42 7 The Rutherford Scattering Cross-section where we note the fact that a solid-angle element in spherical coordinates can be written as dΩ =2π sin ϑdϑ. The differential cross-section is defined as the ratio d (2) σ dΩ : d (2) σ dΩ = b(ϑ)db(ϑ) sin ϑdϑ . (7.1) This expression is often complicated, but its meaning can be visualized, as follows. Consider a stream of particles with current density j 0 per cross- sectional area flowing towards the target and being scattered by the potential V. At a large distance beyond the target, a fraction of the particles enters a counter, where they are recorded. The number of counts in a time Δt is given by ΔN = d (2) σ dΩ ·j 0 · Δt · ΔΩ. The aperture of the counter corresponds to scattering angles in the interval (ϑ, ϑ +dϑ), i.e., to the corresponding solid angle element ΔΩ := 2π sin ϑΔϑ. The differential scattering cross-section is essentially the missing propor- tionality factor in the relation ΔN ∝ j 0 ·Δt · ΔΩ, and (7.1) should only be used for evaluation of this quantity The description of these relations is supported by Fig. 7.1. For A/r-potentials the differential cross-section can be evaluated exactly, with the result d (2) σ dΩ = A 2 16E 2 1 sin 4 ϑ 2 , which is called the Rutherford scattering cross-section. This result was ob- tained by Rutherford in Cambridge, U.K., at the beginning of the twentieth century. At the same time he was able to confirm this formula, motivated by his ground-breaking scintillation experiments with α-particles. In this way he discovered that atoms consist of a negatively-charged electron shell with a ra- dius of the order of 10 −8 cm, and a much smaller, positively-charged nucleus with a radius of the order of 10 −13 cm. In fact, the differential cross-sections for atomic nuclei are of the order of 10 −26 cm 2 , i.e., for α-particles the space between the nuclei is almost empty. Fig. 7.1. Schematic diagram on differential scattering cross-sections. A particle enters the diagram from the left on a path parallel to the x-axis at a perpendicular distance b (the so-called “impact parameter”; here b =0.25). It is then repulsively scattered by a target at the origin (here the scattering occurs for −20 ≤ x ≤ 20, where the interaction is felt) and forced to move along the path y(x)=b +0.00025· (x +20) 2 , until it leaves the diagram asymptotically parallel to the inclined straight line from the origin (here y =0.016 · x). Finally it enters a counter at a scattering angle ϑ =arctan0.016, by which the above asymptote is inclined to the x-axis. If the impact parameter b of the particle is slightly changed (in the (y,z)-plane) to cover an area element d (2) A(b):=dϕ · b · db (where ϕ is the azimuthal angle in that plane), the counter covers a solid-angle element dΩ(b)=dϕ ·sin ϑ(b) ·dϑ.The differential cross-section is the ratio d (2) A dΩ = ˛ ˛ ˛ b·db(ϑ) sin ϑ·dϑ ˛ ˛ ˛ . 8 Lagrange Formalism I: Lagrangian and Hamiltonian In the present context the physical content of the Lagrange formalism (see below) does not essentially go beyond Newton’s principles; however, mathe- matically it is much more general and of central importance for theoretical physics as a whole, not only for theoretical mechanics. 8.1 The Lagrangian Function; Lagrangian Equations of the Second Kind Firstly we shall define the notions of “degrees of freedom”, “generalized co- ordinates”, and the “Lagrangian function” (or simply Lagrangian)assuming asystemofN particles with 3N Cartesian coordinates x α , α =1, ,3N: a) The number of degrees of freedom f is the dimension of the hypersurface 1 in 3N -dimensional space on which the system moves. This hypersurface can be fictitious; in particular, it may deform with time. We assume that we are only dealing with smooth hypersurfaces, such that f only assumes integral values 1, 2, 3, b) The generalized coordinates q 1 (t), ,q f (t) are smooth functions that uniquely indicate the position of the system in a time interval Δt around t; i.e., in this interval x α = f α (q 1 , ,q f ,t), for α =1, ,3N. The gen- eralized coordinates (often they are angular coordinates) are rheonomous, if at least one of the relations f α depends explicitly on t; otherwise they are called skleronomous.TheLagrangian function L is by definition equal to the difference (sic) between the kinetic and potential energy of the system, L = T−V, expressed by q α (t), ˙q α (t)andt, where it is assumed that a potential energy exists such that for all α the relation F α = − ∂V ∂x α holds, and that V can be expressed by the q i ,fori =1, ,f,andt. Frictional forces and Lorentz forces (or Coriolis forces, see above) are not al- lowed with this definition of the Lagrangian function, but the potential energy may depend explicitly on time. However, one can generalize the definition of the Langrangian in such a way that Lorentz forces (or Coriolis forces) are 1 For so-called anholonomous constraints (see below), f is the dimension of an infinitesimal hypersurface element 46 8 Lagrange Formalism I: Lagrangian and Hamiltonian also included (see below). The fact that the Lagrangian contains the differ- ence,andnot the sum of the kinetic and potential energies, has a relativistic origin, as we shall see later. 8.2 An Important Example: The Spherical Pendulum with Variable Length These relations are best explained by a simple example. Consider a pendulum consisting of a weightless thread with a load of mass m at its end. The length of the thread, l(t), is variable, i.e., an external function. The thread hangs from the point x 0 = y 0 =0,z 0 = 0, which is fixed in space, and the load can swing in all directions. In spherical coordinates we thus have (with z measured as positive downwards): ϑ ∈ [0,π], where ϑ = 0 corresponds to the position of rest, and ϕ ∈ [0, 2π): x = l(t) · sin ϑ ·cos ϕ y = l(t) · sin ϑ · sin ϕ z = z 0 − l(t) · cos ϑ. (8.1) The number of degrees of freedom is thus f = 2; the generalized coordinates are rheonomous, although this is not seen at once, since q 1 := ϑ and q 2 = ϕ do not explicitly depend on t, in contrast to the relations between the cartesian and the generalized coordinates; see (8.1). Furthermore, V = mgz = mg ·(z 0 − l(t) · cos ϑ) , whereas the kinetic energy is more complicated. A long, but elementary cal- culation yields T = m 2 ˙x 2 +˙y 2 +˙z 2 ≡ m 2 l 2 · ˙ ϑ 2 +sin 2 ϑ ˙ϕ 2 +4l ˙ l sin ϑ cos ϑ ˙ ϑ + ˙ l 2 sin 2 ϑ . Apart from an additive constant the Lagrangian L = T−Vis thus: L(ϑ, ˙ ϑ, ˙ϕ, t)= m 2 l 2 ˙ ϑ 2 +sin 2 ϑ ˙ϕ 2 +4l ˙ l sin ϑ cos ϑ ˙ ϑ + ˙ l 2 sin 2 ϑ + mg ·(l(t) − z 0 ) · cos ϑ. (8.2) (In the expression for the kinetic energy the inertial mass should be used, and in the expression for the potential energy one should actually use the gravitational mass; g is the acceleration due to gravity.) 8.3 The Lagrangian Equations of the 2nd Kind 47 8.3 The Lagrangian Equations of the 2nd Kind These shall now be derived, for simplicity with the special assumption f =1. Firstly we shall consider an actual orbit q(t), i.e., following the Newtonian equations transformed from the cartesian coordinate x to the generalized coordinate q.Attimet 1 this actual orbit passes through an initial point q 1 , and at t 2 through q 2 . Since Newton’s equation is of second order, i.e., with two arbitrary constants, this is possible for given q 1 and q 2 . In the following, the orbit is varied, i.e., a set of so-called virtual orbits, q v (t):=q(t)+ε · δq(t) , will be considered, where the real number ε ∈ [−1, 1] is a so-called vari- ational parameter and δq(t) a fixed, but arbitrary function (continuously differentiable twice) which vanishes for t = t 1 and t = t 2 .Thus,thevirtual orbits deviate from the actual orbit, except at the initial point and at the end point; naturally, the virtual velocities are defined as ˙q v (t):= ˙q(t)+ε · δ ˙q(t) , where δ ˙q(t)= dq(t) dt . Using the Lagrangian function L(q, ˙q,t) one then defines the so-called action functional S[q v ]:= t 2 t 1 dtL(q v , ˙q v ,t) . For a given function δq(t) this functional depends on the parameter ε,which can serve for differentiation. After differentiating with respect to ε one sets ε = 0. In this way one obtains dS[q v ] dε |ε=0 = t 2 t 1 dt ∂L ∂ ˙q v δ ˙q(t)+ ∂L ∂q v δq(t) . (8.3) In the first term a partial integration can be performed, so that dS[q v ] dε |ε=0 = ∂L ∂ ˙q v |t 1 δq(t 1 ) − ∂L ∂ ˙q v |t 2 δq(t 2 ) + t 2 t 1 dt − d dt ∂L ∂ ˙q v + ∂L ∂q v δq(t) . (8.4) In (8.4) the first two terms on the r.h.s. vanish, and since δq(t) is arbitrary, the action functional S becomes extremal for the actual orbit, q v (t) ≡ q(t), 48 8 Lagrange Formalism I: Lagrangian and Hamiltonian Fig. 8.1. Hamilton’s variational principle. The figure shows the ε-dependent set of virtual orbits q v (t):=t +(t 2 − 1) + ε · sin (t 2 − 1) 3 ,forε = −0.6, −0.4, ,0.6 and times t between t 1 = −1andt 2 =1.Theactual orbit, q(t), corresponds to the central line (ε = 0) and yields an extremum of the action functional. The virtual orbits can also fan out more broadly from the initial and/or end points than in this example iff (i.e., “if, and only if”) the so-called variational derivative δS δq := − d dt ∂L ∂ ˙q v + ∂L ∂q v vanishes. An example is shown in Fig. 8.1. The postulate that S is extremal for the actual orbit is called Hamilton’s variational principle of least 2 action, and the equations of motion, d dt ∂L ∂ ˙q v − ∂L ∂q v =0, are called Lagrangian equations of the 2nd kind (called “2nd kind” by some authors for historical reasons). They are the so-called Euler-Lagrange equa- tions 3 corresponding to Hamilton’s variational principle. (The more compli- cated Lagrangian equations of the 1st kind additionally consider constraints and will be treated in a later section.) For the special case where L = m 2 ˙x 2 − V (x) , Newton’s equation results. (In fact, the Lagrangian equations of the 2nd kind can also be obtained from the Newtonian equations by a general coordinate transformation.) Thus one of the main virtues of the Lagrangian formalism with respect to the Newtonian equations is that the formalisms are physically equivalent; but mathematically the Lagrangian formalism has the essential 2 In general, the term “least” is not true and should be replaced by “extremal”. 3 Of course any function F(L), and also any additive modification of L by a total derivative df (q(t), ˙q(t),t) dt , would lead to the same equations of motion. 8.4 Cyclic Coordinates; Conservation of Generalized Momenta 49 advantage of invariance against general coordinate transformations,whereas Newton’s equations must be transformed from cartesian coordinates, where the formulation is rather simple, to the coordinates used, where the formu- lation at first sight may look complicated and very special. In any case, the index v, corresponding to virtual, may be omitted, since finally ε ≡ 0. For f ≥ 2 the Lagrangian equations of the 2nd kind are, with i =1, ,f: d dt ∂L ∂ ˙q i = ∂L ∂q i . (8.5) 8.4 Cyclic Coordinates; Conservation of Generalized Momenta; Noether’s Theorem The quantity p i := ∂L ∂ ˙q i is called the generalized momentum corresponding to q i .Oftenp i has the physical dimension of angular momentum, in the case when the corresponding generalized coordinate is an angle. One also calls the generalized coordinate cyclic 4 ,iff ∂L ∂q i =0. As a consequence, from (8.5), the following theorem 5 is obtained. If the generalized coordinate q i is cyclic, then the related generalized mo- mentum p i := ∂L ∂ ˙q i is conserved. As an example we again consider a spherical pendulum (see Sect. 8.2). In this example, the azimuthal angle ϕ is cyclic even if the length l(t)ofthe pendulum depends explicitly on time. The corresponding generalized momen- tum, p ϕ = ml 2 · sin ϑ · ˙ϕ, is the z-component of the angular momentum, p i = L z . In the present case, this is in fact a conserved quantity, as one can also show by elementary argu- ments, i.e., by the vanishing of the torque D z . 4 In general relativity this concept becomes enlarged by the notion of a Killing vector. 5 The name cyclic coordinate belongs to the canonical jargon of many centuries and should not be altered. 50 8 Lagrange Formalism I: Lagrangian and Hamiltonian Compared to the Newtonian equations of motion, the Lagrangian formal- ism thus: a) not only has the decisive advantage of optimum simplicity. For suitable coordinates it is usually quite simple to write down the Lagrangian L of the system; then the equations of motion result almost instantly; b) but also one sees almost immediately, because of the cyclic coordinates mentioned above, which quantities are conserved for the system. For Kepler-type problems, for example, in planar polar coordinates we di- rectly obtain the result that L = M 2 v 2 s + m 2 · ˙r 2 + r 2 ˙ϕ 2 − V (r) . The center-of-mass coordinates and the azimuthal angle ϕ are therefore cyclic; thus one has the total linear momentum and the orbital angular mo- mentum as conserved quantities, and because the Lagrangian does not depend on t, one additionally has energy conservation, as we will show immediately. In fact, these are special cases of the basic Noether Theorem, named after the mathematician Emmy Noether, who was a lecturer at the University of G¨ottingen, Germany, immediately after World War I. We shall formulate the theorem without proof (The formulation is consciously quite sloppy): The three conservation theorems for (i) the total momentum, (ii) the to- tal angular momentum and (iii) the total mechanical energy correspond (i) to the homogeneity (= translational invariance) and (ii) the isotropy (rota- tional invariance) of space and (iii) to the homogeneity with respect to time. More generally, to any continuous n-fold global symmetry of the system there correspond n globally conserved quantities and the corresponding so-called continuity equations, as in theoretical electrodynamics (see Part II). For the special dynamic conserved quantities, such as the above-mentioned Runge-Lenz vector, cyclic coordinates do not exist. The fact that these quan- tities are conserved for the cases considered follows only algebraically using so-called Poisson brackets, which we shall treat below. 8.5 The Hamiltonian To treat the conservation of energy, we must enlarge our context somewhat by introducing the so-called Hamiltonian H(p 1 , ,p f ,q 1 , ,q f ,t) . This function is a generalized and transformed version of the Lagrangian, i.e., the Legendre transform of −L, and as mentioned below, it has many important properties. The Hamiltonian is obtained, as follows: 8.6 The Canonical Equations; Energy Conservation II; Poisson Brackets 51 FirstlywenotethattheLagrangian L depends on the generalized ve- locities ˙q i , the generalized coordinates q i and time t. Secondly we form the function ˜ H(p 1 , ,p f , ˙q 1 , , ˙q f ,q 1 , ,q f ,t):= f i=1 p i ˙q i −L(˙q 1 , , ˙q 1 ,q 1 , ,q f ,t) . Thirdly we assume that one can eliminate the generalized velocities ˙q i by replacing these quantities by functions of p i , q k and t with the help of the equations p i ≡ ∂L ∂ ˙q i . This elimination process is almost always possible in nonrelativistic mechanics; it is a basic prerequisite of the method. After this replacement one finally obtains ˜ H(p 1 , , ˙q 1 (q 1 , ,p 1 , ,t), ,q 1 , ,t) ≡H(p 1 , ,p f ,q 1 , ,q f ,t) . As already mentioned, the final result, i.e., only after the elimination process, is called the Hamiltonian of the system. In a subtle way, the Hamil- tonian is somewhat more general than the Lagrangian, since the variables p 1 , ,p f can be treated as independent and equivalent variables in addition to the variables q 1 , ,q f , whereas in the Lagrangian formalism only the gen- eralized coordinates q i are independent, while the generalized velocities, ˙q i , depend on them 6 . But above all, the Hamiltonian, and not the Lagrangian, will become the important quantity in the standard formulation of Quantum Mechanics (see Part III). 8.6 The Canonical Equations; Energy Conservation II; Poisson Brackets As a result of the transformation from L to H one obtains: dH = f i=1 dp i · ˙q i + p i d˙q i − ∂L ∂ ˙q i d˙q i − ∂L ∂q i dq i − ∂L ∂t dt. Here the second and third terms on the r.h.s. compensate for each other, and from (8.5) the penultimate term can be written as −˙p i dq i . Therefore, since dH can also be written as follows: dH = f i=1 dp i ∂H ∂p i +dq i ∂H ∂q i + ∂H ∂t dt, (8.6) 6 Here we remind ourselves of the natural but somewhat arbitrary definition δ ˙q := d(δq) dt in the derivation of the principle of least action. 52 8 Lagrange Formalism I: Lagrangian and Hamiltonian one obtains by comparison of coefficients firstly the remarkable so-called canonical equations (here not only the signs should be noted): ˙q i =+ ∂H ∂p i , ˙p i = − ∂H ∂q i , ∂H ∂t = − ∂L ∂t . (8.7) Secondly, the total derivative is dH dt = f i=1 ˙p i ∂H ∂p i +˙q i ∂H ∂q i + ∂H ∂t , and for a general function F (p 1 (t), ,p f (t),q 1 (t), ,q f (t),t): dF dt = f i=1 ˙p i ∂F ∂p i +˙q i ∂F ∂q i + ∂F ∂t . Insertion of the canonical equations reduces the previous results to: dH dt = ∂H ∂t = − ∂L ∂t and dF dt = f i=1 ∂H ∂p i ∂F ∂q i − ∂H ∂q i ∂F ∂p i + ∂F ∂t , respectively, where we should remember that generally the total and partial time derivatives are different! In both cases the energy theorem (actually the theorem of H conservation) follows: If L (or H) does not depend explicitly on time (e.g., ∂H ∂t ≡ 0), then H is conserved during the motion (i.e., dH dt ≡ 0). Usually, but not always, H equals the total mechanical energy. Thus some caution is in order: H is not always identical to the mechanical energy, and the partial and total time derivatives are also not identical; but if L = T−V, then (if skleronomous generalized coordinates are used) we automatically ob- tain H≡T+ V , as one can easily derive by a straightforward calculation with the above def- initions. Here, in the first case, i.e., with L, one should write T = mv 2 2 , whereas in the second case, i.e., with H, one should write T = p 2 2m . . Schematic diagram on differential scattering cross-sections. A particle enters the diagram from the left on a path parallel to the x-axis at a perpendicular distance b (the so-called “impact parameter”;. Lagrangian and Hamiltonian also included (see below). The fact that the Lagrangian contains the differ- ence,andnot the sum of the kinetic and potential energies, has a relativistic origin, as. center-of-mass coordinates and the azimuthal angle ϕ are therefore cyclic; thus one has the total linear momentum and the orbital angular mo- mentum as conserved quantities, and because the Lagrangian