A Physical Constants1

Constants in Mechanics

−11 3 −1 −2 Gravitational constant G =6.672 59(85) × 10 m kg s (1996) −11 3 −1 −2 =6.673(10) × 10 m kg s (2002) −2 Gravitational acceleration g =9.806 65 m s 30 Solar mass M =1.988 92(25) × 10 kg 8 Solar equatorial radius R =6.96 × 10 m 24 Earth mass M⊕ =5.973 70(76) × 10 kg 6 Earth equatorial radius R⊕ =6.378 140 × 10 m 22 Moon mass M =7.36 () × 10 kg 6 Moon radius R =1.738 × 10 m 9 Distance of Earth from Sun Rmax =0.152 1 × 10 m 9 Rmin =0.147 1 × 10 m 9 Rmean =0.149 6 × 10 m 8 Distance of Moon from Earth Rmean =0.380 × 10 m 7 Period of Earth w.r.t. Sun T⊕ = 365.25 d = 3.16 × 10 s 6 Period of Moon w.r.t. Earth T =27.3d=2.36 × 10 s

1 From Particle Data Group, American Institute of Physics 2002 http://pdg.lbl.gov/2002/contents http://physics.nist.gov/constants 326 A Physical Constants

Constants in Electromagnetism

def Velocity of light c =. 299, 792, 458 ms−1 def. −7 −2 Vacuum permeability μ0 =4π × 10 NA =12.566 370 614 ...× 10−7 NA−2 1 × −12 −1 Vacuum dielectric constant ε0 = 2 =8.854 187 817 ... 10 Fm μ0c Elementary charge e =1.602 177 33 (49) × 10−19 C e2 =1.439 × 10−9 eV m = 2.305 × 10−28 Jm 4πε0

Constants in Thermodynamics

−23 −1 Boltzmann constant kB =1.380 650 3 (24) × 10 JK =8.617 342 (15) × 105 eV K−1 23 −1 Avogadro number NA =6.022 136 7(36) × 10 mole Derived quantities : Gas constant R = NLkB Particle number N,molenumbernN= NLn

Constants in Quantum Mechanics

Planck’s constant h =6.626 068 76 (52) × 10−34 Js h = =1.054 571 596 (82) × 10−34 Js 2π =6.582 118 89 (26) × 10−16 eV s 2 4πε0 −10 Bohr radius a∞ = =0.529 177 208 3 (19) × 10 m me2 # $2 m e2 2 1 e2 Rydberg energy E0 = = 2 = 2 4πε0 2ma0 2 4πε0a0 1 2 2 = 2 mc α =13.605 691 72 (53) eV A Physical Constants 327

−31 Electron rest mass me =9.109 381 88 (72) × 10 kg =0.510 998 902 (21) MeVc−2 −28 Myon rest mass mμ =1.883 566() × 10 kg −27 Proton rest mass mp =1.672 621 58 (13) × 10 kg = 938.271 998(38) MeVc−2 −27 Neutron rest mass mn =1.674 954 3 () × 10 kg 1 g −27 Atomic mass unit (amu) mu = =1.660 538 73 (13) × 10 kg NA mole = 931.494 013(37) MeVc−2 e[c] × −5 −1 Bohr magneton μB = 2mc =5.788 381 749 (43) 10 eV T 2 Fine-structure constant α = e =1/137.035 999 76 (50) 4πε0c

Conversion Factors

1eV= 1.602 176 462 (63) × 10−19 J B Scalars, Vectors, Tensors

In this appendix some relations are collected, which could be useful in non- relativistic mechanics.

B.1 Definitions and Simple Rules

B.1.1 Definitions

(Mathematical) definition: (i) Tensors (of a given rank) are elements of a vector space (i.e., they obey certain operation rules.1 (ii) Under transformation, a tensor of nth rank (with n indices) has, depending on the type of the index, the behavior of the position vectors. (Sloppy, physical) definition: A vector is a quantity with modulus and direc- tion. Comments: • A coordinate transformation (from unprimed to primed coordinates) of a vector a with the (cartesian) components ai is given by  j ai = Ui aj. (B.1) j

Correspondingly, the behavior of a tensor nthr rank under a transforma- tion is given by  j1 ··· jn Ti1...in = Ui1 Uin Tj1...jn . (B.2) j1...jn

1 See Sect. B.2.1 for vectors. 330 B Scalars, Vectors, Tensors

• Scalars and vectors are tensors of zeroth and first rank, respectively. • A scalar is a number. But not each number is a scalar: For example the x component of a vector is a number, but no scalar, because it changes with a coordinate transformation. • Tensors of second rank can be represented by matrices. But not each ma- trix is a tensor: For example the transformation matrix U of (B.1) with the j elements Ui is not a tensor, since it does not refer to a given coordinate system, but gives the relation between two different systems.

B.1.2BehaviorUnderInversion

Under inversion a tensor of even-numbered rank (in particular thus a scalar) transforms into itself; a tensor of odd-numbered rank (thus in particular a vector) transforms into its negative. Pseudo tensors are likewise elements of vector spaces, have but the wrong behavior under transformation: Under inversion a pseudo tensor of even- numbered rank (in particular a pseudo scalar) transforms into its negative; a pseudo tensor of odd-numbered rank (in particular a pseudo vector) trans- forms into itself. Notation: In order to discriminate, a vector is denoted alternatively as a polar vector and a pseudo vector also as an axial vector. Comments: • The cross product of two polar vectors (e.g., in the case of the angular momentum) is an axial vector (pseudo vector).2 • The tensor product (dyadic product) of two polar vectors is a tensor.3 • The scalar product of a vector with a pseudo vector (i.e. the triple scalar product of three vectors) is a pseudo scalar.4

B.2 Vectors

B.2.1 Rules for Vectors

One has the following axioms for vectors as elements of a vector space: A sum of vectors a ∈ Rd and b ∈ Rd is defined, which results in a vector c ∈ Rd, a + b = c ⇔ ai + bi = ci. A multiplication is defined of a vector a ∈ Rd withascalarα ∈ K;thespace is linear: d = αa ⇔ di = αai. 2 See Appendix B.2.3. 3 See Appendix B.2.2. 4 See Appendix B.2.4. B.2 Vectors 331

There is an inner product (scalar product), of two vectors resulting in a scalar s,

a · b = s.

B.2.2 Dyadic Product (Tensor Product) of Vectors

The dyadic product (tensor product)

ab≡ a ⊗ b

of two vectors a, b is a tensor T = ab= a ⊗ b with     c · T = c · a ⊗ b = c · a b     T · d = a ⊗ b · d = a b · d .

Comments: • The behavior of the tensor T under transformation is apparent. • In contrast to the scalar product a · b = b · a the tensor product is not commutative, ab= ba. • The symbol ⊗ is rarely used and is used here only for distinction from the dot product (Fig. B.1). • It is thus of importance, for the distinction from the tensor product to notate the product symbol (dot) in the case of the scalar product!! • The multiplication rule “row times column” for matrices holds likewise for the scalar product of vectors. For the scalar product this is a sum of products (of two terms), while for the tensor product each tensor element consists of just one product.

) *)* . . ... · . . ⊗ ...

Fig. B.1. The scalar product (left) and the tensor product (right) of two vectors with the rule “row times column”

B.2.3 Vector Product (Cross Product) of Vectors

The vector product (cross product) is restricted to vectors in R3;by

c = a × b 332 B Scalars, Vectors, Tensors

Fig. B.2. The outer product of two vectors

a third vector c is assigned to two vectors a, b with

a · c =0=b · c & & & &2& &2 & &2 &a × b& = &a · a& &b · b& − &a · b& a, b, and c = a × b forming a right-handed system (Fig. B.2). The resulting vector c is perpendicular to the vectors a and b; the modulus is equal to the area spanned by the vectors a and b, |a × b| = ab sin α, where α is the angle (in the mathematically positive sense from a to b)be- tween the vectors a and b. The cross product has the following properties: a × b = −b × a a × a =0 a × b =0 ⇒ a =0 or b =0 or a = αb.

B.2.4 Triple Scalar Product of Vectors Like the external product thus the triple scalar product is restricted to vectors 3 in R ;with     a · b × c = a × b · c according to sections B.2.1 and B.2.2 a (pseudo) scalar is assigned to three (polar) vectors. The triple product is the volume of the parallelepiped spanned by the three vectors and positive if the three vectors form a right-handed system.

B.2.5 Multiple Products of Vectors The following relations for vectors in R3 are of use occasionally:       a × b × c = a · c b − a · b c (B.3)           a × b · c × d = a · c b · d − a · d b · c . (B.4) Comment:     a × b × c = a × b × c. C Rectangular Coordinate Systems

C.1 Definitions

3 Let a point in R be represented by the (generally curvilinear) coordinates ξi (i =1, 2, 3).

Definition: The coordinate sheet fi is the sheet fi(ξi)=0(i =1, 2, 3).

Definition: The coordinate line si is the intersecting line of the sheets fj(ξj )=0andfk(ξk)=0(withi, j, k cyclic).

Definition: The unit vector ei is the vector tangential to the coordinate line si in the direction of increasing value of ξi with ei · ei =1.

C.2 Cartesian Coordinates

The position vector is r = x ex + y ey + z ez. The line element is (Fig. C.1)

ds =dx ex +dy ey +dz ez.

z dz

dy dx

y

Fig. C.1. The volume element in cartesian co- x ordinates 334 C Rectangular Coordinate Systems

The surface element is

da =dy dz ex +dz dx ey +dx dy ez. The volume element is (Fig. C.1) d3r =dx dy dz.

C.3 Spherical Polar Coordinates (Spherical Coordinates)

The transformation from cartesian coordinates to spherical polar coordinates is given by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ er sin ϑ cos ϕ sin ϑ sin ϕ cos ϑ ex ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ eϑ = cos ϑ cos ϕ cos ϑ sin ϕ − sin ϑ ey eϕ − sin ϕ cos ϕ 0 ez

Fig. C.2. Spherical polar coordinates (“spherical coordinates”)

The position vector is (Fig. C.2)

r = r er. The line element is

ds =dr er + r dϑ eϑ + r sin ϑ dϕ eϕ. The surface element is 2 da = r sin ϑ dϑ dϕ er + r dr sin ϑ dϕ eϑ + r dr dϑ eϕ. The volume element is d3r = r2 dr sin ϑ dϑ dϕ = −r2 dr d(cos ϑ)dϕ. C.5 Plane Polar Coordinates 335 C.4 Cylindrical Coordinates

The coordinate transformation is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ e cos ϕ sin ϕ 0 ex ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ eϕ = − sin ϕ cos ϕ 0 ey ez 001ez

Fig. C.3. Cylindrical coordinates

The position vector is (Fig. C.3)

r =  e + z ez.

The line element is

ds =d e +  dϕeϕ +dz ez.

The surface element is

da =  dϕ dz e +d dz eϕ +  d dϕ ez.

The volume element is d3r =  d dϕ dz.

C.5 Plane Polar Coordinates

The plane polar coordinates one obtains from the cylindrical coordinates with z = 0 or from the spherical polar coordinates with ϑ = π/2, # $ # $# $ e cos ϕ sin ϕ e r = x . eϕ − sin ϕ cos ϕ ey 336 C Rectangular Coordinate Systems C.6 Inverse Relations

The (rarely needed) inverse of the relations between the unit vectors in short- hand notation is given by

e = T · e ⇔ e = TT · e with the transpose TT = T−1 of the matrix T.

Problems

C.1. Vectors and Coordinate Systems. (a) Write down examples for physical quantities which can be described as vectors. (b) Write down the decomposition of a vector in Cartesian, plane, and spher- ical polar coordinates. What is the difference between the Cartesian and the polar coordinates? What is the relation between spherical and plane polar coordinates? C.2. The Line Element in Spherical Coordinates. For a motion in R3 one can employ Cartesian coordinates r(t)={x(t),y(t),z(t)} or spherical coordinates {r(t),θ(t),ϕ(t)}. Determine the infinitesimal line element (ds)2 =(dx)2 +(dy)2 +(dz)2 in spherical coordinates, and write down the squared modulus of the velocity in these coordinates. C.3. Vector Functions, Trajectory. (a) What is a vector function? Write down examples. (b) The trajectory of a point mass is described by the time-dependent position vector r(t). Write down the decomposition of r(t) in Cartesian and polar coordinates. How are these decompositions different from each other with respect to the time dependence? (c) The set of all points through which r(t) runs for all times t one denotes as space curve or trajectory of the point mass. For the description of the space curve one uses a coordinate system (local trihedron), which is defined by following set of orthogonal unit vectors: dr(s) tangent unit vector: =: e (s) ds t where s is the arc length (along the space curve), & & & & det(s) &det(s)& principal normal: & & =: en(s) ds ds Problems 337

and binormal: eb := et × en & & & & dr(t) &dr(t)& Prove that et(s) can also be written as & &. dt dt

From the definition of en(s) follows: & & & & det &det & = κ en, where κ = & & , ds ds is denoted as the curvature. (d) Determine the unit vectors of the local trihedron for a circular orbit. Em- ploy the parameter representation r(ϕ)=(r cos ϕ, r sin ϕ, 0). Which meaning has κ in this case, and which relation exists to the plane polar coordinates? (e) Prove that de b ∼ e . ds n de One writes b = −τ e and denotes τ as torsion of the space curve. ds n

(f) Determine et, en and eb for the spiral line (helix) h r(ϕ)=(r cos ϕ, r sin ϕ, ϕ), 2π and write down κ and τ. Hint: First express the variable ϕ by the arc length. What does one get for the case h =0? Write down in particular κ and τ for this case. (g) Compare the radius of curvature 1/κ of the helix with that of the circular orbit. D Nabla (Del) Operator and Laplace Operator

Let f be a scalar field and g a vector field. Equivalent notations: ∇f ≡ grad f gradient of f ∇·g ≡ div g divergence of g ∇×g ≡ rot g rotation of g

D.1 Representations of the Nabla and Laplace Operators

Coordinate-Free Representation

 1 ∇◦ = lim da ◦ Δ →0 V ΔV ∂ΔV 1 (n ×∇) ◦ = lim ds ◦, Δ →0 F ΔF ∂ΔF where n is the unit vector normal to the surface element ΔF ; ∂ΔV and ∂ΔF are the surface of the volume ΔV and the boundary of the sheet ΔF , respectively, with the abbreviation

◦ = ·v(r)or◦ = ×v(r)or◦ = f(r).

Representation in Cartesian Coordinates

∂ ∂ ∂ ∇ = e + e + e (D.1) x ∂x y ∂y z ∂z ∂2 ∂2 ∂2 ∇·∇ = + + . (D.2) ∂x2 ∂y2 ∂z2 340 D Nabla (Del) Operator and Laplace Operator

Representation in Spherical Polar Coordinates

∂ 1 ∂ 1 ∂ ∇f = er f + eϑ f + eϕ f (D.3) ∂r r ∂ϑ r sin ϑ ∂ϕ ∇· 1 ∂ 2 ∂ ∂ g = 2 sin ϑ (r gr)+r (sin ϑgϑ)+r gϕ (D.4) r sin ϑ & ∂r ∂ϑ& ∂ϕ & & & er r eϑ r sin ϕ eϕ & 1 & & ∇× g = & ∂ ∂ ∂ & (D.5) r2 sin ϑ & ∂r ∂ϑ ∂ϕ & gr rgϑ r sin ϑgϕ 1 ∂ ∂ 1 ∂2 2 ∂ 1 ∇·∇ = r2 + Λ(ϑ, ϕ)= + + Λ(ϑ, ϕ)(D.6) r2 ∂r ∂r r2 ∂r2 r2 ∂r r2 1 ∂ ∂ 1 ∂2 Λ(ϑ, ϕ)= sin ϑ + . (D.7) sin ϑ ∂ϑ ∂ϑ sin2 ϑ ∂ϕ2

Representation in Cylindrical Coordinates

∂ 1 ∂ ∂ ∇f = e f + e f + e f (D.8)  ∂ ϕ  ∂ϕ z ∂z 1 ∂ 1 ∂ ∂ ∇· g = (g)+ gϕ + gz (D.9)  ∂&  ∂ϕ& ∂z & & & e  eϕ ez & 1 & & ∇× g = & ∂ ∂ ∂ & (D.10)  & ∂ ∂ϕ ∂z & g gϕ gz 1 ∂ ∂ 1 ∂2 ∂2 ∇·∇ =  + + . (D.11)  ∂ ∂ 2 ∂ϕ2 ∂z2

D.2 Standard Relations

r (gradient) ∇r = e = (D.12) r r df (gradient) ∇f(r)= e (D.13) dr r (a =const) ∇(a · r)=a (D.14) (divergence) ∇· r = 3 (D.15) (a =const) ∇·(a × r) = 0 (D.16) (rotation) ∇× r = 0 (D.17) (a =const) ∇×(a × r)=2a (D.18) D.3 Rules 341

(vector gradient) ∇⊗r = 1 (∇α rβ = δαβ) (D.19) ∂ (formal notation) ∇ = (D.20) ∂r ∞ 1 (Taylor series) f(r + a)=ea·∇f(r):= (a ·∇)nf(r) (D.21) n! n=0 1 Δ = −4πδ(r − r0) (D.22) |r − r0|

Comment: According to the last equation

− 1 1 4π |r − r0|

is the Green function to the Laplace operator.

D.3 Rules

One uses the three-dimensional generalization of the rules as known from one dimension. The formal notation

∇≡ ∂ ∂r is of use (which is not defined via the limit of a difference quotient). One must pay attention to the order of the quantities, in particular in the case of the cross product.

Chain and Product Rules

df ∇[f(g(r))] = ∇g(r) (D.23) dg ∇ (fg)=g ∇f + f ∇g (D.24) ∇·(fu)=f ∇·u + u ·∇f (D.25) ∇ (u · v)=u ·∇v + v ·∇u +u × (∇×v)+v × (∇×u) (D.26) ∇·(v × w)=w ·∇×v − v ·∇×w (D.27) ∇×sv = s ∇×v − v ×∇s (D.28) ∇×(v × w)=v ∇·w + w ·∇v − w ∇·v − v ·∇w (D.29) 342 D Nabla (Del) Operator and Laplace Operator

Fig. D.1. Theorem of Gauß: The field E with its source ∇·E (left) outside and (right) inside of the volume V enclosed by the area ∂V ;thepointswhereE cuts the surface ∂V are marked by points; the source of the field (right)ismarkedbya small circle

Multiple Differentiation

∇·∇f =Δf (D.30) ∇×∇s = 0 (D.31) ∇·∇×v = 0 (D.32) ∇×(∇×v)=∇∇·v −∇·∇v (D.33) Δ(fg)=fΔg + gΔf +2(∇f) · (∇g). (D.34)

D.4 Integral Theorems

Analogously to the integration of differentials of functions on R1 one can integrate multi-dimensional integrals once: Abbreviation:

◦ = ·v(r)or◦ = ×v(r)or◦ = f(r).

D.4.1 The Theorem of Gauß

Let f(r), ∇f(r)andv(r), ∇·v(r), respectively be continuous in a volume V . Then one has the theorem of Gauß   d3r ∇◦= da ◦ . (D.35) V ∂V

Comment: The most frequent examples are of the form   d3r ∇·E = da · E, V ∂V with the electric field E(r), see Figs. D.1 and D.2. D.4 Integral Theorems 343

Fig. D.2. Theorem of Gauß: The two-dimensional analogue of Fig. D.1

Fig. D.3. Theorem of Stokes: The field B along a surface with its source ∇×B (left) outside and (right) inside of the (in this example: plane) area F enclosed by the curve ∂F; the point where ∇×B cuts the area F is marked by the open point

D.4.2 Stokes’ Theorem

Let f(r), ∇f(r)andv(r), ∇·v(r), respectively be continuous on a sheet F . Then one has Stokes’ theorem   da ×∇◦= ds ◦ (D.36) F ∂F

Comment: The most frequent examples are of the form   da ∇×B = da × B, F ∂F with the magnetic induction field B(r), see Fig. D.3.

D.4.3 The Theorem of Green

As a special application of the theorem of Gauß with

◦ = f(r)∇g(r) − g(r)∇f(r) 344 D Nabla (Del) Operator and Laplace Operator

one obtains the theorem of Green  d3r [f(r)∇·∇g(r) − g(r)∇·∇f(r)] V (D.37) = da · [f(r)∇g(r) − g(r)∇f(r)] . ∂V

Problems

D.1. Nabla Operator. Gradient, divergence and curl: Determine

∇·r, ∇r, ∇(a · r), ∇f(r), ∇×r, ∇·(a × r). Vector gradient (a) With the help of

α × (β × γ)=(α · γ)β − (α · β)γ

derive a relation for ∇×(a × b). (b) With the help of this procedure understand the operation of a vector gradient. Now determine ∇r.

D.2. Vector Fields.

(a) Sketch the vector fields A(r)=r and A(r)=B × r, by placing arrows with appropriate direction and magnitude of the field on the lines of con- stant field strength |A(r)| =const.B is perpendicular to the drawing plane. (b) A rotational field is described by

v(r)=ω × r.

Determine the vorticity ∇ × v(r). A point charge causes a field of the form r G(r)=γ . r3 Determine the source strength ∇ · G(r). (c) Determine the effect of divergence and rotation on a gradient field A = ∇Φ(r), where Φ(r) is assumed to be twice continuously differentiable but arbitrary otherwise. Problems 345

D.3. Line Integrals.

A. Nonconservative forces: Given be a force field F of the form

F =(3x2 +2y,−9yz, 8xz2). · Determine the work c A dr done an the system along the path (a) C1: straight line from (0,0,0) to (1,1,1) and (b) C2: parabolic arc from (0,0,0) to (1,1,1)

B. Conservative forces: Given be a force F = r/r3. (c) What is special of this force F in comparison to the force field from problem (c) of problem D.2? What conclusion is to be drawn for the work done on the system? · (d) Determine c F dr along the paths (d1) C1:alongthex-axis from (∞;0) to (1;0) (d2) C2: along the bisecting line from (∞; ∞) to (1;1) (d3) C3: along of the arc of the unit circle in the first quadrant from (1;0) to (0;1) E Variational Method

E.1 Functions and Functionals

Definition: A function y = f(x)mapsavalue x to a value y,

x → y.

A functional Y = F [x(t)] maps a function x(t)toavalueY ,

x(t) → Y.

Comments: • In the context of the Hamilton principle of Sect. 3.10, a trajectory q(t)is mapped to the action W . • Here and in the following the functional is a derivative and an integral. Example: (function) y = f(x)= 1+x2  t2 (functional) Y = F [x]= 1+x ˙2 dt. t1 The function is extremal for a value x with df df dy = dx =0 ⇒ =0. dx dx The functional is extremal for a trajectory x(t)with

δY =0. 348 E Variational Method

Here,

f(x + ε dx) − f(x) df dy = lim = dx ε→0 ε dx F [x(t)+εδx(t)] − F [x(t)] δF δY = lim =: δx ε→0 ε δx often: δF = F [x + δx] − F [x]

Here δx is called the variation of the trajectory x(t)andδF is called the variation (or also the differential) of the functional F .

E.2 Variational Problem and Euler Equation

The Hamilton principle treated in Sect. 3.10 is a special case of an integral extremum principle. The extremum principle can be treated in general and shall be touched upon here rudimentary. Let x(t) be a function; the derivatives shall be abbreviated byx ˙ =dx/dt, etc. Furthermore, let g(x(t), x˙(t),t) be a function of multiple variables (u = x, v =˙x and t). Finally, let  t2 Y = F [x]= g(x, x,˙ t)dt t1

be a functional. (In this case the functional is a time derivative and an inte- gral.) Under the constraint of fixed boundary points x(t1)andx(t2)or

δx(t1)=0=δx(t2) (E.1)

Y is extremal (“stationary”) only if the Euler equation holds, # $ ∂g d ∂g δY =0 ⇔ − =0. (E.2) ∂x dt ∂x˙

This relation is basic to many optimizations in science and technology. Proof: With the varied trajectory x(t)+δx(t) one has  t2 δY = [g(x + δx, x˙ + δx,˙ t) − g(x, x,˙ t)] dt t1  # $ t2 ∂g ∂g = δx + δx˙ dt + O(δ∈). t1 ∂x ∂x˙ E.2 Variational Problem and Euler Equation 349

Terms of higher order are neglected in the following, since they vanish in the limit of small variations (δx → 0). The second term can be integrated in parts,  &  # $ t2 &t2 t2 ∂g ∂g & d ∂g δx˙ dt = δx& − δx dt, t1 ∂x˙ ∂x˙ t1 t1 dt ∂x˙ where the integrated (first) term vanishes because of the boundary condition δx(ti) = 0. Then one is left with  # $ t2 ∂g d ∂g δY = − δxdt. t1 ∂x dt ∂x˙

Since this has to hold for arbitrary times t1 (and t2), the integrand (the product of the contents of the square brackets and δx) must vanish, and since this has to hold for arbitrary variations δx, the contents of the square brackets itself must vanish.  Comments: • For 

F [x]= g({xi}, {x˙ i},t)dt

one has  # $ t2 ∂g ∂g δF[x]= δxi + δx˙ i dt ∂xi ∂x˙ i t1 i &  # $ &t2 t2 ∂g & ∂g d ∂g = δxi& + − δxi dt. ∂x˙ ∂x dt ∂x˙ i i t1 i t1 i i

With δxi(t1)=0=δxi(t2) and the independence upon the δxi(t)one obtains in generalization of (E.2) the Euler equations # $ ∂g d ∂g δY =0 ⇔ − =0 ∀i. (E.3) ∂xi dt ∂x˙ i

• For  F [x]= g(x, x,˙ x,...¨ )dt

one has  # $ t2 ∂g ∂g ∂g δF[x]= δx + δx˙ + δx¨ + ... dt ∂x ∂x˙ ∂x¨ t1 # $ & & &t2 &t2 ∂g d ∂g & ∂g & = − + ... δx& + + ... δx˙& + ... ∂x˙ dt ∂x¨ ∂x¨  # $ t1 # $ t1 t2 2 ∂g − d ∂g d ∂g + + 2 + ... δxdt. t1 ∂x dt ∂x˙ dt ∂x¨ 350 E Variational Method

One mustrequire correspondingly more boundary conditions (δx˙(ti)=0, etc.). Then the integrated terms vanish, and the Euler equation is the van- ishing of the square brackets under the integral. More details in Goldstein Sect. 2-2.

Problems

E.1. Dido’s Problem. Given the integral  x2 I = F [y(x),y(x),x]dx. x1 The problem of the variational method is to find such a curve y(x)forwhich the integral I is extremal. This leads to to the Euler–Lagrangian equation d ∂F ∂F − =0 dx ∂y ∂y as a necessary condition for the extremum of the integral I. (a) Often the Integrand F [y,y] does not explicitly depend upon x.Provethat one integration of the Euler–Lagrangian equation in this case leads to ∂F F − y =const. ∂y

(b) Using the result of part (a) solve Dido’s problem which reads: Among all curves y(x)oflengthl, determine that curve y(x) connecting two given points P1 and P2 which, together with the straight line connect- ing these points, is a boundary of the area of largest size. F Linear Differential Equations with Constant Coefficients

The general solution xi(t) of a system of N coupled linear inhomogeneous equations (of nth order)

N D(t) ij xj(t)=fi(t)(i =1,...,N)(F.1) j=1 with # $ n ν ( ) d D t = a (i, j =1,...,N) ij ij,ν dt ν=1 is a superposition of the general solution xhom(t) of the corresponding homo- i part geneous equation and any particular solution xi (t) of the inhomogeneous equation, hom part xi(t)=xi (t)+xi (t). (F.2)

F.1 Homogeneous Linear Differential Equations

The system of the homogeneous linear differential equations D(t) hom ij xj (t)=0 j canbesolvedbyanexponentialansatz

λt xj (t)=wj e . (F.3)

With this ansatz (leading to d/dt → λ) the system of the differential equations is turned into a system of algebraic equations Dij (λ) wj =0 j 352 F Linear Differential Equations with Constant Coefficients with n ν Dij (λ)= aij,ν λ . ν=1

The n × N generally complex zeroes λk of the secular equation

det (D(λk)) = 0

(the matrix D has the matrix elements Dij (λ)) lead to eigen solutions

(k) (k) λkt xi (t)=wi e (F.4) for single zeroes or, for K-fold zeroes λκ = λ (κ =2,...,K), to eigen solutions

( ) ( ) κ κ κ−1 λκt ≤ xi (t)=wi t e (2 κ

F.2 Inhomogeneous Linear Differential Equations

A particular (also: special) solution of the inhomogeneous differential (F.1) one can obtain with the help of the Green matrix g(t) with the elements gij (t),  part  −   xi (t)= dt gij (t t )fj(t )(F.6) j with D(t) ij gJl(t)=δi,lδ(t). (F.7) j Since the matrix of the Green functions in many cases is not known, one can obtain a particular solution alternatively by Fourier transformation of the inhomogeneity  ∞ dω −iωt fi(t)= e Fi(ω)(F.8) −∞ 2π — or possibly by Laplace transformation  ∞ −λt fi(t)= dλ e Fi(λ). 0 The Fourier transform of the differential equation (F.1) is an algebraic equa- tion (from the convolution theorem of the Fourier transformation) F.3 Stability of Solutions 353 Dij (ω)Xj(ω)=Fi(ω)(F.9) j

for Xi(ω),  −1 Xi(ω)= D (ω) ij Fj (ω). (F.10)

The back transformation of Xi(ω) leads to a particular solution  ∞ part dω −iωt xi (t)= e Xi(ω). (F.11) −∞ 2π

(This is equivalent to the determination of the Green matrix by Fourier trans- formation.)

F.3 Stability of Solutions

To investigate the stability of a solutions x(t)ofanon-linear differential equa- tion (or of a system of differential equations) (here mostly equations of mo- tion), one considers a small deviations (t)ofthesolutionx(t). Thus one sets x(t)+(t) in the differential equation (or in the system of differential equations) and expands in terms of small . To zeroth order one obtains the differential equation (or the system of differential equations) for x(t)((t)=0). To first order one obtains a linear differential equation for i(t), which one solves with the usual exponential ansatz eλt. With this the differential equation for (t) is turned into an algebraic equation in the form of an eigen value equation. The eigen values λ are obtained from the secular equation. Definition: A stationary solution is called stable, if a small deviation  as a function of time remains in the neighborhood of  = 0; a stationary solution is called unstable,ifasmalldeviation from the increases as a function of time. Asolutionx(t) is thus unstable, if one has Re λ>0. Also a multiple zero λ = 0 indicates an instability. G Quadratic Matrices and Their Eigen Solutions

G.1 The Eigen Value Problem

Let M be a complex (N × N)-matrix and e a complex (column) vector (this is an (N × 1)-matrix). The eigen value problem reads

M · e = λe.

The eigen solutions this equation consist of eigen values λi and eigen vectors ei with M · ei = λiei.

G.2 Definitions

Let M be an (N × N)-matrix. The transposed matrix MT has the elements

T T T (M )αβ = Mβα (M ) = M.

The complex conjugate matrix M∗ has the elements

∗ ∗ ∗ ∗ (M )αβ = Mαβ (M ) = M.

The adjoint matrix is

† ∗ T T ∗ † ∗ M =(M ) =(M ) (M )αβ = Mβα.

A matrix is called real, if one has

M = M∗. 356 G Quadratic Matrices and Their Eigen Solutions

A matrix is called symmetric, if one has

M = MT.

A matrix is called anti-symmetric or skew-symmetric, if one has

M = −MT.

A matrix is called hermitian or self-adjoint, if one has

M = M†.

A matrix is called orthogonal, if one has

M−1 = MT.

A matrix is called unitary, if one has

M−1 = M†.

Even though a is a (column) vector (an (N × 1)-matrix) with the elements † aα, then the adjoint vector a is a (row) vector (a (1 × N)-matrix) with the ∗ elements aα. The scalar product of two (complex) vectors a and b is

N † · ∗ † · ∗ a b = aαbα =(b a) . α=1 The unitary matrix U describes a length- and angle-conserving coordinate transformation: Let a1, a2, b1,andb2 be vectors with

a2 = U · a1 b2 = U · b1.

Then one has † † a2 · b2 = a1 · b1. Proof: With (U · a)† = a† · U† one obtains

† † † † † −1 † † a2 ·b2 =(U·a1) ·(U·b1)=a1 ·U ·U·b1 = a1 ·U ·U·b1 = a1 ·1·b1 = a1 ·b1.  G.3 Properties of the Eigen Values 357 G.3 Properties of the Eigen Values

Let M be a complex (N × N)-matrix with the eigen values λi. 1. There are exactly N eigen values

λi i =1,...,N. 2. The matrices M and MT have the same eigen values. ∗ ∗ 3. The matrices M and M have the eigen values λi and λi , respectively. 4. The eigen values of a hermitian matrix are real, † ⇒ ∗ M = M λi = λi . 5. The modulus of the eigen values of a unitary matrix is equal to unity, † −1 M = M ⇒|λi| =1. Proof: (1) The solution condition for the equation (M − λ1) e =0 is the vanishing of the secular determinant, det(M − λ1)=0 If M is an (N × N)-matrix, the secular determinant is a polynomial of Nth order in λ with N (complex) zeroes. These N zeroes are the N eigen values. (2) Because of det(MT − λ1)=det[(M − λ1)T]=det[(M − λ1)] the matrices MT and M have the same eigen values. (3) Because of ∗ det (M∗ − λ∗1)=det[(M − λ1)] =0 ∗ ∗ the matrices M and M have the eigen values λi and λi , respectively. (4) See the proof to (6) further below. (5) Because of M† =(M∗)T the matrices M† and M∗ have the same eigen values according to the property (2). Let ei be an eigen vector of M with eigen value λi. Then on the one hand one has † −1 M · M · ei = M · M · ei = ei and on the other hand † · · † · ∗ | |2 M M ei = M λiei = λi λiei = λi ei. By comparison one finds |λi| =1.  358 G Quadratic Matrices and Their Eigen Solutions G.4 Properties of the Eigen Vectors of Hermitian Matrices

† × ∗ Let M = M be a hermitian (N N)-matrix with the eigen values λi = λi and the eigen vectors ei. The eigen vectors shall be normalized to unity, i.e., the modulus of the eigen vectors shall have the value 1,

|ei| =1.

6. The eigen vectors belonging to different eigen vectors are orthogonal,  ⇒ † · λi = λj ei ej =0. The eigen vectors belonging to equal eigen values can be chosen as or- thogonal. If the eigen vectors are normalized (to unity), one has the orthonor- mality relation † · ei ej = δi,j . 7. One has the completeness relation1 ⊗ † ei ei = 1, i 8. For the eigen values one has † · · λi = ei M ei. 9. The matrix M can be represented by its eigen solutions, ⊗ † M = λiei ei . i 10. Analogously one has n n ⊗ † ∈ N M = λi ei ei with n , i and with this for a function represented by the Laurent series ∞ ∞ n ⇔ n ⊗ † ⊗ † f(M)= anM f(M)= an λi ei ei = f(λi)ei ei . n=−∞ n=−∞ i i

1 † Thetensorproduct(ofthevectorsa and b) is denoted here by ⊗,(a ⊗ b)αβ = ∗ aαbβ. G.4 Properties of the Eigen Vectors 359

Proof: (6) Let

M · ei = λiei

M · ej = λjej.

One forms the scalar product of the equations with ej and ei, † · · † · ej M ei = λiej ei † · · † · ei M ej = λjei ej. By conjugation of the second equation one obtains † · · ∗ † · ej M ei = λj ej ei with M = M†. Subtraction of the two equations leads to   − ∗ † · 0= λi λj ej ei. † · | |2  In the case i = j one obtains with ei ei = ei =1=0 ∗ λi = λi , see the property (4).  ∗ In the case λi = λj = λj the expression can vanish only if † · ej ei =0 holds. (7) With † · δi,j = ej ei from property (6) one obtains by inserting the unit matrix † · † · · † · ⊗ † · ej ei = ej 1 ei = ej ek ek ei k † † 1 i = j = (e · e )(e · e )= δ δ = δ = . j k k i j,k k,i i,j 0 i = j k k (8) The result for the eigen values one obtains by scalar multiplication with the eigen vector and by use of the orthonormality property (6), · ⇒ † · · † · M ei = eiλi ei M ei = ei eiλi = λi. (9) From λiei = M · ei one obtains by tensor product and summation ⊗ † · ⊗ † · λiei ei = M ei ei = M 1 = M i i with the property (7). (10) By consecutive multiplication with M.  H Dirac δ-Function and Heaviside Step Function

H.1 Properties of the Dirac δ-Function and of the Heaviside Step Function

Properly speaking, the Dirac δ function is not a function, but a so-called distribution. The δ function has the following properties (left panel of Fig. H.1):

δ(x − x0)=δ(x0 − x)(H.1) δ(x − x0)=0 f¨ur x = x0 (H.2)  b f(x0) x0 ∈]a, b[ f(x) δ(x − x0)dx = (H.3) 0 x0 ∈ [a, b] a  f(x ) f(x) δ(g(x)) dx = i with g(x )=0 |g(x )| i i i 1 or δ(g(x)) = δ(x − x )(H.4) |g(x )| i i i 1 in particular δ(ax)= δ(x)(H.5) |a| d δ(x − x0)= θ(x − x0). (H.6) dx

Fig. H.1. Dirac δ function (left, schematic) and Heaviside step function (right) 362 H Dirac δ-Function and Heaviside Step Function

Fig. H.2. Representation of the Dirac delta function by one function from various functional sequences

Here, θ(x − x0) is the Heaviside step function (right panel of Fig. H.1), 4 0 xx0 Comment on (H.4): With the condition that g(x) can be inverted (i.e., that g(x) is injective), g(x) ⇒ x(g), one has    dx 1 f(x) δ(g(x)) dx = f(x) δ(g(x)) dg = f(x(g)) δ(g)dg dg dg dx and contributions to the integral come from the points with g =0,with

δ(g(x)) = δ(−g(x)) = δ(|g(x)|).

H.2 Representation of the δ-Function by Functional Sequences

The Dirac δ function can be represented as the limit of various functional sequences, δ(x) = lim δn(x). n→∞ The following functional sequences are the most common ones, see Fig. H.2, 1 n |x| < 2 δn(x)= n box function 0const H.5 The δ-Function in R3 363

n 2 2 δ (x)=√ e−n x Gauß function n π n 1 δ (x)= Lorentz function n π 1+n2x2  n sin nx 1 ±ikx δn(x)= = dk e cf. (H.7) πx 2π −n

H.3 Integral Representation of the δ-Function

 1 ∞ δ(x)= dt e±ixt. (H.7) 2π −∞

H.4 Periodic δ-Function

1 sin[(2M +1)πy/L] δ(x − nL) = lim (H.8) M→∞ L sin(πy/L) n Proof: See (I.30). 

H.5 The δ-Function in R3

Coordinate-Free Representation

The three-dimensional generalization of (H.3) is  ∈ 3 − f(r0)forr V d rf(r) δ(r r0)= ∈ (H.9) V 0forr V.

Integral Representation



1 3 ik·(r−r0) δ(r − r0)= d k e . (H.10) (2π)3

Representation in Cartesian Coordinates

δ(r − r0)=δ(x − x0) δ(y − y0) δ(z − z0). (H.11) 364 H Dirac δ-Function and Heaviside Step Function

Representation in Spherical Polar Coordinates

δ(r − r0) δ(ϑ − ϑ0) δ(r − r0)= δ(ϕ − ϕ0) r2 sin ϑ δ(r − r0) = − δ(cos ϑ − cos ϑ0) δ(ϕ − ϕ0). (H.12) r2

Representation in Cylindrical Coordinates

δ( − 0) δ(r − r0)= δ(ϕ − ϕ0) δ(z − z0). (H.13) 

H.6 The δ-Function as an Inhomogeneity of the Poisson Equation

1 Δ = −4πδ(r − r). (H.14) |r − r|

Proof: Case r = r: If one places the origin of the coordinate system at the point r, one has for r =0

1 (D.13) 1 r ∇ = − e = − r r2 r r3 1 r (D.25) 1 1 ∇·∇ = −∇ · = − ∇·r − r ·∇ r r3 r3 r3 1 −3r = − 3 − r · =0. r3 r5 Case r = r: Integration over a (small) sphere K with its center at the origin yields with the Gauß theorem    ( 35) 3 ∇·∇1 D. ·∇1 − · er d r = da = da 2 . K r ∂K r ∂K r 2 With da = r erdΩ (in spherical polar coordinates) one obtains thus on the one hand   1 d3r ∇·∇ = − dΩ = −4π K r ∂K and on the other hand  −4π d3rδ(r)=−4π. K  I Fourier Transformation

I.1 The Transformation: Fourier Integral

Let f(x) be a function with the properties 1. f(x) is piecewise smooth; 2. at discontinuities xi one has

1 f(xi)= 2 [f(xi +0)+f(xi − 0)];

3. f(x) is absolute integrable, i.e., the integral

 ∞ |f(x)|dx −∞

exists. Then one has ∞ 1 f(x)= dk eikxF (k)(I.1) 2π −∞ ∞ F (k)= dx e−ikxf(x). (I.2) −∞

Notation: F is called Fourier transform of f (and vice versa). Comments: • The choice of the factor 2π is the common choice in (solid-state) physics. In mathematical treatments mostly the symmetrical convention is used,1

1 See also the comment in footnote 2 on p. 375 further below. 366 I Fourier Transformation  1 f(x)=√ dy eixyF (y) 2π  1 F (y)=√ dx e−ixyf(x). 2π • F (k = 0) is the mean value of the function f(x). • For the generalization to functions of multiple variables (to higher dimen- sions) see Sect. I.2.

I.1.1 Examples and Applications (i) Real Functions

f(x)=f ∗(x) ⇔ F (k)=F ∗(−k). Proof: From  ∞ F (k)= dx e−ikxf(x) −∞ one obtains with f(x)=f ∗(x)  ∞ F ∗(k)= dx e+ikxf(x)=F (−k). −∞ 

(ii) Real Even and Odd Functions

f(x)=±f(−x) ⇔ F (k)=±F (−k)=±F ∗(k)  ∞ 1 cos kx ⇔ F (k)=2 dxf(x) × 0 −isinkx Proof:  ∞ F (k)= dxf(x)e−ikx ∞  ∞  0 = dxf(x)e−ikx + dxf(x)e−ikx 0 −∞  ∞  ∞ = dxf(x)e−ikx + dxf(−x)eikx 0 0 ∞   = dxf(x) e−ikx ± eikx 0  ∞ 1 2coskx = dxf(x) × . 0 −2i sin kx  I.1 The Transformation: Fourier Integral 367

(iii) Dirac δ-Function

−ikx0 f(x)=δ(x − x0) ⇔ F (k)=e . Proof: ∞

−ikx −ikx0 F (k)= dx e δ(x − x0)=e . −∞  In particular one has thus the reverse

 ∞ dk δ(x − x)= eik(x−x ). (I.3) −∞ 2π

(iv) A Constant

 ∞ (I.3) f(x)=1 ⇔ F (k)= dx e−ikx =2πδ(k). (I.4) −∞ Comment: The (infinitely extended) function f(x) = 1 does not fulfill con- ∞ | | dition (3), namely that the integral −∞ f(x) dx exists. Thus the result is not a usual function (but a so-called distribution).

(v) Lorentz Function

1 F (t)=iθ(t)e−iω0te−γt ⇔ f(ω)= (I.5) ω0 − ω − iγ Proof:  ∞  ∞ f(ω)=i dtθ(t)ei(ω−ω0+iγ)t =i dt ei(ω−ω0+iγ)t −∞ 0 0 − 1 1 =i = . i(ω − ω0 +iγ) ω0 − ω − iγ For the reverse of the proof the integral  ∞ dω e−iωt −∞ 2π ω − ω0 +iγ has to be evaluated. This is done most comfortably with the integral theorem of Cauchy. The pole lies at ω = ω0 − iγ in the lower complex ω half-plane. Case t<0: For t<0 one closes the integration path as in Fig. I.1 by a semi-circle in the upper half-plane with ω = R eiϕ 368 I Fourier Transformation

= -

Fig. I.1. The integration path in the complex frequency plane for t<0

= -

Fig. I.2. The integration path in the complex frequency plane for t>0 and lets the radius R of the semi-circle tend towards ∞,   1 ∞ e−iωt dω 1 e−iωt dω = 2π −∞ ω − ω0 +iγ 2π ω − ω0 +iγ  π −iReiϕt iϕ − 1 e R e idϕ lim iϕ R→∞ 2π 0 R e − ω0 +iγ  π i iϕ =0− lim e−iRe t dϕ R→∞ 2π 0  i π =0− lim eR(−icosϕ+sin ϕ)t dϕ =0. R→∞ 2π 0 From the Cauchy integral theorem the closed integral vanishes, since no pole is encircled, and the second integral vanishes, since the integrand with eR sin ϕt, sin ϕ>0andt<0 with increasing R vanishes exponentially. Case t>0: For t>0 one closes the integration path as in Fig. I.2 by a semi-circle in the lower half-plane,   1 ∞ dω e−iωt 1 e−iωt dω = 2π −∞ ω − ω0 +iγ 2π ω − ω0 +iγ  −π −iR eiϕt iϕ − 1 e R e idϕ lim iϕ R→∞ 2π 0 Re − ω0 +iγ = I1 + I2.

For the closed integral (the path taken in negative sense) one obtains  −iωt 1 dω e 1 −i(ω0−iγ)t −iω0t −γt I1 = = (−2πi) e = −ie e . 2π ω − ω0 +iγ 2π I.1 The Transformation: Fourier Integral 369

=++

Fig. I.3. Deformation of the integration path in the complex z-plane; the path on the left is the sum of the paths on the right

The integral over the semi-circle vanishes for the reasons analogous to the case t<0:  −π −iReiϕt iϕ − 1 e R e idϕ I2 = lim i 2π R→∞ Re ϕ − ω0 +iγ 0 i −π = lim eR(−icosϕ+sin ϕ)t dϕ =0. 2π R→∞ 0

The integral vanishes, since the integrand with eR sin ϕt,sinϕ<0andt>0 vanishes exponentially. 

(vi) Gauß Function

1 − 1 2 2 − 1 2 2 f(x)= √ e 2 x /a ⇔ F (k)=e 2 k a . (I.6) a 2π The Fourier transform of a Gauß function is thus again a Gauß function. Proof:  ∞ 1 −i − 1 2 2 F (k)= √ dx e kxe 2 x /a a 2π −∞  ∞ 1 − 1 ( −i )2 − 1 2 2 = √ dx e 2 x/a ka e 2 k a , a 2π −∞ where in the exponent one has made a quadratic complement. Now one makes a transition to the complex z-plane. The integrand is an analytic function; the integration path can thus be deformed and partitioned as sketched in Fig. I.3. For the first integral one obtains

 2 ∞+ika − 1 k2 a2 1 − 1 (z/a)2 F1(k)=e 2 √ dz e 2 a 2π −∞+ika2  2 ∞+ika − 1 2 2 1 − 1 ( −i )2 =e 2 k a √ dz e 2 z/a ka a 2π −∞+ika2  ∞ − 1 2 2 1 − 1 ( )2 − 1 2 2 =e 2 k a √ dz e 2 z/a =e 2 k a . a 2π −∞ 370 I Fourier Transformation

Since the closed integral does not encircle a pole, the second integral vanishes. The two last integrals along the paths parallel to the imaginary axis vanish, if one writes z = z +iz in the exponent,

− 1 ( −i )2 − 1 [ +i( − )]2 e 2 z/a ka =e 2 z /a z /a ka , and takes the limit z →∞. 

(vii) Periodic Functions with the Periodicity L

The functions f(x)=f(x + L) likewise do not fulfill condition (3). See Appendix I.3 on Fourier series.The result is ∞ − F (k)= 2πδ(k kn)Fkn n=−∞

L 1 F = e−ikxf(x)dx k L 0 2π k = n n L ∞ (I.24) ⇒ iknx f(x) = e Fkn . n=−∞

I.1.2 Convolution Theorem

Let f(t), g(t), and h(t) be functions with the properties (1)–(3) with the Fourier transforms F (ω), G(ω), and H(ω), respectively, (and vice versa) with  F (ω)= dt eiωtf(t)(I.7)  G(ω)= dt eiωtg(t)(I.8)  H(ω)= dt eiωth(t). (I.9)

Then one has the convolution theorem

F (ω)=G(ω) H(ω) (I.10) f(t)= dtg(t − t) h(t)= dtg(t) h(t − t). I.1 The Transformation: Fourier Integral 371

Proof:   dω dω f(t)= e−iωtF (ω)= e−iωtG(ω)H(ω)  2π  2π  dω = e−iωt dt eiωt g(t) dt eiωt h(t)  2π   dω = dtg(t) dth(t) e−iω(t−t −t )   2π  = dtg(t) dth(t) δ(t − t − t)= dtg(t) h(t − t).

Here the presumptions have made sure that the functions have a sufficiently nonpathological behavior, such that the sequence of the integrations can be interchanged.  Comments: • The Fourier transform of a product of functions is thus a convolution of the Fourier transforms (and vice versa). • Example: Forced oscillator (response theory). Let the displacement x(t), the Green function g(t), and the driving force k(t)begivenwith  x(t)= dtg(t − t) k(t).

Then the corresponding Fourier transforms are given by X(ω), G(ω), K(ω) with X(ω)=G(ω)K(ω). • The theorem can easily be generalized to higher dimensions.

I.1.3 Parseval’s Equation Let f(t)andg(t) be functions with the properties (1)–(3) and F (ω)andG(ω) their Fourier transforms as in (I.7) and (I.8). Then one has Parseval’s theorem   dω dtf∗(t) g(t)= F ∗(ω) G(ω). (I.11) 2π

Proof:      dω dω dtf∗(t) g(t)= dt F ∗(ω)eiωt G(ω)e−iω t 2π 2π     dω dω = F ∗(ω) G(ω) dt ei(ω−ω )t  2π  2π dω dω = F ∗(ω) G(ω)2πδ(ω − ω)  2π 2π dω = F ∗(ω) G(ω). 2π 372 I Fourier Transformation

With the presumptions one has made sure that the functions have a sufficiently nonpathological behavior, such that the sequence of the integrations can be interchanged.  Comments: • The theorem can be generalized easily to higher dimensions. • The “scalar product” of two functions is thus independent of the repre- sentation (here in (I.11) in t-space and ω-space, respectively). • In generalization to three dimensions and in application to the quantum- mechanical wave functions as a function of three coordinates the Parseval theorem expresses the equality of the scalar product of the wave function in different representations.

I.1.4 Uncertainty Relation

From the examples (v) of the Lorentz function and (vi) of the Gauß function as well as from the examples (iii) and (iv) one can observe that functions, the main contributions of which are concentrated in a narrow regime, lead to Fourier transforms, which are distributed over a broad regime, and vice versa. The “widths” of the Gauß function and its Fourier transform

1 − 1 2 2 (I.6) − 1 2 2 f(x)= √ e 2 x /a ⇔ F (k)=e 2 k a . a 2π are √ 1 √ b ≈ a 2andb ≈ 2 x k a with a product of the order of 1, independent of a.

bxbk ≈ 2;

For more details see the following. – The widths of the imaginary part of the Lorentz function and its Fourier transform

(I.5) 1 F (t)=−iθ(t)e−iω0te−γt ⇔ f(ω)= ω − ω0 +iγ are 1 b ≈ and b ≈ γ, t γ ω respectively, with a product of the order of 1, independent of γ.

btbω ≈ 1.

Definition : The mean square deviation Δx from the mean value x of the one-dimensional function f(x)is I.1 The Transformation: Fourier Integral 373

 1 2 2 /  1/2 Δx = (x − x ) = x2 − x 2 (a)  1 2 with xn = dxxn|f(x)| (b) (I.12) N N = dx |f(x)|2 (c)

Example: Gauß Function

For the mean square deviations Δx and Δk of the Gauß function and its Fourier transform, respectively, one finds

1 Δx · Δk = 2 . (I.13)

Proof: With 1 (I.6) 1 − 2 2 f(x) = √ e 2 x /a a 2π one finds  ∞ √ 2 2 (GR 3 321 3) 1 −x /a . . 1 a π 1√ Nx = 2 dx e = 2 = 2πa −∞ 2πa 2 4a π  ∞ 1 −x2/a2 Nx x = 2 dxxe =0 2πa −∞  ∞ 3√ 2 2 (GR 3 461 2) 2 1 2 −x /a . . 1 a π √a Nx x = 2 dxx e = 2 = 2πa −∞ 2πa 4 8 π √ a ⇒ (Δx)2 =4a π √ = 1 a2 8 π 2

With (I.6) 2 2 F (k) =e−k a one obtains by the replacement a → 1/a

 ∞ √ −k2a2 π Nk = dk e = (I.14) −∞ 2a

Nk k =0  ∞ √ 2 2 π N k2 = dkk2e−k a = (I.15) k 4a3 −∞√ 2a π 1 ⇒ (Δk)2 = √ = π 4a3 2a2 374 I Fourier Transformation

With this one obtains 1 (Δx)2(Δk)2 = . 4  Comments: • While one can determine the wave length and thus the corresponding wave vector for an infinitely extended plane wave precisely, one can determine the wave vector of a Gauß wave packet of the (finite) width Δx only up to an error (called uncertainty in quantum mechanics) determined by Δk =(2Δx)−1 (and vice versa). A similar relation holds for other wave packets and/or other observables (like frequency and time). • The precision of a measurement is subject to an uncertainty relation found here, which relates the minimum precision of two observables intercon- nected by Fourier transformation. In the present context this turns out to be a classical phenomenon. In quantum mechanics the uncertainty relation is applied to observables, which are conjugate to each other (like position and momentum).

I.2 Fourier Transformation in R4:PlaneWaves

I.2.1 The Whole R3

The canonical convention, at least within solid-state physics, for the Fourier transformation in space (R3)andtimeis

 3  ∞ d k ik·r dω −iωt f(r,t)= 3 e e F (k,ω). (I.16) R3 (2π) −∞ 2π

Notice the different sign in the phases. The time dependence is of no further interest here (but may be in another context). The reverse is

  ∞ F (k,ω)= d3r e−ik·r dt eiωtf(r,t). (I.17) R3 −∞

Comment: F and f differ by the units of a volume (and time). One has in particular analogously to (I.3)

 ∞  dω dt eiωt =2πδ(ω) ⇔ e−iωt = δ(t) (I.18) −∞ 2π

  3 3 −ik·r 3 ⇔ d k ik·r d r e =(2π) δ(k) 3 e = δ(r) (I.19) R3 (2π) I.2 Fourier Transformation in R4: Plane Waves 375

with the one- and three-dimensional Dirac δ-functions δ(t), δ(ω)andδ(r), δ(k), respectively.

I.2.2 Normalization Volume V

Often, instead of the R3, a large but finite (possibly periodically continued) volume V is considered. Then one has discrete wave vectors k, and one has to replace the integration over k by a summation over k,  d3k 1 3 F (k)= F (k) (I.20) (2π) V k

for a function F (k).2 Analogously to (I.3) one has  3 −ik·r d r e = Vδk,0 (I.21) V 1 eik·r = δ(r). (I.22) V k

For more details on Fourier series in R3 see Sect. I.3.6.

Example: Coulomb and Yukawa potential

For the Yukawa potential one has 1 −αr 1 4π ik·r e = 2 2 e . (I.23) r V k k + α

The Coulomb potential is obtained taking the limit α → 0. Proof: 1 f(r)= F (k)eik·r V  k 1 F (k)= d3r e−ik·r e−αr. r

The representation of the integral in spherical polar coordinates (with the direction of k as the specific direction) leads to

 ∞  2 −ikr cos ϑ 1 −αr F (k)= drr dΩr e e . 0 r

2 3 The density of states of the wave vectors√ in k-space is V/(2π) .Itisthispoint,at which the symmetrical use of the factor 2π as in the mathematical treatments would be unpractical. 376 I Fourier Transformation

The integration over the angles yields    2π 1 −ikr − ikr −ikr cos ϑ −ikrt e e dΩr e = dϕ dt e =2π 0 −1 −ikr and thus  ∞ ! "   2π 2π −1 −1 F (k)= dr e(ik−α)r − e−(ik+α)r = − ik 0 ik ik − α −ik + α 2π 2ik 4π = = . ik k2 + α2 k2 + α2  Comments: • Corresponding to (I.10) and (I.11), the convolution theorem and Parseval’s theorem for functions of one variable are generalized to functions of several variables. • As an example and in generalization to three dimensions one obtains, with the Coulomb potential 1 e2 V (r)= , 4πε0 |r| the relation between charge density ρ(r) and electrostatic potential φ(r),  φ(r)= d3rV (r − r) ρ(r) ⇔ φ%(k)=V% (k) ρ%(k).

• In the case of the Coulomb potential the Fourier coefficient F (k)isnot 3 defined for k =0.Thisisbecausein R3 d r/r the potential decreases with r slower than the volume increases. In application to solid-state physics, screening effects remove this divergence.3

I.3 Fourier Series

In solid-state physics one must distinguish between quantities, which are de- fined in the continuous space, for example the electron density, and those, which are defined only on lattice points, for example nuclear positions or atomic displacements.

I.3.1 The Series Let f(x) be a periodic function with the properties 1. f(x) is periodic, f(x)=f(x + L); 2. f(x) is piecewise smooth; 3 See the Course on Solid-State Physics. I.3 Fourier Series 377

3. At discontinuities xi one has

1 f(x )= [f(x +0)+f(x − 0)]; i 2 i i Then the function f can be expanded into a Fourier series,

∞ iknx f(x)= cn e (I.24) n=−∞ ∞ a0 alternatively = + (a cos k x + b sin k x) (I.25) 2 n n n n n=1  2 L with an = dxf(x)cosknx = cn + c−n (I.26) L 0  2 L bn = dxf(x)sinknx =i(cn − c−n) (I.27) L 0  L 1 −ikn x cn = dxf(x)e (I.28) L 0 2π k = n. (I.29) n L

Proof: One performs a Fourier transformation, cuts the integration regime into the single periodicity intervals, and obtains

 ∞ F (k)= dx e−ikxf(x) −∞  ∞ (m+1)L = dx e−ikxf(x) m=−∞ mL  ∞ L = dxe−ikx e−ikmLf(x + mL)(x = x − mL) 0 m=−∞  ∞ L = e−ikmL dx e−ikxf(x)(x → x) 0 m=−∞  ∞ L (I.30) = 2πδ(kL − n2π) dx e−ikxf(x) 0 n=−∞  ∞ L (H.5) 2π = δ(k − n2π/L) dx e−ikxf(x) L 0 n=−∞ 378 I Fourier Transformation

with y = kL in (I.30) of Sect. I.3.2 further below. Thus one obtains  ∞ 1 f(x)= dk eikxF (k) 2π −∞   ∞ ∞ L 1 2π = dk eikx δ(k − n2π/L) dxe−ikx f(x) 2π −∞ L 0 n=−∞  ∞ L iknx 1  −iknx  = e dx e f(x )(kn = n2π/L) L 0 n=−∞ ∞ iknx = e cn n=−∞ with  L 1 −iknx cn = dx e f(x). L 0  Comments:

• The coefficients an and bn are real, if f is real, while the coefficients cn are complex. • The relation to the Fourier integral is given by ∞ 1 F (k)= 2πδ(k − k )c c = F (k ). n n n 2π n n=−∞ • The convergence is uniform for f(x) continuous. The definition of uniform convergence of a sum N SN (x)= sn(x) n=−N is: |SN (x) − S(x)| < for N>N0 is independent of x. Physically meaningful functions are continuous. For strongly simplified models (e.g., the quantum-mechanical potential barrier) functions are pos- sibly discontinuous, and the convergence of the Fourier series of discon- tinuous functions shows, among others, the so-called Gibbs phenomenon (this is an oscillation around the original function, the amplitude of which cannot be reduced by extending the interval.) • For the generalization to higher dimensions see Sect. I.3.6.

I.3.2 Examples and Applications (i) Even Periodic Functions

f(x)=f(−x) ⇔ bn =0. I.3 Fourier Series 379

(ii) Odd Periodic Functions

f(x)=−f(−x) ⇔ an =0.

(iii) Periodic δ-Function

∞ ∞ eimy =2π δ(y − n2π). (I.30) m=−∞ n=−∞ Proof: Consider the (finite) sum

M 2M imy −iMy imy SM = e =e e . m=−M m=0

This is a geometrical series, which can be summed,

2M −iMy imy SM =e e m=0 i(2 +1) −i( + 1 ) i( + 1 ) − M y M 2 y − M 2 y −iMy 1 e e e =e iy = −i 1 i 1 1 − e e 2 y − e 2 y 1 sin((2M +1)2 y) = 1 sin( 2 y) ∞ 1 = π δ2M+1( 2 y − nπ), cf. Sect. H.2. n=−∞

Now one takes the limit ∞ ∞ imy 1 e = lim SM = π lim δ2M+1( 2 y − nπ) M→∞ M→∞ m=−∞ n=−∞ ∞ ∞ 1 = π δ( 2 y − nπ)=2π δ(y − n2π). n=−∞ n=−∞ 

Comment: The quantity SM is the representation of the periodic δ-function by a functional sequence similar to the one in Appendix H.2, the behavior of which at y = 0 is the same as that of the last example of the functional sequence of the non-periodic δ function of Appendix H.2. The expression is a periodic function of y. 380 I Fourier Transformation

(iv) Orthogonality (m ∈ Z and n ∈ Z)

  2π L 1 1 2π iy(m−n) iz L (m−n) dy e = δm,n ⇔ dz e = δm,n. (I.31) 2π 0 L 0

Proof: In the case m = n one has  1 2π dy =1, 2π 0 and in the case m = n one has

 2 1 π 1 ei2π(m−n) − 1 dy eiy(m−n) = =0. 2π 0 2π i(m − n) 

(v) Completeness

For y and y both from the same periodicity interval (of the length 2π)one has from example (iii)

∞ ei(y−y )m = 2πδ(y − y − 2πn). (I.32) m=−∞ n

The completeness relation expresses the fact that each well-behaved function withtheproperties(1)–(3)(periodicinaninterval)canbeexpandedintoa Fourier series,  f(y)= dyδ(y − y)f(y)  1 = dy ei(y−y )mf(y) 2π m  1 = eiym dye−iy mf(y) 2π m iym = e cn. m

I.3.3 Convolution Theorem

For functions f, g,andh with the properties (1)–(3), with the same period L and with the series ∞ iknx f(x)= Fne n=−∞ I.3 Fourier Series 381

and analogously for g and h one has

h(x)=f(x) g(x) ∞ ∞ ⇔ Hn = FmGn−m = Fn−mGm (I.33) m=−∞ m=−∞ H = F G n n n  1 L 1 L ⇔ h(x)= dyf(x − y) g(y)= dyf(y) g(x − y). (I.34) L 0 L 0

Proof: (i) By inserting one obtains   1 1 H = dx e−iknxh(x)= dx e−iknxf(x) g(x) n L L  ∞ ∞ 1 = dx e−iknx F eikmx G eiklx L m l m=−∞ l=−∞ ∞ ∞  1 = F G dx e−i(kn−km−kl)x m l L m=−∞ l=−∞ ∞ ∞ (I.31) = FmGl δn,m+l m=−∞ l=−∞ ∞ ∞ = FmGn−m = Fn−lGl. m=−∞ l=−∞

(ii) By inserting one obtains ∞ ∞ −iknx −iknx h(x)= e Hn = e FnGn n=−∞ n=−∞ ∞   1 L 1 L = e−iknx dy e−iknyf(y) dz e−iknzg(z) L 0 L 0 n=−∞   ∞ 1 L 1 L = dxf(y) dyg(z) e−ikn(x−y−z) L 0 L 0 n=−∞   L L (I.32) 1 1 = dyf(y) dzg(z)Lδ(x − y − z) L 0 L 0   1 L 1 L = dyf(y) g(x − y)= dzf(x − z) g(z). L 0 L 0  382 I Fourier Transformation

I.3.4 Parseval’s Equation

For two functions f and g with the properties (1)–(3) with equal periods and with the series ∞ ikny f(y)= e Fn n=−∞ ∞ ikny g(y)= e Gn n=−∞

one has  L ∞ 1 ∗ ∗ dyf (y) g(y)= Fn Gn. (I.35) L 0 n=−∞ Proof: By inserting one obtains   L L 1 ∗ 1 −ikmy ∗ ikny dyf (y) g(y)= dy e Fm e Gn L 0 L 0 m  n L ∗ 1 −i(km−kn)y = FmGn dy e L 0 m n (I.31) ∗ = FmGnδm,n m n ∗ = FmGm. m 

I.3.5 Fourier Series in R3:Lattices

Functions, which are defined on a periodic lattice (with N lattice points h in the periodicity volume V ), can be described by wave vectors q defined from within a so-called Brillouin zone (BZ), 1 f(h)= eiq·hF (q) (I.36) V q∈BZ V F (q)= e−iq·hf(h) (I.37) N h

with the replacement  V d3rf(r)= f(h). V N h I.3 Fourier Series 383

Comments: • Occasionally one diverges from this replacement and defines f(h)= eiq·hF˜(q) q∈BZ 1 F˜(q)= e−iq·hf(h). N h

Then f and F˜ have the same units. • In particular one has 1 iq·h e = δh,0 (I.38) N q∈BZ 1 −iq·h e = δq,G =: Δ(q) (I.39) N h G

with reciprocal lattice vectors G.4

I.3.6 Functions with Lattice Periodicity

Functions with lattice periodicity

f(r)=f(r + h)

with lattice vectors h can be expanded analogously to (I.24)pp. into a Fourier series, f(r)= F (G)eiG·r (I.40) G  1 F (G)= d3rf(r)e−iG·r (I.41) Va Va with reciprocal lattice vectors G and the volume

Va = V/N of the unit cell. One has in particular

iG·h  e = 1 (I.42) 1 3 −iG·r d r e = δG 0 (I.43) V , a Va eiG·r = N δ(r − h). (I.44) G h

4 This is of key importance for the structure determination in crystallography, see the Course on Solid-State Physics. 384 I Fourier Transformation Summary: Fourier Integral and Fourier Series

Notation: r position vector h lattice vector G reciprocal lattice vector k vector from the reciprocal space q vector from the (first) Brillouin zone The reciprocal space is discrete for V finite, continuous for V = R3 Fourier integrals:  3 (I.16) d k (I.20) 1 f(r) = eik·rF (k) = eik·rF (k) (2π)3 V  k (I.17) F (k) = d3r e−ik·rf(r)  3 −ik·r (I.19) 3 d r e =(2π) δ(k)=Vδk,0  3 (I.19) d k ik·r 1 ik·r δ(r) = 3 e = e . (2π) V k

Fourier series for lattice-periodic functions f(r)=f(r + h): (I.40) f(r) = eiG·rF (G) G  (I.41) 1 F (G) = d3r e−iG·rf(r) Va Va (I.42) iG·h  1 =e 1 3 −iG·r (I.43) d r e = δG,0 Va Va (I.44) eiG·r = δ(r − h) G h also 1 −ik·h (I.39) e = δk,G =: Δ(k) N h G 1 iq·h (I.38) e = δh,0. N q∈BZ Problems 385 Problems

I.1. Fourier Transformation of a Rectangular Function. Given be a periodic function f(t) with the period length T , t1 t1 h − 2

(a) Sketch this function. (b) Determine the Fourier coefficients Fn of f(t), +∞ iωnt f(t)= Fne . n=−∞

T (c) Write down the real Fourier series for t1 = 2 . (d) How does the frequency spectrum change for t1  T ?

I.2. Fourier Series: Properties. For the functions f,g,h with the same period L andwiththeseries +∞ iknx f(x)= Fne n=−∞ and analogously for g and h one has: (a) Parseval’s equation  L +∞ 1 ∗ ∗ f (x)g(x)dx = Fn Gn T 0 n=−∞

(b) Convolution theorem (i) +∞ +∞ h(x)=f(x)g(x) ↔ Hn = FmGn−m = Fn−mGm m=−∞ m=−∞ (ii)   1 L 1 L Hn = FnGn ↔ h(x)= f(x − y) g(y)dy = f(y) g(x − y)dy L 0 L 0 Prove these claims. J Change of Variables: Legendre Transformation

Here the transition from the Lagrangian function to the Hamiltonian function will be performed, which results in a change of the variables, namely from the generalized velocitiesq ˙ to the canonical momenta p. Let f(x,y,...) be a function of multiple variables with # $ # $ # $ ∂f ∂f ∂f df = dx + dy + dz + ... ∂x y,z,... ∂y x,z,... ∂z x,y,... = X dx + Y dy + Z dz + ... with # $ ∂f X = = X(x,y,z,...) ⇒ x = x(X,y,z,...) ∂x y,z,... and analogously for y, z, etc. A Legendre transformation is a transformation of variables; for example the Legendre transformation with respect to the variable x has the form ∂f g(X,y,...)=f(x,y,...) − x = f − xX ∂x with x = x(X,y,...). Namely, then one has

dg =df − d(xX)=(X dx + Y dy + Z dz + ...) − (X dx − x dX) − = #x dX$+ Y dy + #Z dz $+ ... # $ ∂g ∂g ∂g = dX + dy + dz + ... ∂X y,z,... ∂y X,z,... ∂z X,y,... with # $ ∂g = −x ∂X y,z,... and # $ # $ ∂g ∂f Y = = ∂y X,z,... ∂y x,z,... 388 J Change of Variables: Legendre Transformation

etc. While x is one of the variables (in thermodynamics so-called natural variables) of the function f, X is one of the natural variables of the function g.

Comments: • Analogous results are obtained for the transformation from the coordinate pair y and Y etc. instead of the coordinate pair x and X. • The two consecutive Legendre transformation (of the same pair of coordi- nates) leads back to the original function,

∂g h = g − X = g + Xx = f. ∂X • dL H = L − pq,˙ p = . d˙q • Legendre transformations play a vital role in thermodynamics for the tran- sition of one thermodynamic potential to another. As an example of ther- modynamical potentials, the internal energy U and the free energy F are related to each other via the Legendre transformation F = U − TS,and for example compressible systems one has

dU(S, V )=T dS − p dV dF (T,V )=−S dT − p dV.

The canonical pairs of variables here are (T,S)and(p, V ), while in the Lagrangian theory the pair is (p, q˙). References

A. Textbooks on Classical Mechanics

There is a vast literature on classical mechanics. In the following a subjective selection of text books is given.

1. H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd edn. (Addison-Wesley, San Francisco, 2002) 638 pp. The first and even more so the second edition was the prime mechanics textbook with a vast range of contents. 2. W. Greiner, in Theoretical Physics, vols. 1 and 2. Mechanics, 6th edn. A text book with numerous worked-out examples. 3. L. D. Landau, I. M. Lifshitz, in Theoretical Physics vol. 1. Mechanics, The textbooks by Landau and Lifshitz always present unconventional and ex- citing views. The first volume starts with the Lagrangian formulation. 4. G. Ludwig, in Einf¨uhrung in the Grundlagen der Theoretischen Physik vol. 1. Raum, Zeit, Mechanik, 2. Aufl. (Vieweg, Darmstadt, 1978) 449 pp., out of print (in German). First of four volumes with the mathematical and philosophical treatment of theoretical physics. 5. F. Scheck, Mechanics (Springer, Berlin, 1999). A very modern book with treatments extending to deterministic chaos. 6. A. Sommerfeld, in Lectures on Theoretical Physics vol. I. Mechanics reprint (Harri Deutsch, Frankfurt a. M.)

B. Texbooks on Special Relativity

Concerning the literature on Special Relativity: A series of textbooks on me- chanics contains a chapter on Special Relativity theory (SR), in particular the books by Goldstein et al. [1], Greiner [2], and Scheck [5], given above. In addi- tion there is a separate Volume 3A on SR by Greiner and Rafelski [7]. A subjec- tive selection follows, see also the listing with Comments at the end of Chap. 7 (p. 332ff) in Goldstein [1]. 390 References

7. W. Greiner, J. Rafelski, Theoretische Physik Band 3A Special Relativity theory This is one of the volumes of the text book series, which I personally do not like particularly; there is material overlapping with that of other volumes (Vol. 1 Mechanics I,Vol.3Electrodynamics), such that this volume is thus an (occa- sionally interesting) collection; with problems and solutions. 8. W. Rindler, Essential Relativity, 2nd edn. (Springer, Berlin, 1979) DM 72, 284 pp. This is a very beautifully and clearly written book, in which Special (104 pp.) and General Relativity theory (180 pp.) is treated in a compact overview. 9. W. Rindler, Introduction to Special Relativity, 2nd edn. (Clarendon, Oxford, 1992) pp. 59–185. A somewhat more elementary version of the first part of the preceding book, it contains in addition continuum mechanics.

C. Special References

10. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). 11. I.S. Gradshteyn, I.M. Ryzhik (abbreviated by GR), Tables of Integrals, Series, and Products, 2nd edn. (Academic, New York, 1980). 12. G. Herzberg, Molecular Spectra and Molecular Structure II: Infrared and Raman Spectra of Polyatomic Molecules, (van Nostrand, New York, 1945). 13. E.L. Hill, Rev. Mod. Phys. 23, 253 (1951). 14. K. Jung, Figur der Erde, Handbook of Physics, vol. 17 (Springer, Heidelberg, 1956) p. 606. 15. E. Noether, Nachr. Akad. Wiss. G¨ottingen II, Math. P. Kl. 235 (1918). 16. H. W¨anke The earth in the planetary system, Handbook of Physics (New Series) vol. V/2a (1984), Fig. 5 (p. 29). Index

absolute space, 5 orbital, 240 absolute time, 5 plane polar coordinates, 45 as principle, 219 quantum mechanics, 171 absorption, oscillator, 136 rigid body, 230, 239 acceleration rough collision, 190 definition, 5 spin, 240 plane polar coordinates, 44 Steiner’s theorem, 239 action aphelion, 166 continuum formulation, 314 arc length, 336 definition, 92 areal velocity function, 296 conservation, 158, 164, 171 adjoint matrix, definition, 356 atomic mass unit (amu) advanced Green function, 134 numerical value, 327 agreement Atwood machine, 86, 104 conservative forces, 11 force of constraint, 87 constant mass, 7 modified, 105 del operator, 11 Avogadro number, numerical value, 326 reference system, 7 axial vector, 330 standard configuration, 219 definition, 15 time derivatives, 5 axiom, Newton’s, 207 velocity-independent forces, 13 axioms, Newton’s, 4, 56 amu, numerical value, 327 angular acceleration, 44 ballistic trajectory, 181 angular momentum basic experience conservation, 21, 157 charges in e.m. fields, 12 central force, 18 interacting masses, 10 central potential, 28 basis closed system, 17, 50 fixed and moving, 41 collision, 185 bead Kepler problem, 164 on rotating rod, 107 scattering, 185, 189, 198 bead on rotating rod, 59, 67, 99, 106 definition, 14 conserved Hamiltonian function, 99 equation of motion, 15 constraint, 70 392 Index

Lagrange, 73 catenary curve, 103 virtual displacement, 65 Cauchy integral theorem, 132, 368 behavior under transformation causality, 93, 133, 134 pseudo scalar, 330 center of mass, 14, 185, 228 pseudo tensor, 330 hemisphere, 233 tensor, 329 center-of-mass coordinate, 49 vector, 329 center-of-mass motion, 50, 77, 184 binormal, 336 center-of-mass system, 185 Bloch ansatz, 154 central collision, 189, 190 Bloch oscillation, 305 central force, 53, 157 body and central potential, 28 deformable, 309 and trajectory, 160 rigid, 309 angular-momentum conservation, 18 spherical, 243 field, definition, 9 symmetrical, 243 body-fixed coordinates, 253 motion in a plane, 41 Bohr magneton, numerical value, 327 scattering, 189 Bohr radius, numerical value, 327 central potential, 28, 157, 187 Boltzmann constant, numerical value, and trajectory, 160 326 angular-momentum conservation, 28 Born approximation, 197 characteristic function, 300 bound motion, 33, 157 definition, 12 bound state, 170 energy conservation, 28 brachistochrone, 102, 103 generalized quantities, 79 scattering, 189, 198 c.c., 119 centrifugal force, 45, 79, 214 calculus of residues, 131 centrifugal potential, 159 canonical equations centripetal force, 45 and conserved quantity, 304 cgs system, 10 Bloch oscillation, 305 CH4 molecule, 151 canonical equations of motion, 271, 276 chain, linear, 316 harmonic oscillator, 277 Chandler period, 260 modified Hamilton principle, 282 characteristic function, 299 pendulum, 272 central potential, 300 canonical momentum, 78, 81, 269 free particle, 299 charged particle, 81, 273 harmonic oscillator, 301 conservation, 96 charge density, electrical, 232 gauge transformation, 111, 274 charged particle canonical momentum density, 321 canonical system, 273 canonical momentum, 81, 273 canonical transformation, 282, 304 Hamiltonian function, 273 and time development, 287 Lagrangian function, 81, 273 definition, 284 closed system, 50 magnetic field, 286 angular-momentum conservation, 17 oscillator, 285 conservation of momentum, 17 phase-space volume, 294 conserved quantities, 50 cartesian coordinates, 333 definition, 2 Laplace operator, 339 CO2-molecule, 138 nabla operator, 339 collinear collision, 190 Index 393 collision, 33, 183, 184 conservative force, 18, 27, 68, 71, 72, 80 central, 189, 190 definition, 11 collinear, 190 one-dimensional, 32 eccentric, 190 conserved quantity, 17, 276 elastic, 190 in static electro-magnetic field, 28 definition, 187 constant of the motion, 17 hard spheres, 184, 190 constraint, 34 inelastic and generalized coordinates, 57 definition, 187 Atwood machine, 87 laboratory and center-of-mass system, dumbbell, 58 204 holonomic, 58, 71, 84, 86 oblique, 189, 190 motion with, 55 rough, 190 nonholonomic, 58 smooth, 190 pendulum, 61 Special Relativity theory, 189 skate, 60 collision normal, 190 pendulum, 34, 57, 59 commutator brackets, 278 reduction of the degrees of freedom, completeness 55, 61 eigen vectors, 146, 358 rheonomous, 58, 67, 271 Fourier series, 380 rigid body, 227 configuration space, 8, 61, 278 scleronomous, 58, 67 point transformation, 282 sliding bead, 70 conics, 165 spherical pendulum, 62 conjugate momentum, 78 wheel, 90 conservation continuity equation angular momentum, 21, 96, 100, 157 energy current density, 317 central force, 18 energy–momentum, 317 central potential, 28 continuum closed system, 17, 50 limit, 232 areal velocity, 158, 164, 171 rigid body, 232 canonical momentum, 96 continuum mechanics energy, 18, 21, 96–98 as field theory, 309 central potential, 28 continuum theory, 309 closed system, 50 convergence Hamiltonian function, 98 uniform, 378 bead, 99 convolution theorem momentum, 17, 21, 96, 100 Fourier integrals, 370 closed system, 17, 50 Fourier series, 380 collision, 185 oscillator, 131 scattering, 185, 186 coordinate line, 333 space homogeneity, 96 coordinate sheet, 333 strain–momentum, 323 coordinate systems, 333 string, 324 coordinate transformation conservation laws and symmetry, 23 infinitesimal rotation, 210 conservation of areal velocity, 158 Legendre, 98, 269, 387 conservation of parity moving, 207 space inversion, 96 rotation, 209 conservation theorems, 28, 317 translation, 208 394 Index coordinates cyclic coordinate, 96, 272, 296 as observables, 5 and Hamiltonian function, 272 body system, 253 cycloid pendulum, 152 cartesian, 333 cyclotron frequency, 287 Laplace operator, 339 cylinder nabla operator, 339 in hollow cylinder, 265 cyclic, 96, 272 on inclined plane, 265 cylindrical, 335 cylindrical coordinates, 335 Laplace operator, 340 Laplace operator, 340 nabla operator, 340 nabla operator, 340 generalized, 57, 62, 278 continuum, 310 d’Alembert’s principle, 67–70, 92 moving, 83, 207, 283 decay, nuclear, 187 rotation, 207 deflection towards east, 223 spherical, 334 deformation, elastic, 309 Laplace operator, 340 deformation, plastic, 309 nabla operator, 340 degree of freedom, 61 translation, 207 delta function, Dirac, 361 R3 Coriolis force, 6, 45, 79, 214 in , 363 integral representation, 363 meteorology, 224 periodic, 363 on earth, 214, 224 determinism, 279 corpuscular nature of matter, 5, 232 quantum mechanics, 279 Coulomb force, 5 differential cross-section, 197, 198 Coulomb friction, 13, 53 Rutherford, 200 Coulomb interaction, 6 differential equation, linear, 113, 351 Coulomb potential, 184 eigen solution, 352 Fourier transformation, 375 Dirac δ function, 361 coupled pendulums, 150 displacement field, 310 coupled vibrations, 138, 145, 151 displacement, virtual, 64 (in)stability, 141 dissipation function coupling, minimal, 274 free case with friction, 111 crank, 109 distribution, 361 cross product, 331 double pendulum, 62 cross-section, 185, 195, 196 Lagrange, 74 differential, 197, 198 double star, 179 Rutherford, 200 dumbbell sphere, 201 constraint, 58 total, 200 dyadic product, 331 Rutherford, 201 dynamical momentum, 81 crystal, as oscillator, 115 dynamics, 5, 25 crystallography, 383 cube E¨otv¨os, 9 inertia tensor, 244 earth with obstacle, 267 as top, 260 cuboid Coriolis force, 214, 224 inertia tensor, 236, 237, 241 rotating, 223 Steiner’s theorem, 241 earth mass moment of inertia, 264 numerical value, 325 Index 395 earth radius energy density, 310, 311, 318 numerical value, 325 potential, 312 eccentric collision, 190 string, 312, 313 eccentricity, conic, 165 energy dissipation, oscillator, 134 effective potential, 159 energy–current density, 319 eigen angular momentum, 240 energy–momentum tensor, 317 eigen solution continuity equation, 317 CO2-molecule, 140 energy current density, 320 coupled vibrations, 146 equation of motion inertia tensor, 243 canonical, 271, 276 linear differential equation, 352 pendulum, 272 matrix, 355 Euler, 257 oscillator, 118, 121 Hamiltonian, 269, 271 eigen vector Lagrangian, first kind, 84 orthogonality, 143 Lagrangian, second kind, 70 properties, 358 Newtonian, 4, 56, 71 observable, 276 Einstein pendulum, 36 equivalence principle, 8 mathematical, 116 summation convention, 288, 311 Poisson bracket, 276 Einstein mechanics total angular momentum, 15 unified, 222 total energy, 16 elastic collision, 190 total momentum, 15 elastic deformation, 309 equivalence principle, 8, 9 electro-magnetic field, 9, 109 Euklidian space, 5 particle in –, 80, 273 Euler angles, 254, 255 static, conserved quantity, 28 Euler equation, 92 electron mass, numerical value, 327 variational method, 348 elementary charge, numerical value, 326 Euler equations of motion, 257 energy Euler period, 260 equation of motion, 16 Euler–Lagrangian equations, 71 inner, 14 external force kinetic, 14 definition, 2 mass distribution, 311 extremum principle, 55 rigid body, 230 potential, 11, 18 field, 309 energy conservation, 18, 21, 98, 320 definition, 9 central potential, 28 displacement –, 310 closed system, 50 electro-magnetic, 9 collision, 185 fine-structure const., num. value, 327 in magnetic field, 28 force scattering, 185 centrifugal, 45, 214 string, 320 centripetal, 45 time homogeneity, 96 conservative, 11, 18, 27, 68, 71, 72, 80 energy current density Coriolis, 6, 45, 79, 214 continuity equation, 317 on earth, 214, 224 energy–momentum tensor, 320 Coulomb, 5 string, 319 definition, 8 396 Index

external rectangular function, 385 definition, 2 Yukawa potential, 375 frictional, 20, 64, 72 free fall Newtonian, 13 from large height, 52 Stokes, 13 into center, 180 generalized (type I), 68 with friction, 111 generalized (type II), 78 free motion, 33, 52 gravitational, 5, 6 free particle impulsive, 184 characteristic function, 299 inertial, 5, 45, 208, 209, 214 friction, 53, 305 inner Coulomb, 13, 53 definition, 2 driven oscillator, 155 Lorentz-, 12 Newton, 53 Lagrange, 80 sliding, 13 of constraint, 34, 56, 63, 66 static, 13 Atwood machine, 87 Stokes, 53, 120 Lagrangian equations, first kind, 84 friction coefficient, 13 pendulum, 34 frictional force, 20, 64 rigid body, 227 Lagrange, 72 virtual work, 66 Newtonian, 13 wheel, 91 Stokes, 13 rotation-free, 18 frictional heat, 135 velocity dependent, 13, 80 oscillator, 134 velocity-dependent, 31 functional, 347 Lorentz, 80 action as –, 93 force density, 312, 322 definition, 92 force field, 312 FWHM, 137 form (in)variance Newton equations, 56, 83 Galilean invariance, 220 form invariance Galilean transformation, 84, 217–219 Galilean, 7 acceleration, 213 Lagrangian equations, 56, 83 and Lorentz transformation, 217 Lagrangian function, 81 as limit, 222 Foucault pendulum, 214 center of mass conservation, 96 four-gradient, 318 form invariance, 7 Fourier coefficients general, 210 periodic structures, 22 special, 210, 219 Fourier integral, 384 velocity, 212 Fourier series, 376, 384 gas constant, numerical value, 326 completeness, 380 gauge invariance convolution theorem, 385 Lagrangian function, 81 orthogonality, 380 gauge transformation, 81 Parseval’s equation, 382, 385 canonical momentum, 111, 274 Fourier transformation, 113, 365 Hamiltonian function, 274 Coulomb potential, 375 Lagrangian function, 82, 274 forced oscillator, 126 Gauß function, 363 functions with lattice periodicity, 383 Fourier transformation, 369 lattice, 382 uncertainty relation, 373 phase-space volume, 295 Gauß, integral theorem, 342 Index 397 general Galilean transformation, 210 charged particle, 112, 273 General relativity theory, 9, 175, 218 conservation, 98 general transformation bead, 99 Galilean and Lorentzian, 218 cyclic variable, 272 generalized coordinate, 57, 62, 278 gauge transformation, 274 continuum, 310 pendulum, 270 generalized force (type I), 68 separation, 273 generalized force (type II), 78 Hamiltonian mechanics, 78, 269 generalized momentum, 78 hard-core potential, 184 generalized potential, 80 harmonic oscillator, see oscillator, 113 generalized susceptibility, 136 heat, frictional, 135 generating function, 290, 297 Heaviside step function, 131, 361 harmonic oscillator, 291 Heisenberg uncertainty relation, 232 geosynchronous satellite, 181 hemisphere, center of mass, 233 Gibbs’ phenomenon, 378 hermitian matrix, definition, 356 gradient holonomic constraint, 58, 71, 84, 86 four-, 318 homogeneity gravitational acceleration in time, 21 numerical value, 325 energy conservation, 96, 97 gravitational constant, 10, 165 space, 21 numerical value, 325 conservation of momentum, 96, 100 gravitational force, 5, 6 homogeneous function, 109 gravitational mass, 8 Hooke’s law, 113, 114 gravitational potential, 12, 165, 184 Huygens, 152 perturbation of the, 175 scattering, 195 impact parameter, 189 Green function, 113, 352 inclined plane, 53, 251, 263 advanced, 134 moving, 88, 108 Laplace operator, 341 force of constraint, 89 oscillator, 127, 129, 131 with cylinder, 265 poles, 130 with mass, 108 retarded, 133 inertia moment, 248 Green, integral theorem, 343 inertia tensor, 229, 231 cube, 244 cuboid, 236, 237, 241 half width Steiner’s theorem, 241 absorption, oscillator, 137 pyramid, 263 Hamilton principle, 92, 93, 298, 314 sphere, 234, 242, 264 Lagrangian density, 314 Steiner’s theorem, 242 modified, 281 Steiner’s theorem, 238 Hamilton–Jacobi equation, 296, 297 inertial force, 5, 45, 69, 79, 207–209, 214 time independent, 299 inertial mass, 8 Hamiltonian density, 310 inertial system, 5, 83, 84, 207, 208, 210, continuity equation, 317 218 string, 319 definition, 4 Hamiltonian equations of motion, 271 examples, 6 Hamiltonian function, 97, 269 inertial systems as Legendre transformation, 269, 387 transformation, 222 398 Index inner product laboratory system, 185 definition, 331 Lagrange multiplier, 64, 84 instability Lagrangian density, 96, 310, 311, 314 coupled oscillators, 141 Hamilton principle, 314 definition, 353 string, 314, 316 symmetrical top, 260 Lagrangian equations, first kind, 84, 85 integral of the motion, 17, 23 Lagrangian equations, second kind, 67, integral theorem 70, 71, 94 Cauchy, 368 form invariance, 83 Gauß, 342 Lagrangian function, 71, 269 Green, 343 charged particle, 81, 273 Stokes, 343 CO2 molecule, 138 internal energy, 14 form invariance, 81 internal force gauge invariance, 81 definition, 2 gauge transformation, 82, 274 invariance separability, 76 Galilean, 7, 220 top, 268 lattice translation, 22 Lagrangian Mechanics, 55 space rotations, 21, 96 Landau levels, 287 space translation, 21, 96 Laplace operator time translation, 21, 96 cartesian coordinates, 339 invariance and conservation quantity, 28 cylindrical coordinates, 340 isotropy, space polar coordinates, 340 angular-momentum conservation, 21, spherical coordinates, 340 96 Laplace–Runge–Lenz vector, 173 lattice translation invariance, 22 Kepler lattice vibrations laws, 163, 178, 179 absorption, 136 and force, 178 Laue, 22 double star, 179 law of inertia, 4 first, 163 law of motion, 4 second, 164 Legendre transformation, 98, 269, 387 third law, 164, 167 thermodynamics, 388 as approximation, 172 length contraction, 5, 222 Kepler problem, 157 Lenz vector, 173, 182 angular-momentum conservation, 164 line element trajectory, 281 cartesian coord., 333 Kepler trajectories cylindrical coordinates, 335 polar representation, 165 spherical coordinates, 334, 336 kinematics, 5, 25 line integral, 18, 345 kinetic energy closed, 18 definition, 14 linear chain, 316 mass distribution, 311 diatomic, 153 relativistic, 69 monatomic, 154 rigid body, 230 Liouville, theorem of –, 295 kinetic momentum, 81 local trihedron charged particle, 273 spiral line, 336 Index 399

Lorentz force, 12 Maxwell theory, 218 in mechanics and electrodynamics, unified, 222 222 mean value, in time Lagrange, 80 definition, 23 Lorentz function, 136, 363 virial theorem, 23 Fourier transformation, 367 mechanical similarity, 172 Lorentz transformation, 96, 217, 218, meteorology and Coriolis force, 224 221 minimal coupling, 81, 274 and Galilean transformation, 217 model, oscillator as –, 114 linearity, 221 molecular crystal, 117 special, 221 molecule CH4, 151 magnetic field diatomic, 150 canonical transformation, 286 molecule, as oscillator, 115 particle in –, 80, 273 moment of inertia magneton, Bohr cuboid, 264 numerical value, 327 momentum many-body system, 27 canonical, 78, 81, 269 mass gauge transformation, 111 electron conjugate, 78 numerical value, 327 definition, 14 gravitational, 8 dynamical, 81 in the electromagnetic field, 109 equation of motion, 15 inertial, 8 generalized, 78 on a cone, 109 in electromagnetic field, 81 on a sphere, 110 kinetic, 81 on inclined plane, 108 quasi-, 22 on logarithmic spiral, 106 strain–, 320 on parabola, 106 uncertainty relation, 374 on rotation paraboloid, 111 momentum density, 310, 322 on spiral, 106 canonical, 321 reduced, 49, 157 string, 321 mass density, 5, 232, 310 moon mass mathematical pendulum, 116 numerical value, 325 period, 306 moon radius matrix numerical value, 325 adjoint, definition, 356 motion definition, 356 bound, 33 eigen solutions, 113 free, 33, 183 eigen value problem, 355 in a plane, 41, 157 hermitian, definition, 356 unbound, 33, 183 orthogonal, definition, 356 real, definition, 356 nabla operator self-adjoint, definition, 356 cartesian coordinates, 339 skew-symmetric, definition, 356 cylindrical coordinates, 340 symmetric, definition, 356 grad, div, rot, 344 transposed, definition, 356 polar coordinates, 340 unitary, definition, 356 spherical coordinates, 340 400 Index

Newton coupled, 138, 151 axiom I, 207 (in)stability, 141 axiom II, 56 crystal, 115 axioms, 4, 92 damped, 120 equation of motion, 71 driven first law, 55 with friction, 155 Newtonian equations eigen solutions, 118, 121 form (in)variance, 83 energy dissipation, 134 Newtonian friction, 13, 53 forced, 125 Newtonian mechanics, 222 generating function, 291 Noether’s theorem, 20, 95 Green function, 127, 129, 131 invariances, 95 half width, 137 non-holonomic constraint harmonic, 113, 149 skate, 60 molecule, 115 nonholonomic constraint, 58 overdamped case, 124 pendulum, 61 phase portrait, 280 normal coordinate, 147, 149 power, 135 normal vibration, 147 RLC circuit, 116 definition, 143 simple, 117 nuclear excitation, 203 three-dimensional, 115 trajectory, 280 with friction, 305 oblique collision, 189, 190 oscillator, harmonic observable trajectory, 280 definition, 276 overdamped case (damped oscillator), equation of motion, 276 124 open system, 51 definition, 2 parity operator identity, 213 conservation of, 96 optics normal vibration, 142, 144 phase-space volume, 295 Parseval’s equation, 135 orbit, 280 Fourier integrals, 371 orbital angular momentum, 240 Fourier series, 382 orthogonal matrix, definition, 356 Parseval’s theorem orthogonality quantum mechanics, 372 eigen vectors, 143, 146, 358 particle decay, 203 Fourier series, 380 pendulum, 34 plane waves, 380 canonical equations of motion, 272 oscillation constraint, 34, 57, 59 period of the general, 34 force of constraint, 46 transient, 135 forces, 34 oscillator Foucault, 214 n-dimensional, 149 Hamiltonian function, 270 absorption, 136 inertial force, 46 half width, 137 large amplitude, 38 as model, 114 mathematical, 116 canonical equations, 277 period, 306 canonical transformation, 285 physical, 249 characteristic function, 301 plane, 53 Index 401

spherical properties, 277 constraint, 62 quantum mechanics trajectory, 281, 305 commutator, 277 with mobile suspension, 152 Poisson equation, 364 pendulums polar coordinates coupled, 150 Laplace operator, 340 perihelion, 166, 194 moving basis, 41 conserved quantity, 173 nabla operator, 340 rotation, 163, 171, 175, 181 plane, 335 period acceleration, 44 Chandler, 260 velocity, 44 Euler, 260 spherical, 334 general oscillation, 34 polar representation periodic delta function, 363 Kepler trajectories, 165 phase portrait polar vector, 330 oscillator, 280 definition, 15 phase space, 278–280 pole motion of the earth, 260 point transformation, 283 position phase-space volume uncertainty relation, 374 canonical invariance, 294 potential, 12 and trajectory, 178, 180 optics, 295 central, 12 quantum mechanics, 295 Coulomb, 184 time invariance, 295 effective, 159 physical pendulum, 249 from trajectory, 178, 179 Planck constant, numerical value, 327 generalized, 80 plane motion gravitational, 165, 184 Lagrange, 72 hard-core, 184 plane pendulum, 53 scalar, 80 plane polar coordinates, 335 vector, 80 acceleration, 44 potential energy, 11, 18 velocity, 44 density, 312 plane waves potential step orthogonality, 380 scattering at a –, 205 planetary motion, 157 power, 16 plastic deformation, 309 damped oscillator, 135 point mass, 5, 17 definition, 14, 19 as approximation, 1 principal axes, 231, 243 point masses and rigid body, 227 principal moments of inertia, 243 point transformation and (un)stable rotation, 266 configuration space, 83, 282 circular cone, 264 phase space, 283 cube, 264 Poisson brackets, 304 cylinder, 264 and quantum mechanics, 278 sphere, 264 canonical invariance, 288 tetrahedral molecule, 264 definition, 275 principal normal, 336 equation of motion, 276 principle fundamental, 277 absolute time, 219 problem, 304 d’Alembert, 67–70, 92 402 Index

Hamilton, 92, 93 collision, 189 Lagrangian density, 314 kinetic energy, 69 relativity, 218, 221 representation virtual work, 66 different coordinate systems, 211 process, 279 acceleration, 213 product position, 212 cross, 331 velocity, 212 dyadic, 331 response function tensor, 331 oscillator, 127, 129, 131 triple scalar, 332 rest mass, 6 vector, 331 retarded Green function, 133 pseudo scalar rheonomous constraint, 58, 67 behavior under transformation, 330 rigid body, 227, 309 pseudo tensor angular momentum, 230, 239 behavior under transformation, 330 constraints, 227 pseudo vector, 330 continuum, 232 definition, 15 force of constraint, 227 pyramid kinetic energy, 230 inertia tensor, 263 torque, 239 RLC circuit as oscillator, 116 quantization, 278 rocket, 54 quantization of vibrations, 148 rolling pendulum, 266 quantum mechanics, 78, 114, 117, 120, rotating pendulum, 107 232, 274, 278 stability, 108 and Poisson brackets, 278 rotation, 210 angular momentum, 171 (un)stable, 266 commutator, 277 coordinates, 207 determinism, 279 fixed axis, 247 magnetic field, 287 fixed point, 253 Parseval’s theorem, 372 space, 20 phase-space volume, 295 with friction, 266 spin, 240 rotation vector, 21, 247 tunneling effect, 34 rotational motion, 28 quarks, 5 stability of the –, 260 quasi-momentum, 22 rough collision, 190 Runge–Lenz vector, 173 radius, Bohr Rutherford scattering, 184, 205 numerical value, 327 cross-section, 200 reaction law, 4 scattering angle, 195 real matrix, definition, 356 Rydberg energy, numerical value, 327 reduced mass, 49, 157 reference frame, moving, 207 satellite, geosynchronous, 181 relative coordinate, 49 scalar, 329 relative motion, 50, 77, 184 scalar field relativity principle, 218, 221 definition, 9 Relativity theory scalar potential, 80 General, 9, 175, 218 scalar product, 356 Special, 5, 6, 96, 218, 221 scattering, 33, 183, 184 capture reaction, 187 at a potential step., 205 Index 403

central potential, 198 space, Euklidian, 5 elastic, 188 special Galilean transformation, 210 definition, 187 special Lorentz transformation, 221 from reflecting sphere, 204 Special Relativity theory, 5, 6, 96, 218, gravitational potential, 195 221 inelastic, 203 capture reaction, 187 definition, 187 collision, 189 Rutherford, 184, 205 kinetic energy, 69 with nuclear excitation, 203 special transformation scattering angle Galilean and Lorentzian, 218 definition, 194 sphere Rutherford scattering, 195 inertia tensor, 234, 242, 264 scattering state, 170 Steiner’s theorem, 242 scleronomous constraint, 58, 67 rolling and sliding, 266 secular determinant, 140 scattering from –, 204 secular equation, 118, 121 spherical body, 243 self-adjoint matrix, definition, 356 spherical coordinates, 334 separation Laplace operator, 340 Hamiltonian function, 273 line element, 336 Lagrangian function, 76 nabla operator, 340 separation of variables, 31 spherical top, 244 separatrix, 305 spin angular momentum, 240, 256 SI system, 10 spiral line similarity, mechanical, 172 local trihedron, 336 skate, 60, 90 spool constraint, 60 with thread, 264 skew-symmetric matrix, definition, 356 spring constant, 114 sliding friction, 13 stability smooth collision, 190 coupled oscillators, 141 soccer ball, 266 solar equatorial radius definition, 353 numerical value, 325 rotating pendulum, 108 solar mass rotation, 260 numerical value, 325 top, 268 solid-state physics, 22, 365, 374, 376 standard configuration, 219 sound velocity, string, 316 state space bound, 170 absolute, 5 definition, 278 configuration, 8 scattering, 170 space homogeneity static friction, 13 conservation of momentum, 96, 100 statistical mechanics, 113 space inversion Steiner’s theorem, 231, 238, 249 angular-momentum conservation, 100 angular momentum, 239 conservation of parity, 96 cuboid, 241 space isotropy sphere, 242 angular-momentum conservation, torque, 239 96, 100 step function, Heaviside, 131, 361 space rotation, 20 Stokes space translation, 20 friction, 53, 120 404 Index

frictional force, 13 target, 185, 197 integral theorem, 343 tensor, 329 strain momentum behavior under transformation, 329 conservation, 323 definition, 329 string, 324 energy-momentum, 317 strain–momentum density, 320 inertia, 229 stress, 320 stress, 322 stress tensor, 322 tensor product, 331 string, 323 theorem string, 309 Noether, 20 energy conservation, 320 Parseval energy current density, 319 Fourier integrals, 371 energy density, 312, 313 Fourier series, 382 Hamiltonian density, 319 thermodynamics, 20, 279, 388 Lagrangian density, 314, 316 Legendre transformation, 388 momentum density, 321 throw sound velocity, 316 inclined, 53 strain–momentum conservation, 324 vertical, 52 stress tensor, 323 time summation convention, Einstein, 288, absolute, 5 311 as principle, 219 sun radius development numerical value, 325 as canonical transformation, 287 superposition principle, 4 dilation, 5, 222 surface element homogeneity cartesian coord., 334 energy conservation, 96, 97 translation, 20 cylindrical coordinates, 335 top, 243, 253 spherical coordinates, 334 earth as, 260 susceptibility, generalized, 136 Lagrangian function, 268 symmetrical body, 243 spherical, 244 symmetrical matrix, definition, 356 stability, 268 symmetrical top, 258 symmetrical, 258 (in)stability, 260 (in)stability, 260 symmetry and conservation laws, 23 torque, 15, 18, 27 system rigid body, 239 canonical, 273 Steiner’s theorem, 239 closed, 50 total angular momentum angular-momentum definition, 14 conservation, 17 total cross-section, 200 conservation of momentum, 17 Rutherford, 201 conservation quantities, 50 total energy definition, 2 definition, 14 open, 51 total momentum definition, 2 definition, 14 trajectory, 280, 336 tangent unit vector, 336 and potential, 178, 179 tangential space, 280 ballistic, 181 tangential vector, 333 from potential, 178, 180 Index 405

harmonic oscillator, 280 axial, 330 Kepler problem, 281 definition, 15 pendulum, 281, 305 behavior under transformation, 329 trajectory and central force, 160 decomposition in orthog. coord., 336 trajectory and central potential, 160 polar, 330 trajectory of a particle, 5 definition, 15 transformation pseudo, 330 canonical, 282 definition, 15 Galilean, 84, 217–219 Runge–Lenz, 173 general vector field Galilean and Lorentzian, 218 definition, 9 Legendre, 98, 269, 387 div, rot, 344 Lorentz, 96, 217, 218, 221 vector potential, 80 special vector product, 331 Galilean and Lorentzian, 218 velocity transient oscillation, 135 definition, 5 translation plane polar coordinates, 44 coordinates, 207 velocity dependent force, 80 in time, 20 velocity of light, 10, 218 space, 20 as SR principle, 221 transposed matrix, definition, 356 numerical value, 326 triple scalar product, 332 velocity-dependent force, 31 tunneling effect, 34, 114 Lorentz, 80 turning points, 33 vibrations, coupled, 145 two-particle interaction, 6 virial theorem, 23, 28 two-particle system, 48 virtual displacement, 64 definition, 64 uncertainty relation, 372 virtual work, 99 Gauß function, 373 of the force of constraint, 66 Heisenberg, 232 principle of –, 66 momentum, 374 Volterra differential equation, 306 position, 374 volume element uniform convergence, 378 cartesian coord., 334 unit system, electromagnetism, 10 cylindrical coordinates, 335 unitary matrix, definition, 356 spherical coordinates, 334 universe, 54 wheel, 90 vacuum diel. constant, num. value, 326 constraint, 90 vacuum permeability, numerical value, force of constraint, 91 326 wheel and axis, 105 variational method, 71, 347 work brachistochrone, 102, 103 definition, 14, 19 catenary curve, 103 virtual, 66, 99 Dido’s problem, 350 world line, 8 Euler equation, 348 variational principle, 310 Yukawa potential vector, 329 Fourier transformation, 375