}\) and the second because \(\vect{L}\) is orthogonal to \(\vect{x}\).

Derives Proposition 27.39. Take the scalar product of Equation (27.54) with \(\vect{x}\). On the left, \(\vect{x}\cdot\vect{A}=Ar\cos\phi\) with \(\phi\) measured from \(\vect{A}\). On the right, the cyclic property of the scalar triple product gives \(\vect{x}\cdot\left(\vect{p}\times\vect{L}\right) =\vect{L}\cdot\left(\vect{x}\times\vect{p}\right)=L^{2}\), while \(\vect{x}\cdot\mu k\hat{\vect{r}}=\mu k r\). Hence \(Ar\cos\phi=L^{2}-\mu k r\), which rearranged is the first member of Equation (27.55). Comparing with Equation (27.47) identifies \(A=\mu k e\), and substituting Equation (27.48) gives \(A^{2}=\mu^{2}k^{2}e^{2}=\mu^{2}k^{2}+2\mu EL^{2}\). Since \(r\) is least where \(\cos\phi=1\), the vector points at the perihelion.

Derivation. Derives Phenomenon 27.42. Within this chapter the derivation can go only as far as the conditional statement, and that is the honest content of the phenomenon. Theorem 27.41 shows that closure of every bounded orbit is equivalent to the force being \(-k/r^{2}\) or \(-\kappa r\), and the second is excluded by Phenomenon 27.32, since a linear attraction puts the centre of force at the centre of the ellipse and not at a focus. A non-closing bound orbit therefore falsifies the exact inverse square. What it does not do is say what replaces it. The first attempt was to keep a central force and modify the exponent, writing \(f\propto r^{-(2+\delta)}\); for a nearly circular orbit a small \(\delta\) produces an apsidal angle \(\Phi\approx\pi\left(1+\delta/2\right)\) and hence a slow rotation of the line of apsides, and \(\delta\) can always be chosen to fit one planet. The proposal fails because the \(\delta\) that fits Mercury does not fit the others, and because it has no independent motivation whatever.

The correct account is not a modified force law at all. In general relativity the orbit is a geodesic of a curved spacetime and its equation acquires the term shown in Equation (53.2), quadratic in \(u=1/r\) and suppressed by \(c^{-2}\); the resulting advance is Equation (53.1), with no adjustable parameter, and it predicts \(42.98\) seconds of arc per century for Mercury against the observed residue. That derivation belongs to Experiment: The Classical Tests of General Relativity, where the measurement and its error budget are set out, and to Phenomenon 53.1, which states the result. What this chapter contributes is the logic that makes the measurement mean anything: without Theorem 27.41 a non-closing orbit would be an anomaly of no particular significance, and with it, it is a bound on the force law.

Derives Theorem 27.44. Consider \(G = \sum_{a}\vect{p}_{a}\cdot\vect{x}_{a}\). Along the motion,

since \(\vect{p}_{a}\cdot\dot{\vect{x}}_{a} = m_{a}\abs{\dot{\vect{x}}_{a}}^{2}\) sums to twice the kinetic energy. Averaging over \([0,\tau]\), \(\avg{\dd G/\dd t} = [G(\tau)-G(0)]/\tau\), and boundedness of positions and momenta bounds \(G\), so the limit \(\tau\to\infty\) vanishes (exactly, already at one period, for periodic motion). This is Equation (27.56). When \(\vect{F}_{a} = -\pp V/\pp\vect{x}_{a}\) with \(V\) homogeneous of degree \(n\), Euler's homogeneous-function theorem (Remark 21.40) gives \(\sum_{a}\vect{x}_{a}\cdot\pp V/\pp\vect{x}_{a} = nV\), whence Equation (27.57); \(n=-1\) is the gravitational and Coulomb case.

Derives Corollary 27.45. \(E\) is constant, so \(E = \avg{T} + \avg{V}\); combine with Equation (27.58).