8.6 The Canonical Equations; Energy Conservation II; Poisson Brackets 53 In the second case the sum of the braced terms yields a definition for the so-called Poisson brackets: [H,F] P := f i=1 ∂H ∂p i ∂F ∂q i − ∂H ∂q i ∂F ∂p i . (8.8) For the three components of the above-mentioned Runge-Lenz vector F := (L e ) j , with j = x, y, z , it can be shown that with the particular (but most important) Hamilto- nian for the Kepler problem (i.e., with -A/r potentials) the Poisson brackets [H,F] P vanish, while the Poisson brackets of F with the other conserved quantities (total momentum and total angular momentum) do not vanish. This means that the Runge-Lenz vector is not only an additional conserved quantity for Kepler potentials, but is actually independent of the usual con- served quantities. The equations of motion related to the names of Newton, Lagrange and Hamilton (i.e., the canonical equations in the last case) are essentially all equivalent, but ordered in ascending degree of flexibility, although the full power of the respective formalisms has not yet been (and will not be) ex- ploited. We only mention here that there is a large class of transformations, the so-called canonical transformations, leading from the old (generalized) coordinates and momenta to new quantities, such that Hamilton’s formal- ism is preserved, although generally with a new Hamiltonian. In quantum mechanics (see Part III) these transformations correspond to the important class of unitary operations. Additionally we mention another relation to quantum mechanics. The Poisson bracket [A, B] P of two measurable quantities A and B is intimately related to the so-called commutator of the quantum mechanical operators ˆ A and ˆ B, i.e., [A, B] P → i ˆ A ˆ B − ˆ B ˆ A . 9 Relativity I: The Principle of Maximal Proper Time (Eigenzeit) Obviously one could ask at this point whether Hamilton’s principle of least action 1 is related to similar variational principles in other fields of theoretical physics, e.g., to Fermat’s principle of the shortest optical path Δl opt . 2 The answer to this question is of course affirmative. However, we shall abstain from making the relations explicit, except for the particular interpre- tation and relativistic generalization of Hamilton’s principle of least-action by Einstein’s principle of maximal proper time (or maximal eigenzeit). 3 According to Einstein’s principle, the motion of small particles under the influence of the effects of special and general relativity takes place in a curved four-dimensional “space-time” (three space dimensions plus one time dimension) and has the particular property that the proper time (eigenzeit) of the particle between the starting point and the end point under the influence of gravitational forces should be a maximum; the eigenzeit of the particle is simply the time measured by a conventional clock co-moving with the particle. As shown below, Einstein’s postulate corresponds not only to Hamilton’s principle, but it also yields the correct formula for the Lagrange function L, i.e., L = T−V, of course always in the limit of small velocities, i.e., for v 2 c 2 ,wherec is the velocity of light in vacuo. 1 The name least action is erroneous (as already mentioned), since in fact only an extremum of the action functional is postulated, and only in exceptional cases (e.g., for straight-line motion without any force) is this a minimum. 2 The differential dl opt is the product of the differential dl geom. of the geometrical path multiplied by the refractive index. 3 As mentioned, one should be cautious with the terms “least”, “shortest” or “maximal”. In fact for gravitational forces in the non-relativistic limit the action, W := R t B t A dtL, can be identified with −m 0 c 2 R t B t A dτ (see below) and one simply obtains complementary extrema for the proper time τ and for the Hamiltonian action W . Actually, however, in general relativity, see [7] and [8], the equations of motion of a particle with finite m 0 under the influence of gravitation are time- like geodesics in a curved Minkowski manifold, i.e., with maximal (sic) proper time. However, the distinction between “timelike” and “spacelike” does not make sense in Newtonian mechanics, where formally c →∞(see below). 56 9 Relativity I: The Principle of Maximal Proper Time (Eigenzeit) In the next section we shall firstly define some necessary concepts, such as the Lorentz transformation. 9.1 Galilean versus Lorentz Transformations The perception of space and time underlying Newtonian mechanics corre- sponds to the so-called Galilean transformation: All inertial frames of Newtonian mechanics are equivalent, i.e., Newton’s equations have the same form in these systems. The transition between two different inertial systems is performed via a Galilean transformation: If the origin of a second inertial frame, i.e., primed system, moves with a velocity v in the x direction, then x = x + vt and t = t , and of course y = y ,z = z . For the Galilean transformation, space and time are thus decoupled.New- ton’s equations of motion have the same form in both the primed and un- primed inertial frames, where for the forces we have of course F x ≡ F x . The above relation implies a simple addition of velocities, i.e., if motion occurs in the unprimed system with velocity u, and in the primed system with velocity u (in both cases in the x direction) then u = u + v. According to Newtonian mechanics, an event which took place in the unprimed coordinate system with exactly the vacuum velocity of light u = c would thus have a velocity u = u −v (= c) in the primed system. Thus Maxwell’s theory of electrodynamics (see Part II), which describes the propagation of light with velocity c = 1 √ ε 0 μ 0 (see below), is not invariant under a Galilean transformation. On the suggestion of Maxwell himself, the hypothesis of “additivity of ve- locities” was tested for light with great precision, firstly by Michelson (1881) and then by Michelson and Morley (1887), with negative result. They found that u = c ⇔ u = c, implying that something fundamental was wrong. 9.1 Galilean versus Lorentz Transformations 57 Fig. 9.1. The Lorentz transformation. The Lorentz transformation 8 < : x = x + v c ct r 1− v 2 c 2 ,ct= ct + v c x r 1− v 2 c 2 9 = ; transforms the square {0 ≤ x ≤ 1, 0 ≤ ct ≤ 1} into a rhombus with unchanged diagonal (x = ct ↔ x = ct ). The angle of inclination α between the primed and non-primed axes is given by α =tanh v c As mentioned previously, it had already been established before Ein- stein’s time that the basic equations of electrodynamics (Maxwell’s equa- tions), which inter alia describe the propagation of light, were simply not invariant under a Galilean transformation, in contrast to Newtonian mechan- ics. Thus, it was concluded that with respect to electrodynamics the inertial systems were not all equivalent, i.e., there was a particular frame, the aether, in which Maxwell’s equations had their usual form, whereas in other inertial frames they would be different. However, Maxwellian electrodynamics can be shown to be invariant with respect to a so-called Lorentz transformation, 4 which transforms space and time coordinates in a similar way, as follows: x = x + v c ct 1 − v 2 c 2 ; ct = ct + v c x 1 − v 2 c 2 . (Additionally one has of course y = y and z = z .) These equations have been specifically written in such a way that the above-mentioned similarity with respect to space and time becomes obvious. (We should keep these equations in mind in this form.) Additionally, Fig. 9.1 may be useful. 4 Hendryk A. Lorentz, Leiden, NL; 1904 58 9 Relativity I: The Principle of Maximal Proper Time (Eigenzeit) Furthermore a nontrivial velocity transformation follows from the Lorentz transformations. Since u := dx dt (not:= dx dt ), one obtains u = u + v 1+ u v c 2 , which implies that u = c ⇔ u = c, independently of v. Before Einstein, these relations were only considered to be strange mathematical properties of the aether, i.e., one believed erroneously that the Newtonian and Galilean considerations on space and time needed no modification. Einstein’s special theory of relativity (1905) 5 then changed our perception radically. It transpired that Newton’s theory, not that of Maxwell, had to be modified and refined. The modifications involved our basic perception of space and time (but fortunately, since v 2 c 2 , Newtonian theory still remains valid in most practical cases of everyday life). However, an important new paradigm now entered science: a theory could be true only under certain quantitative constraints and could be refined or modified in other cases. The main implications of the new theory may be restated, as follows: For all physical events all inertial frames are equivalent (i.e., there is no need for a special inertial frame called the aether). However, the transforma- tion between different inertial frames must be made via a Lorentz transforma- tion not a Galilean transformation. As a consequence, as already mentioned 6 , with these new insights into space and time Newtonian mechanics, in con- trast to Maxwell’s theory of electromagnetism, had to be modified and refined, but fortunately only for very high velocities when the condition v 2 c 2 is violated. 9.2 Minkowski Four-vectors and Their Pseudo-lengths; Proper Time Two years after Einstein’s epochal work of 1905, the mathematician Hermann Minkowski introduced the notion of a so-called four-vector ˜v := (v 1 ,v 2 ,v 3 , i · v 4 ) . Here, all four variables v α , α =1, ,4, are real quantities (i.e., the fourth component of ˜v is imaginary 7 ). 5 Einstein’s special theory of relativity was published in 1905 under the title “ Zur Elektrodynamik bewegter K¨orper ” in the journal Annalen der Physik (see [5] or perform an internet search). 6 Sometimes an important statement may be repeated! 7 Many authors avoid the introduction of imaginary quantities, by using instead of ˜v the equivalent all-real definition ˜v := ` v 0 ,v 1 ,v 2 ,v 3 ´ ,withv 0 := v 4 ;how- 9.2 Minkowski Four-vectors and Their Pseudo-lengths; Proper Time 59 Moreover, these real variables v 1 , ,v 4 are assumed to transform by a Lorentz transformation combined with rotation in three-space, just as the variables x, y, z and ct. The union of such Minkowski four-vectors is the Minkowski space M 4 . A typical member is the four-differential d˜x := (dx, dy, dz,i ·cdt) . It is easy to show that under Lorentz transformations the so-called pseudo- length ˜v 2 := v 2 1 + v 2 2 + v 2 3 − v 2 4 of a Minkowski four-vector is invariant (e.g., the invariance of the speed of light in a Lorentz transformation results simply from the fact that for a Lorentz transformation one has: x 2 = c 2 t 2 ⇔ x 2 = c 2 t 2 ). In addition, the so-called pseudo-scalar product of two Minkowski four-vectors, ˜v · ˜w := v 1 w 1 + v 2 w 2 + v 3 w 3 − v 4 w 4 , is also invariant for all Lorentz transformations, which means that Lorentz transformations play the role of pseudo-rotations in Minkowski space. Among the invariants thus obtained is the so-called proper time (eigen- zeit) dτ, which corresponds to the pseudo-length of the above-mentioned Minkowski vector dτ := d˜x 2 −c 2 =dt · 1 − v 2 c 2 . Here we have assumed dx = v x dt etc. and consider only events with v 2 <c 2 . Atimeintervaldτ measured with a co-moving clock, the proper time,is thus always shorter than the time interval dt measured in any other frame. This means for example that a co-moving clock transported in an aeroplane around the earth, ticks more slowly than an earth-based clock remaining at the airport, where the round-trip around the earth start and ends. This is a measurable effect, although very small! (More drastic effects result from the cascades of μ particles in cosmic radiation. The numerous decay products of these cascades, which have their origin at a height of ∼ 30 km above the surface of the earth, have only a proper lifetime of Δτ ≈ 10 −6 s. Nevertheless, showers of these particles reach the surface of the earth, even though with v ≈ c in 10 −6 s they should only cover a distance of 300 m before decaying. The solution for this apparent discrepancy is the gross difference between dτ and dt for velocities approaching the speed of light 8 .) ever, avoiding the imaginary unit i, one must pay some kind of penalty,being forced instead to distinguish between covariant and contravariant four-vector components, which is not necessary with the “imaginary” definition. 8 Here one could introduce the terms time dilation, i.e., from dτ → dt,andlength contraction, e.g., from 30 km to 300 m. 60 9 Relativity I: The Principle of Maximal Proper Time (Eigenzeit) In a curved space the ticking speed of clocks is not only influenced by v, but also by gravitating bodies, as described by Einstein’s general theory of relativity (which goes far beyond the scope of our text). Here, we only note that for sufficiently weak gravitation potentials V(r) one has the relation dτ =dt· 1 − v 2 c 2 + 2V m 0 c 2 . The quantity m 0 is the rest mass of the considered particle, as already men- tioned. Thus, Einstein’s principle of the maximal proper time implies that af- ter multiplication with (−m 0 c 2 )theactual path yields an extremum of the action 9 W := −m 0 c 2 t 2 t 1 dt 1 − v 2 c 2 + 2V m 0 c 2 . A Taylor expansion 10 of this expression yields (in the lowest nontrivial order w.r.t. v 2 and V) the usual Hamilton principle of least action. In addition one should note that the first three spatial components and the fourth timelike component of a Minkowski four-vector (because of the factor i 2 = −1) enter the final result with different signs. Ultimately, this different behavior of space-like and time-like components of a Minkowski four-vector (i.e., the square of the imaginary unit i appearing with the time- like component) is the genuine reason why in the formula L = T−V the kinetic energy T and potential energy V enter with different signs, i.e., we are dealing here with an intrinsically relativistic phenomenon. 9.3 The Lorentz Force and its Lagrangian If, additionally, an electric field E and a magnetic induction B are present, then the force F q exerted by these fields on a particle with an electric charge q is given by F q = q ·(E + v ×B) , where the last term describes the Lorentz force. The magnetic induction B can be calulated from a vector potential A, where B =curlA , 9 Dimensional analysis: action := energy × time. 10 ‘Sufficiently weak’, see above, means that a Taylor expansion w.r.t. the lowest nontrivial order makes sense. 9.4 The Hamiltonian for the Lorentz Force;Kinetic versus Canonical Momentum 61 whereas an electric field E can be mainly calculated from the scalar poten- tial Φ,since E = −gradΦ − ∂A ∂t . The main content of the relativistic invariance of Maxwell’s theory is, as we shall only mention, that Φ and A can be combined into a Minkowski four- vector ˜ A := A x ,A y ,A z , i Φ c . Furthermore, the quadruple ˜u := 1 1 − v 2 c 2 (v x ,v y ,v z , ic) (≡ d˜x dτ , i.e., the Minkowski vector d˜x divided by the Minkowski scalar dτ) constitutes a Minkowki four-vector, so that it is almost obvious to insert the invariant pseudo-scalar product q ˜ A · ˜u into the Lagrangian, i.e., q ˜ A · ˜u = q 1 − v 2 c 2 ·(A ·v −Φ) . In fact, for v 2 c 2 this expression yields an obvious addition to the La- grangian. Thus, by the postulate of relativistic invariance, since without the Lorentz force we would have L ∼ = T−V−qΦ , we obtain by inclusion of the Lorentz force: L≡T−V− q 1 − v 2 c 2 (Φ −v · A ) , (9.1) and in the nonrelativistic approximation we can finally replace the compli- cated factor ∝ q simply by q itself to obtain a Lagrangian including the Lorentz force. 9.4 The Hamiltonian for the Lorentz Force; Kinetic versus Canonical Momentum; Gauge Transformations One can also evaluate the Hamiltonian from the Lagrangian of the Lorentz force. The details are not quite trivial, but straightforward. The following 62 9 Relativity I: The Principle of Maximal Proper Time (Eigenzeit) result is obtained: H = (p −q · A(r,t)) 2 2m + V (r)+q · Φ. (9.2) Here, by means of a so-called gauge transformation, not only A, but simultaneously also p, can be transformed as follows: q · A → q · A +gradf (r,t) , p → p +gradf(r,t) , and q · Φ → q · Φ −∂ t f(r,t) . The gauge function f(r,t) is arbitrary. What should be kept in mind here is the so-called minimal substitution p → p −q ·A , which also plays a part in quantum mechanics. From the first series of canonical equations, ˙x = ∂H ∂p x etc., it follows that mv ≡ p − qA . This quantity is called the kinetic momentum, in contrast to the canonical momentum p, which, as mentioned above, must be gauged too, if one gauges the vector potential A. In contrast, the kinetic momentum mv is directly measurable and gauge-invariant. After lengthy and subtle calculations (→ a typical exercise), the second series of canonical equations, ˙p x = − ∂H ∂x etc., yields the equation corresponding to the Lorentz force, m ˙ v = q ·(E + v × B) . 10 Coupled Small Oscillations 10.1 Definitions; Normal Frequencies (Eigenfrequencies) and Normal Modes Let our system be described by a Lagrangian L = 3N α=1 m α 2 ˙x α − V (x 1 , ,x 3N ) , and let x (0) := x (0) 1 , ,x (0) 3N be a stable second-order equilibrium configuration, i.e., V (x (0) ) corresponds to a local minimum of 2nd order, the forces F α (x 0 ):=− ∂V ∂x α |x (0) vanish and the quadratic form Q := 3N i,k=1 ∂ 2 V ∂x i ∂x k Δx i Δx k is “positive definite”, Q>0, as long as Δx := x −x (0) =0, except for the six cases where the Δx i correspond to a homogeneous transla- tion or rotation of the system. In these exceptional cases the above-mentioned quadratic form should yield a vanishing result. The (3N) × (3N)-matrix V α,β := ∂ 2 V ∂x i ∂x k is therefore not only symmetric (V α,β ≡ V β,α ), such that it can be diagonal- ized by a rotation in R 3N , with real eigenvalues, but is also “positive”, i.e., . time,is thus always shorter than the time interval dt measured in any other frame. This means for example that a co-moving clock transported in an aeroplane around the earth, ticks more slowly than an earth-based. can be mainly calculated from the scalar poten- tial Φ,since E = −gradΦ − A ∂t . The main content of the relativistic invariance of Maxwell’s theory is, as we shall only mention, that Φ and A. important class of unitary operations. Additionally we mention another relation to quantum mechanics. The Poisson bracket [A, B] P of two measurable quantities A and B is intimately related to