Physics 105 (Fall 2013) Notes: The Two-Body Conservative Central Force Problem

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1 Physics 5 (Fall 3) Notes: The Two-Body Conservative Central Force Problem A.E. Charman Department of Physics, University of California, Berkeley (Dated: /5/3; revised //3; revised /8/3) Here we analyze the two body central force problem, or two-body problem for short, describing the dynamics of two point particles subject to conservative, central (and reciprocal) forces. As an important example, Kepler s Laws are derived. I. OVERVIEW The Two-Body Conservative Central Force Problem (usually just abbreviated as the Two-Body Problem ) is one of the most important in classical mechanics. It concerns the motion of two point particles subject to internal forces that are conservative (derivable from a time-independent potential energy), central (acting in a direction along the line joining the particles), and hence reciprocal (satisfying the Third Law). We refer to the two-body problem and not the two-particle problem because the resulting framework applies somewhat more generally, for example to the gravitational interactions between two uniform spherical masses. The two-body problem undergirds the derivation of Kepler s Laws directly from the Laws of Motion and of Universal Gravitation, one of the early triumph s of Newtonian mechanics. It is one of the few known integrable systems in three spatial dimensions, illustrating the connections between symmetries, conservations laws, and reduction of the degrees of freedom for which we must simultaneously solve, and also forms the starting point for a variety of perturbation and other approximation techniques used to tackle more complicated and realistic problems in celestial mechanics, orbital dynamics, and atomic and molecular physics. As such, this model, like the harmonic oscillator, is one of the primary workhorses of classical mechanics, and is a true cornerstone of physics and high water mark in intellectual history. The general strategy for solving the two-body problem is illuminating, as it takes advantage of some of the most important ideas in classical mechanics: the decomposition of the dynamics into COM and relative motion, use of symmetries, and mechanical analogic reasoning, whereby we reduce problems we do not at first know how to solve to problems we already do know how to solve. r = (R R)!F m m Rcom R R origin (in inertial frame) FIG. The two-body central force problem. The forces on each particle are equal in magnitude but opposite in direction, parallel or antiparallel to the relative displacement r = R R between the particles. The relative positions (npot shown) R and R with respect to the center of mass (COM) are also oppositely directed along this line. Electronic address: acharman@physics.berkeley.edu

2 To see where we are headed, we outline our strategy and tactics: decompose the dynamics into motion of the center of mass and motion about the center of mass; use momentum conservation to deduce the COM motion; change variables to map dynamics of the two particles in the COM frame to the analogous dynamics of one particle of reduced mass; use angular momentum conservation to restrict relative motion to one plane, and to decouple angular and radial motion; map the radial motion to the analogous dynamics of a single particle moving in one-dimensional effective potential use energy conservation to solve for the radial dynamics; use angular momentum conservation to solve for the angular dynamics; map back to the original variables if needed. Also, we will look at the important special case of attractive inverse-square force laws like the gravitational or Coloumb forces, and derive Kepler s Law s from Newton s Laws. II. DYNAMICAL SETTING In some chosen inertial frame, suppose a point particle of mass m, position R, and velocity Ṙ interacts with a second point particle of mass m, position R, and velocity Ṙ, according to a conservative, central pair-interaction potential U( R R ) which is a function only of the distance between the particles (and of course possibly invariant parameters such as their masses or charges). This means the instantaneous force F on particle due to particle (equal and opposite to the force F on particle due to particle ) is F = F (R, R ) = R U ( R R ) = (R R) R R U ( R R ) = F, () where U (R) = d dru(r) is the negative derivative of the potential with respect to distance between the particles. The corrresponding Lagrangian governing the dynamics is simply L(R, R, Ṙ, Ṙ ) = T (Ṙ, Ṙ ) U( R R ), () where the kinetic energy is given by the usual non-relativistic expression: T (Ṙ, Ṙ ) = m Ṙ + m Ṙ, (3) and we are implicitly assuming that any other forces on these particles, internal or external, are negligible. III. CHANGE OF VARIABLES We seek to change variables from those specifying the individual positions of the two particles to a pair of collective variables specifying, respectively, the position of the center of mass (COM) and the relative displacement of the particles. But before transforming kinematic variables, we first transform some parameters for subsequent convenience:

3 3 A. Total Mass and Reduced Mass From the individual particle masses m and m, we can define the total mass M = m + m, (4) which is twice the arithmetic mean of the two particle masses, and also the reduced mass: ( µ = + ) m m = m m m + m = m m M, (5) defined as half the harmonic mean of the two particles masses the same rule as you use to add resistance or inductance in parallel, or to add capacitance in series, or to combine spring constants for ideal springs connected in series, etc. Note that m m = µ M, implying the geometric mean of the individual particle masses m and m equals the geometric mean of µ and M. Because m, m, and (m ± m ), it is easy to verify that µ m m 4M, (6) where µ = if and only if both m = and m =, and µ = 4 M if and only if m = m = M. We see that in the limit as m, we have M m and µ, while as m, we have M and µ m. Similarly, as m, we see that M m and µ, while as m, it can be shown that M and µ m. In particular, in the limit as one of the particles possesses much more mass than the other, the reduced mass µ becomes numerically close to the mass of the lighter particle, while the total mass M will be dominated by the mass of the heavier particle. Because M and µ are both symmetric functions of m and m, they cannot be fully inverted to find m and m separately. The best we can do is: [ ] max[m, m ] = M + 4 µ M (7a) min[m, m ] = M [ 4 µ M ], (7b) determining the two individual particle masses but not which is associated with which particle. B. Center of Mass Position and Relative Displacement We can transform in an invertible and coordinate-independent manner from a pair of individual particle positions to the center-of-mass position and the relative displacement. We define the center of mass (COM) position as usual: R com = mr+mr m +m and then the relative displacement between particles: = m M R + m M R, (8) r = R R = R R, (9) where the relative positions of each particle with respect to the COM are given by R = R R com = ( ) m M R m M R = m M (R R ) = + m M r R = R R com = m M R + ( ) m M R = m M (R R ) = m M r, (a) (b) so each particle s displacement relative to the COM is proportional to the full inter-particle displacement r but also to the other particle s relative mass fraction.

4 Equivalently, these relative positions R and R can be interpreted as the particle positions as measured in the so-called barycentric or center-of-mass (COM) frame, i.e., a frame of reference co-moving with the COM, whose origin coincides with the COM, but with unrotated axes remaining parallel to the original inertial frame s axes. This will be an inertial frame, because in the absence of external forces, we know that the COM must move at constant velocity. Note that the direction of the displacements of the particles are in opposite directions in this frame, as must be the case since the COM lies at the origin in the COM frame. As a quick check on our algebra, notice that while as expected. R com = mr +mr M R R = m M = mm M r mm M r = µ M (r r) =, () r ( m M m+m r) = M r = r, () 4 C. Velocities The corresponding velocities are for the COM, and for the relative motion, where v com = Ṙcom = mṙ+mṙ m +m = m M Ṙ + m M Ṙ, (3) ṙ = Ṙ Ṙ = Ṙ Ṙ, (4) Ṙ = Ṙ Ṙcom = + m M ṙ (5a) Ṙ = Ṙ Ṙcom = m M ṙ (5b) are the velocities of the particles relative to the COM, or equivalently, the velocities in the COM frame. Notice that each particle s motion in the COM frame is proportional to the relative velocity but also to the other particle s mass fraction that is, the particles move in opposite directions in the COM frame, but the heavier particle moves more slowly in this frame, in order to maintain zero total momentum. D. Linear Momenta The total linear momentum is given as usual by where P tot = Mv com = MṘcom = m Ṙ + m Ṙ = p + p, (6) p = m Ṙ p = m Ṙ (7a) (7b) are the usual linear momenta of each particle as measured in the original inertial frame. The corresponding momenta measured in the COM frame are: p = m Ṙ = m (+ m M ṙ) = + mm M ṙ = +µ ṙ (8a) p = m Ṙ = m ( m M ṙ) = mm M ṙ = µ ṙ, (8b) and we may define (without any prime, by convention) a momentum p = µ ṙ = +p = p. As expected, the particles have opposite momenta in the COM frame, so that P tot = p + p =. Notice that p = µṙ is equal to the linear momentum that a single point particle of mass µ would have if it were moving in the COM frame with velocity ṙ.

5 5 E. Kinetic and Potential Energies The potential energy (which assumes the same value in either the original inertial frame or COM frame) can obviously be written as where U = U(r), (9) r = r = R R = R R () is the instantaneous distance between the particles, as measured in either (in fact any) frame. The kinetic energy can be decomposed as where is the COM kinetic energy, and T = T com + T, () T com = M Ṙcom = Ptot M () T = m Ṙ + m Ṙ = m m M ṙ + m m M ṙ = mm +mm M ṙ = m m (m +m ) M ṙ = µ ṙ = p µ (3) is the kinetic energy relative to the COM, or equivalently the kinetic energy as measured in the COM frame. Notice that T = µ ṙ is equal to the kinetic energy that a single point particle of mass µ would have if it were moving with moving with velocity ṙ. F. Angular Momenta Similarly, we can decompose the total angular momentum about the origin as where is the angular momentum of the COM, and L tot = L com + L, (4) L com = R com P tot = R com MṘcom (5) L = R m Ṙ + R m Ṙ = ( m M r) m ( m m M ṙ) + ( M r) m ( m M ṙ) = mm +m m M r ṙ = mm(m+m) M r ṙ = µ r ṙ = r µṙ = r p is the angular momentum relative to the COM. Equivalently, L can be interpreted as the full angular momentum as measured in the COM frame, with respect to the origin in that frame. Notice that L = r µṙ is equal to the angular momentum about that origin that a single point particle of mass µ would possess, were it to be located at r and moving with velocity ṙ. (6)

6 6 G. Transformed Lagrangian Hence, the full Lagrangian can be written as: L = L(R com, r, Ṙ com, ṙ) = T U = T com + T U = M Ṙcom + µ ṙ U(r) (7) in terms of the COM and relative variables. is just L Ṙ com = MṘcom = P tot, while the canonical momen- The canonical momentum conjugate to R com tum conjugate to r is L r = µ ṙ = p. The full Lagrangian can be decomposed as where L = L com + L, (8) L com = L com (Ṙcom) = M Ṙcom (9) governs the COM motion, analgous to the free motion of a single fictitious particle of mass M located at R com ; and L = L (r, ṙ) = µ ṙ U(r) (3) governs the COM-frame relative motion, which we see is analogous to the motion of a single fictitious point particle of mass µ and position r moving in a conservative, central potential U(r). Note that in the context of the two-body problem, L (r, ṙ) can be interpreted as describing the COMframe motion of a fictitious single particle. But the same Lagrangian will of course apply to the dynamics of an actual single particle if it is subject to a conservative, central (radially-directed, radially symmetric in magnitude) force field in some inertial frame. The two problems are actually exactly equivalent in the limit as m, such that particle will just be stuck at what we can take to be the origin, in which case we we have µ m, v com, and r R = R. IV. EULER-LAGRANGE EQUATIONS OF MOTION AND THEIR SOLUTION We see that in (7) we have completely decoupled the COM and relative variables in terms of separate additive contributions to the Lagrangian. We can solve separately for the respective COM and relative trajectories, and moreover, both the COM and relative dynamics are separately analogous to those of a single fictitious point particle, of mass M and µ, respectively. That is, rather than having to solve directly for the coupled motion of two physical particles, we can solve for the uncoupled effective motion of two fictitious point particles, one a free particle of mass M, and the other a particle of mass µ subject to a spherically-symmetric force F (r) = ˆr F (r) = ˆr U (r), (3) where we define F (r) = U (r) = d dr U(r) as the radial component of the force (which is in fact the only non-vanishing component). A. Choice of Coordinates for R com and r So far we have not actually adopted any specific choice of coordinate systems with which to represent R com and r. A suitable choice will simplify the subsequent mathematics significantly. Because R com will

7 just follow a rectilinear trajectory at fixed velocity, the choice of coordinates does not matter too much, but Cartesian coordinates X com = (X com, X com, Z com ) T (with respect to some fixed orthogonal axes in the original inertial frame) are perhaps the most convenient. Our main focus will be on the relative motion in the COM frame, which in fact must be confined to a single plane because of angular momentum conservation. Even though the relative Lagrangian L is spherically symmetric, it therefore makes sense to actually choose cylindrical coordinates for r, with the z-axis taken to be perpendicular to the plane in which the relative motion is confined. That is, because both the magnitude and direction of L = r µṙ are conserved for all time, as long as L, we can choose a COM-frame cylindrical coordinate system such that 7 ẑ = r ṙ r ṙ. (3) Otherwise, if L =, meaning r ṙ = and hence r and ṙ are parallel (or else one or both vanish), then any acceleration r arising from U(r) will also be parallel to r. so we can choose as ẑ any direction such that ẑ r = ẑ ṙ = initially, and these orthogonality conditions will continue to hold for all subsequent time. In either case, the particle can only move transversely to ẑ, and therefore if we choose z = initially, the subsequent motion must remain confined to the z = plane for all time. In fact, since r ẑ r (r ṙ) =, we naturally choose the origin of our coordinate system at r =, so the local transverse radial direction can be taken as and finally, the local azimuthal direction is then chosen to be ˆr = ˆr = r r, (33) ˆφ = ẑ ˆr = r ṙ (ṙ r)r. (34) r r ṙ With these conventions, the transverse radial coordinate is just r = r = r, while the vertical coordinate always vanishes: z =, since, once again, angular momentum conservation guarantees that the motion remains confined for all time to the z = plane, i.e., the plane containing the initial displacement r() and initial relative velocity ṙ(). For definiteness, we could then choose a particular ˆx direction, say ˆx = r() if it is non-zero, and then measure the azimuthal coordinate φ counterclockwise from this axis. Alternatively, we can choose φ = to coincide with some privileged point on the r(t) trajectory for example, a radial turning point. The relative velocity, or equivalently COM-frame velocity of the fictitious particle, is in fact at most two-dimensional, because ż =. ṙ = ṙ ˆr + r φ ˆφ + ż ẑ = ṙ ˆr + r φ ˆφ (35) In these coordinates, the vertical component l of relative angular momentum, l = L z = ẑ L = ẑ [r µṙ] = ẑ r µṙ ẑ = r µṙ = L (36) is equal to the full magnitude of the COM-frame angular momentum. Of course, this angular momentum can also be written explicitly as l = ẑ [r µṙ] = µ[ẑ r] ṙ = µ[ẑ (rˆr + zẑ)] [ṙˆr + r φ ˆφ + żẑ] = µ[ẑ rˆr] [ṙˆr + r φ ˆφ] = + µr ˆφ [r φ ˆφ] = µr φ (37) in terms of the radial distance r and angular velocity φ of the fictitious particle.

8 8 B. Transformed Lagrangian, Conservation Laws, and Equations of Motion In these chosen coordinates, the Lagrangian can be re-written as L = L com + L = M Ẋcom + µ[ ṙ + r φ + ż ] U(r) (38) Physically, there are two point particles of mass m and m and positions R and R, respectively. Dynamically, we therefore should have a total of 3+3 = 6 position coordinates and 3+3 = 6 components of velocity. After transforming, we do in fact have 3 COM position coordinates and 3 relative displacement coordinates, and well as 3 components of COM velocity and 3 components of relative velocity. But mathematically, we can now think of the system as describing two fictitious uncoupled particles, one of mass M at a position described by X com = (X com, Y com, Z com ) T, and one of mass µ at r at a position described by r, φ, and z =. Note this Lagrangian has several symmetries, and hence associated conservation laws: translational invariance, rotational invariance, and boost invariance (up to a gauge term) with respect to the COM, rotational invariance with respect to the relative displacement, and explicit time translation invariance. In particular, the position X com is cyclic, so d L com dt Ẋcom = Lcom X com = MẌcom =. (39) which can be immediately integrated once to yield Ẋ com (t) = Ẋcom() (4) and again to yield: X com (t) = X com () + Ẋcom() t. (4) So we turn to the less-trivial relative motion, as governed by L, analogous to the motion of a single fictitious particle of mass µ located at r. Note that the z coordinate is cyclic, so and hence d L dt ż = L z = µ z =, (4) ż(t) = ż() =, (43) and z(t) = z() + ż() t = + =, (44) confirming what we had already concluded based on angular momentum conservation, namely that the motion in the COM frame is confined to a single plane for all time. (Analogously, the motion of the fictitious particle is confined to the z = plane for all time). The φ coordinate is also cyclic, so d L dt φ = L φ = d dt [µr φ] = d dtl =, (45) confirming that the angular momentum about the COM is conserved. (Analogously, the vertical component of angular momentum about the origin of the fictitious particle is conserved). In addition, time-translation invariance guarantees that H = M Ẋcom + µ[ ṙ + r φ + ż ] + U(r) (46)

9 is conserved, but we already know that ż = is separately conserved, and also Ẋcom and hence T com = M Ẋcom are separately conserved, so in fact we can conclude that E = µ[ ṙ + r φ ] + U(r) (47) is conserved, which can be interpreted as the total COM-frame energy, or equivalently as the total energy of the fictitious particle of mass µ. Using the conservation of l and E, we can now work out the dynamics for the radial coordinate r(t). The Euler-Lagrange equation for r is d L dt ṙ L r = µ r µr φ = d dr U(r), (48) where as usual the transverse radial acceleration involves two terms, one reflecting changes in radial speed and one reflecting centripetal effects. Using the conservation of angular momentum l = µr φ, this can be rewritten as µ r = µr φ du(r) dr = l µr du(r) [ 3 dr = d l dr µr + U(r) ] (49) which we recognize as being analogous to the motion of a single point particle in one dimension, subject to the effective potential V eff (r) = U(r) + l µr (5) which includes the original potential energy U(r) plus a so-called centrifugal barrier term l µr, which acts like a repulsive contribution to the effective radial potential energy. Why? If l = µr φ is to remain constant, as r decreases, φ must increase, and since total energy in the COM frame is conserved, this angular kinetic energy must grow at the expense of either true potential energy or radial kinetic energy (as there remains no vertical motion, and hence nowhere else from which to draw the energy). The effective dynamics of this fictitious one-dimensional particle are therefore governed by which has as an obvious constant of the motion H eff = µṙ + V eff (r) = µṙ + equal to the actual COM-frame energy of the original particles. µ r = d dt T = d dr V eff(r), (5) l µr + U(r) = µṙ + µr φ + U(r) = E, (5) 9. Formal solution for the Radial Motion as a Function of Time In the equivalent one-dimensional problem, note that r is acting like the one-dimensional position of the fictitious particle, not the distance from the origin. So in order to avoid possible jumps in angle φ by π if r reaches r =, it is convenient to allow for the possibility that r may take on positive or negative values, while being careful to now write V eff (r) = U( r ) + l µr (53) to ensure the correct physical behavior of the potential energy. However, as we will see momentarily, usually r cannot ever reach r =. Consider the evolution of r(t) for given values of the constants of motion E and l. Because of their conservation, the radial coordinate r must be confined to a classically allowed region, consisting of a continuous, connected interval bracketing r() and for which V eff (r) E. This interval may be finite, half-infinite, or doubly-infinite, or even reduce to a single point in the case of a circular orbit. Generically, solutions to E = V eff (r) will lead to radial turning points r ± between the radial motion will be confined.

10 Actually, we can always assume r r(t) r + +, where we agree to set r = if there is no finite turning point for r r(), and set r + = + if there is no finite turning point for any r r(). The motion will be bounded if r ± < strictly, and unbounded otherwise. Typically specifically, unless l = exactly or else U(r) has a singularity as r that cancels that in the centrifugal barrier term l µr we will have V eff (r) + as r, leading to an inner radial turning point at some r. Smaller values of r are not possible, because in order to conserve angular momentum the system would require more rotational kinetic energy than the total energy available. In exceptional cases without a sufficient centrifugal barrier, there will be a regimes for which r = r + = + r + for certain values of E, but there may be other classically allowed radial ranges as well. Given the radial turning points r ± defining the boundaries of the relevant classically allowed region, we can calculate the time intervals from the most recently encountered turning point via an integral of the form: t t = + r(t) dr µ(e Veff(r)) (54) r or t t + = r(t) r+ dr = + dr µ(e Veff(r)) µ(e Veff(r)) (55) r + r(t) where t is the most recent time for which r(t ) = r, and t + is the most recent time for which r(t + ) = r +. In principle, these relations can then be inverted over each branch of the motion to obtain the radial trajectory r(t).. Circular Orbits If r = r + = r, then the classically allowed region shrinks to a single point, and the trajectory corresponds to a circular orbit r(t) = r, requiring ṙ = and r =. But r = is equivalent to V eff(r ) = d dt V eff(r) r=r =, meaning r is a local extrema (or at least critical point), while ṙ = implies that energy is chosen just right, so that E = V eff (r ). The circular orbit r = r is stable with respect to small radial perturbations if V eff(r ) > and unstable if V eff(r ) <. Note however, that this only tests for stability with respect to small displacements in radial position or radial velocity. Other perturbations could change the functional form of V eff (r) or shift the plane in which the motion is confined, and cannot be assessed by examining the effective radial potential alone. 3. Formal solution for the Angular Motion Once r(t) is determined, the angular motion can in principle be determined using angular momentum conservation: φ(t) φ() = t dt φ(t ) = t dt t l µ r(t ) = l µ dt [r(t )], (56) so we formally have reduced to quadratures the problem of finding both r(t)and φ(t).

11 4. Kepler s Second Law Note that, regardless of the functional form of U(r), the angular motion satisfies Kepler s Second Law, which says the displacement vector r(t) sweeps out equal areas in equal times. indeed, Kepler s Second Law is just a restatement of the conservation of angular momentum in this context. Recall that the magnitude of the vector a b can be interpreted as the area of the parallelogram whose sides are specified by the three-dimensional vectors a and b, while the vector itself points in a direction normal to this parallelogram, which is the usual convention for ascribing a direction to a small area. So, neglecting O(dt ) terms, the vector area da swept out by r(t) between t and t + dt is just one-half the area of the parallelogram defined by r(t) and r(t + dt): so da = r(t) r(t + dt) = r(t) [r(t) + dt ṙ(t)] = r(t) r(t) + dt r(t) ṙ(t) = dt r(t) ṙ(t), (57) which is a constant vector. Perhaps a bit more simply, in cylindrical coordinates, we see that da dt = r ṙ = µ L, (58) l = µr dφ dt, (59) but r rdφ = r φ dt can be interpreted as the scalar area da of an infinitesimal triangle of height r and base r dφ = r φ dt which is swept out by r(t) between t and t + dt, so that l = µ da dt. (6) Again, note that this holds for any central force, not just for gravitational attraction. 5. Caveat Notice that the equation of motion (5) could be derived from an effective Lagrangian L eff (r, ṙ) = µṙ V eff (r) = µṙ l µr U( r ). (6) However, this is not equal to the Lagrangian L in fact the centrifugal term enters with opposite sign. This is because what we have reinterpreted as a contribution to an effective radial potential energy what started out as an angular contribution to the kinetic energy, and kinetic and potential energies appear in a Lagrangian with opposite sign. We can also think in terms of the nature of the constraint µr φ l =. This is a non-holonomic constraint, so we cannot simply substitute it into the Lagrangian L before taking variational derivatives. When we study Hamiltonian methods, this subtlety regarding the sign of the centrifugal term in the radial dynamics can largely be avoided. 6. Binet Method and Shape of Orbits If l =, the the motion is confined to a single line through the origin. so φ = and φ(t) = φ does not advance, Otherwise, if l, then φ(t), and we can use the azimuthal angle φ as an independent variable instead of the time t. In fact, it is often more useful to look at the spatial shape of the orbit, that is, at r(φ) regarded as a function of the angle variable φ.

12 In order for the angle φ(t) to be an invertible function of time t, we need to allow φ to grow continuously and to assume arbitrary values as t increases, rather than wrapping back to at ±π. An orbit is closed if and only if r(φ) is a periodic function of φ for some integral multiple of π (and multiples thereof). Now instead of solving for r(t) then φ(t), we can solve directly for r(φ) by using a clever transformation introduced by Jacques Philippe Marie Binet, a 9th century French mathematician and astronomer. Introducing the change of variable u = r, (6) and using l = µr φ along with the chain rule, we can rewrite the differential operator d dt in terms of d dφ : Using this relation, we can write and then so the radial equation of motion can be written equivalently as d dt = dφ dt ṙ = d dt r = l µ u r = d dtṙ = l µ u d dφ = l d µr dφ = l µ u d dφ u = l ( ) µ u u [ d dφ l µ d d dφ. (63) dφ u = l µ d dφu, (64) d dφ u] = l µ u d dφ u, (65) µ r = l µr 3 U (r) = l µr 3 + F (r) (66) l µ u d u dφ = l µ u3 + F (/u) (67) or d dφ u = u µ l u F ( ) u (68) where F (r) = U (r) = d dr U(r). Provided l, this second-order differential equation can be solved for u(φ) given appropriate initial conditions, thereby determining r(φ) = /u(φ). If we can solve this differential equation for appropriate initial conditions, then we can deduce the shape of the orbit. Conversely, if given the shape of the orbit, we can deduce the force law without having to know anything about the velocity along the orbit. V. INVERSE SQUARE FORCE LAWS AND KEPLER S LAWS An very important special case of conservative central force motion, applicable in gravitational and Coulomb problems, is that of the inverse-square force law, corresponding to the interaction potential energy U(r) = κ r (69) for all r >, or equivalently, to the force F (r) = ˆr κ r, (7) for some constant parameter κ, where the force is attractive if κ > and repulsive if κ <. For example, for the gravitational interaction between two point masses, κ = Gm m = GMµ.

13 3 c ϕ f FIG. The bounded orbits of the attractive Kepler (inverse square law) problem are ellipses in the COM frame, with the center of force at one focus, specified in polar cylindrical coordinates r and φ. Shown are the lengths of the semimajor axis (a), the semiminor axis b, the semilatus rectum (c), and also the length f from the geometric center to a focus, as well as the minimal radius r min (achieved at the periapsis) and the maximal radius r max (reached at the apoapsis). All these lengths are determined by the COM-frame energy E and magnitude of angular momentum l. A. Kepler s First Law We can now establish Kepler s First Law, which its most general form says that all relative trajectories r(t) for an inverse-square force law are conic sections - circles, ellipses, parabolas, or hyperbolas. If l =, then r and ṙ remain parallel, and the motion is confined to a straight line (or line segment) through the origin, which can be thought of as a degenerate (limiting) case of an ellipse (for motion along a segment) or a hyperbola (for unbounded motion along a line). Otherwise, if l, then we can use Binet s transformation. For the inverse square force law, for all r >, and the Binet equation (68) becomes F (r) = d dr U(r) = κ r = κ u, (7) d dφ u = u µ l u F ( ) u = u + µ l u κu = u + µ κ l, (7) which is a linear differential equation in u which can be integrated to obtain: u(φ) = r(φ) = α cos(φ φ ) + µκ l (73) for some integration constants α and φ. For later convenience, we define new constants c = l µκ (74a) ɛ = αc = αl µκ, (74b) so that r(φ) = c + ɛ cos(φ φ ), (75) where c is fully determined once l is fixed, and ɛ and φ are dimensionless constants to be chosen based on further initial conditions.

14 4. Bounded Kepler Orbits First, assume an attractive force such that κ >, and hence c >. So bounded (and in fact, closed) orbits are possible. As long as we are interested in the shape of single orbits, and not, say, transfers between orbits, we can always orient the in-plane (ˆx and ŷ) axes so that: (i) r(φ) is at a radial turning point when φ =, and (ii) ɛ. This means that we are choosing φ = nπ for some integer n, so the orbit can be written as for some ɛ, The radial turning points must be r(φ) = r min = r max = c + ɛ cos(φ), (76) c +ɛ = l µκ c ɛ = +ɛ ɛ r min, and so we can relate the parameter ɛ to the energy E by equating E = V eff (r min ) = l µ rmin [ κ r min = l µ r min r min c +ɛ (77a) ] = l µ (77b) [ +ɛ +ɛ ] c c c = µκ l (ɛ ), (78) and we see that ɛ is a function of the conserved quantities l and E : ɛ = + + µκ E l, (79) which also establishes the non-obvious inequality E l > µκ for bounded Kepler orbits. This orbit will be truly bounded if r max < strictly, meaning the system does not have enough energy to achieve arbitrarily large separations r. Since lim U(r) =, this is equivalent to saying the energy is r negative: E <, which from (78) implies ɛ <. But it is not difficult to show that for c > and ɛ <, (89) describes an ellipse with one focus at the origin r =, with eccentricity ɛ, and with a length c of the so-called semilatus rectum, but we will first need to review some geometric facts about the ellipse. The eccentricity ɛ of the ellipse is a measure of the aspect ratio of the ellipse, and is equal to: ɛ = r max r min r max + r min. (8) so clearly satisfies ɛ <, where ɛ = corresponds to a circle of constant radius r min = r max. The ellipse is defined such that the sum of the distances between any point on the ellipse to each of two specified foci is some constant, which we will denote a. By center of the ellipse we mean the intersection of the two mutually perpendicular axes of reflection symmetry, the longer of which is called the major axis and contains the foci located symmetrically about the center, and the shorter is the minor axis. Except for the special case of a circle, the center does not coincide with the origin r = of our coordinates; rather the origin is taken to be one of the foci (specifically the one on the right looking down at a diagram of the ellipse on the page, according to our usual conventions for relating Cartesian and polar coordinates). The line segment between the center and either vertex on the ellipse, that is, between points of maximal distance from the center, is a semimajor axis, and is of length a, which is the arithmetic mean of the minimal and maximal distances between the ellipse and the origin/focus: a = (r min + r max ). (8)

15 The length a of the semimajor axis is also equal to the mean radius weighted by arc length around the ellipse: 5 π ds r = dφ ( dr dφ ) + r r(φ) = a. (8) Both foci of the ellipse lie along the major axis, equidistant from the center. The length f is defined as the distance between one focus and the center of the ellipse. A semiminor axis is the line segment between the ellipse and its center which is perpendicular to the semimajor axes; its length b is the geometric mean of the the minimal and maximal distances between the ellipse and the origin/focus: b = r min r max (83) The point on the ellipse closest to the origin, at the edge of one semimajor axis and corresponding to r = r min, is called the periapsis (from the Greek, via Latin, meaning near arch ). The point on the ellipse furthest from the origin, which must lie on the far edge of the other semimajor axis, and corresponding to r = r max, is called the apoapsis ( far arch ). Together these points are called the apsides. (For the particular case of a body orbiting the sun, they are called the perihelion and aphelion, respectively; for a body orbiting the Earth, the perigee and apogee). The area enclosed by the ellipse is just A = πab. The semilatus rectum (Latin for half of a straight side ) is a line segment between one focus and a point on the ellipse, oriented transverse to the major axis (and therefore parallel to the minor axis). Its length, denoted by c (some books will use d, or p, or various other symbols) is the harmonic mean of the minimal and maximal radii measured from the origin/focus: c = ( r min + r max ). (84) For an ellipse, the various characteristic distances satisfy a number of additional elementary relationships, including: f = a b = aɛ, c = a( ɛ ) = b ɛ = r min ( + ɛ) = r max ( ɛ) = b a. (85a) (85b) For f =, or equivalently ɛ =, the ellipse is in fact just a circle. As f or equivalently ɛ, the ellipse becomes a parabola. In polar coordinates centered on the right focus, with angle measured counterclockwise from the ˆx axis, the ellipse must satisfy the defining relation specifying that the distances from the foci to any point on the ellipse sum to exactly a: or which is equivalent to: as claimed. r ˆr + r ˆr + f ˆx = a (86) (r a) = r ˆr + f ˆx = r + (f) + 4fr cos φ (87) r = a f a + f cos φ = a ɛ a a + ɛa cos φ = a( ɛ ) + ɛ cos φ = c + ɛ cos φ, (88) Note that for a given energy E <, the eccentricity decreases as l increases. Conversely, the circular orbit has the most angular momentum for a given energy.

16 6. Unbounded (Scattering) Orbits for Attractive Potentials Supposing still that κ > but that ɛ >, we see that c r(φ) = + ɛ cos(φ), (89) will still have an inner radial turning point at r min = c + ɛ = l µκ + ɛ, (9) so this inner turning point still allows us to connect ɛ to the other conserved quantities: E = V eff (r min ) = µκ l (ɛ ) >, (9) but here the energy is positive for ɛ >, meaning that the kinetic energy is sufficiently large so that the orbit is said to be unbounded, or equivalently to be a scattering orbit. In fact, we see that r as cos φ ɛ or equivalently as φ ± arccos( ɛ ). Thus we expect a pair of asymptotes at certain finite angles with respect to the origin, and we will find hyperbolic trajectories. Although only one branch of the hyperbola wil be associated with a single trajectory, it is convenient to visualize both branches, and the center of reflection symmetry between them. The hyperbola can be defined as the locus of all points such that the difference in distance between two foci (located symmetrically along the major axis) is equal to the constant a, where f = ɛa is the distance between the center and one of the foci, and it turns out a is also the distance between the center and a vertex, which is defined as a point where the hyperbola crosses the major axis, meaning it is the closest point on a branch of the hyperbola to the center. The semilatus rectum is again defined as a line segment from the focus to the ellipse, transverse to the major axis; but now has length c = a(ɛ ). Centering our origin (r = ) on the left focus, the defining relation after rearranging and squaring becomes which is equivalent to: as claimed. f ˆx r ˆr r ˆr = a (9) (r + a) = f ˆx r ˆr = (f) + r 4fr cos φ, (93) r = f a a + f cos φ = ɛ a a a + ɛa cos φ = a(ɛ ) + ɛ cos φ = c + ɛ cos φ, (94) If ɛ = exactly, meaning E = exactly, the particles have just enough energy to achieve arbitrary large separation (albeit in infinite time), and we claim the relative trajectory is then a parabola, which can be thought of as the limiting case of an ellipse where the foci become infinitely separated, or as the limiting case of an hyperbola where the angle between the asymptotes approaches π. In this case, by rearranging and squaring we find which is equivalent to r = c +cos φ, (95) r = (c r cos φ) = c + r cos φ cr cos φ = c + r r sin φ cr cos φ, (96) r sin φ = c cr cos φ, (97) or in Cartesian coordinates (x = r cos φ, y = r sin φ) centered at the same focus, which indeed describes a parabola. y = c cx, (98)

17 7 3. Unbounded (Scattering) Orbits for Repulsive Potentials In the case of repulsive potentials (corresponding to κ < ), obviously no bound orbits are possible since E. The easiest way to work out the orbits is to choose ɛ (it is then the negative eccentricity) but insist that r, leading to hyperbolic (or in the limiting case, parabolic) orbits, but with the focus/origin outside rather than inside the convex hull of the orbit. That is, the incoming fictitious particle gets turned around before crossing the plane which passes through the center of force and is transverse to the initial velocity. B. Kepler s Second Law and Orbital Speeds We have already proved Kepler s Second Law for the general two-body central force problem. For closed Kepler orbits, it confirms that the angular speed is maximal at the periapsis and minimal at the apoapsis. Energy conservation additionally tells us that the total speed is also maximal at the periapsis and minimal at the apoapsis. However, at these apsides, where the total speed is extremal, the radial velocity must vanish, because they are radial turning points. The radial speed is maximal in between, at points where the effective potential V eff (r) is minimal specifically at each end of the latus rectum. C. Kepler s Third Law Closed (and therefore elliptical) orbits are necessarily periodic. Integrating Kepler s Second Law over one period τ, we find A = τ dt da dt = τ dt l µ = τl µ (99) but for an ellipse, A = πab, () so τ = πabµ l () and hence τ = 4π a b µ l = 4π µ l a a ( ɛ ) = 4π µ a( κ a3 ɛ ) = 4π µ l κ a3 c c = µ 4π κ a3, () µκ so the square of the period is proportional to the the cube of the length a of the semimajor axis. Note that a is a natural measure of the average radius of the obit, since it is both the arithmetic mean of the minimal and maximal radii, and also the arclength-weighted average of the orbital radius. This length a, and therefore the temporal period of the orbit, depend only on the COM-frame energy E, and not on the COM-frame angular momentum: a = c l ɛ = µκ l µκ ( E ) = κ E, (3) which itself a beautiful and useful result for the inverse-square central force problem, related to the integrability properties of this system (see below) and the fact that in the quantum mechanics of the

18 hydrogen atom, the energy eigenvalues depend (in the absence of relativistic and spin effects) only on the principal quantum number and not on the angular momentum quantum number. For the specific case of the gravitational two-body problem, note that so µ κ = µ = µ Gm m GµM = GM, (4) 8 τ = 4π GM a3, (5) where Kepler s constant 4π GM is not quite constant across all planets, since it depends on the total mass M and not just the solar mass. VI. DYNAMICAL INVARIANTS AND INTEGRABILITY Because the bound orbits for the Kepler problem are all closed, with the shape of the orbit described by a single-valued mapping between angle and radius, it must be the case that in cylindrical coordinate the period of the radial motion and the the period of the angular motion are equal. Moreover, the period of motion depends only on the energy and not the angular momentum. For a general problem, even for a completely integrable one, there is no reason that either of these properties would necessarily be the case, unless some sort of additional constraint is present. Indeed, such a constraint arises from another conservation law, connected to a so-called hidden symmetry of the Kepler problem.. Laplace-Runge-Lenz Vector In addition to E and L, it can be shown directly from the equations of motion for the Kepler problem that an additional vector-valued quantity is also conserved, the so-called Laplace-Runge-Lenz Vector: A = p L µ κ ˆr, (6) which includes a term which is bilinear in the angular and linear momenta. Since A is a vector, this might seem to provide an additional three conserved quantities, but really only one additional independent constant of the motion has been found, not three, since A, L, and E are related by two additional equations: A L = A A = µ κ + µe l, (7a) (7b) but the vector A does necessarily lie in the plane of motion, and its direction  in this plane does provide an additional, genuinely independent constant of the motion always pointing in the direction from the COM-frame origin (center of force) to the periapsis of the orbit. Therefore its conservation is what guarantees that the bounded orbits must be closed non-inverse-square-law central forces, for example, will have non-closed trajectories, where the apsides precess over each successive orbit. In fact, we can derive the Kepler orbits quite simply from the conservation of A. We have A r = Ar cos φ = r (p L ) µκr = (r p) L µκr = L L µκr = l µκr, (8) where we are measuring φ from the periapsis, and A = A µκɛ is the constant magnitude of A. Rearranging, this becomes: r = l µκ + A µκ cos φ = c + ɛ cos φ (9)

19 which is the equation for a conic section, with eccentricity ɛ = A l µκ and length of semilatus rectum c = µκ. From the above expression for A A we can rewrite the eccentricity in terms of the energy and angular momentum: which is equivalent to our earlier result. 9 ɛ = l µκ E, () It turns out that this additional conservation law is not associated with a symmetry under a transformation of the configuration space coordinates by functions of these coordinates (and/or time) alone, but is associated with a more general sort of symmetry under transformations which mix the generalized positions and their conjugate momenta. Equivalently, the symmetry can be interpreted as a higher-dimensional geometric symmetry, where the usual motion in three dimensions is regarded as the projection of a motion in four dimensions. (For group theory aficionados: this hidden symmetry is associated with SO(4) for bound orbits and SO(3, ) for unbound orbits). As is often the case in the history of science, the accepted name does not reflect the actual history of discovery, but more the history of popularization. The conservation of A for a inverse-square central force was apparently first discovered for a special case by Jakob Hermann, then generalized by Johann Bernoulli in 7, then rediscovered by Laplace, then re-derived by Hamilton, then by Gibbs. The latter s derivation was used in a popular textbook in Germany by Carl Runge, which was then referenced by Wilhelm Lenz in a widely-read paper on the quantum mechanics of the hydrogen atom, and gained further notoriety when used by Pauli to solve for the energy spectrum of hydrogen by algebraic means. The Laplace-Runge-Lenz (LRL) vector also reveals an interesting fact about the momentum p in the case of Kepler motion, namely that its tip traces out an offset circle in momentum space. We define Cartesian axes ˆx A and ŷ = ẑ ˆx in the plane of motion (transverse to L ẑ). Then by equating the squared-magnitude of both sides of we can deduce that µκˆr = p L A, () p x + (p y A/l) = (µκ/l), () which is the equation for an offset circle of radius µκ/l in momentum space. A. Subintegrability, Integrability, Superintegrability, and Maximal Superintegrability Recall that we say a system is integrable when the problem of specifying any trajectory given the initial conditions can be reduced to quadratures, meaning we can in principle express the trajectories using one-dimensional definite integrals as well as various purely algebraic relations involving root-finding or function inversion (even if, in practice, we cannot necessarily perform the needed integrations or algebra analytically). This notion of integrability is closely related to whether the system possesses a sufficient number of autonomous constants of the motion that are independent (in ways that we will make more precise later in the course). A single-degree-of-freedom conservative problem is always integrable, because formally we can express the trajectories in terms of one-dimensional integrals involving the conserved value of the energy and the potential energy function. Conservation of a single function of the (generalized) position and (generalized) velocity determines the shape of the trajectory in the two-dimensional position/momentum phase space, and further initial conditions specify where on this curve the system lies at a specified time. Even though we have two dynamical variables (say q and q), we only need one conserved quantity to a determine a ( ) = dimensional curve (trajectory), since we still must have the freedom to choose initial conditions, and therefore to specify where along this curve we are to start at time t =.

20 Similarly, an n-dof holonomic system has n dynamical variables (n generalized coordinates and n generalized velocities), but it turns out that we only need n independent, autonomous (i.e., explicitly time independent) constants of the motion for the system to be integrable. The conserved quantities restrict the trajectory to an n-dimensional surface in the n-dimensional space of dynamical variables, and the remaining initial conditions pick out a particular starting point and subsequent curve on this surface. In principle, the solution at any time can be determined using algebra at performing n definite integrals. (We will explore this in more detail after introducing the Hamiltonian formalism). But generically, a conservative n-dof system will have one conserved quantity, namely the energy, but perhaps no more. If it has more than but less than n independent (and autonomous) conserved quantities, it is said to be subintegrable or partially integrable. These count as non-integrable, and such systems can exhibit chaotic motion which we will discuss in more detail at the end of the course. If the system has n independent (and autonomous) constants of motion, it is (completely) integrable, leading to regular trajectories which are quasi-periodic if bounded. And if it has more than n independent (autonomous) constants of the motion, it is said to be superintegrable, with the additional dynamical invariants further constraining the possible motion. How many independent constant of motion can an n-dof system have? It can have up to n, and their intersecting level surfaces would then define a one dimensional curve in the space of dynamical variables. Any more, and no actual dynamics would be possible we need the freedom to specify initial conditions which then determine where on this curve the system is located at a given time. For a two-body central force problem, in the relative coordinates in the COM frame, we have 3 degrees of freedom and = 6 dynamical variables (the components of r and ṙ), and also 4 independent, autonomous conserved quantities: the energy E, and 3 components of the relative angular momentum L. Generically, these systems are therefore superintegrable, since 4 > 3. But for the inverse square force law, we have in addition independent component of the Laplace-Runge-Lenz vector, say A ˆx in our notation. So the Kepler problem is in fact maximally superintegrable, because 5 = 3 = 6. We have seen above that we could deduce the shape of the Kepler orbits algebraically, without any actual integration just by manipulating the Laplace-Runge-Lenz Vector. This is a consequence, and sign, of maximal superintegrability, in which -dimensional trajectories are determined entirely by conservation laws, and not by calculus. The intersection of the level hypersurfaces associated with the conserved quantities determines the one-dimensional shape of the trajectory. Specifically, in the full two-body problem we have 6 degrees of freedom for the relative motion, meaning 6 generalized coordinates and 6 corresponding generalized velocities. A total of 6 initial conditions specifying the COM position and COM velocity determine the trivial (rectilinear) motion of the COM. Then, 6 additional initial conditions are needed to specify the relative position and relative velocity should determine the motion in the COM frame. Indeed, we need: numbers to specify the orientation of the plane to which the relative motion is confined (for example, a unit normal vector); parameters to specify the intrinsic shape of the ellipse (for example, the length of the semimajor axis and the eccentricity); another parameter to specify the angular orientation of the ellipse (or hyperbola or parabola) in that plane; and finally number to specify the location along the ellipse at some reference time. For the Kepler problem, the first 5 of these are related to time-independent constants of the motion, respectively: the direction of the angular momentum; the COM-frame energy and magnitude of COM-frame angular momentum; and the in-plane direction of the Laplace-Runge-Lenz vector, leaving only the freedom of choosing the time of arrival at periapsis. What happens when we add more particles? The full three-body problem has 9 degrees of freedom, but investigations by Bruns, Poincaré, Painlevé, and others have established that there are no well-behaved extra autonomous constants of the motion beyond energy, linear momentum, and angular momentum. So there are 9 degrees of freedom but only 7 autonomous invariants, so the three-body problem is not integrable. By adding just one more particle, we go from a maximally superintegrable system (in the relative coordinates) to a subintegrable, chaotic system.