1 Survey of Elementary Principles

1 Survey of elementary principles

Some history: 1600: Galileo Galilei 1564 – 1642 cf. section 7.0 ∼ Johannes Kepler 1571 – 1630 cf. section 3.7 1700: IsaacNewton 1643–1727 cf. section 1.1 ∼ 1750–1800: LeonhardEuler 1707–1783 cf. section 1.4 ∼ JeanLeRondd’Alembert 1717–1783 cf. section 1.4 Joseph-LouisLagrange 1736–1813 cf. section 1.4, 2.3 1850: CarlGustavJacobJacobi 1804–1851 cf. section 10.1 ∼ William Rowan Hamilton 1805 – 1865 cf. section 2.1 Joseph Liouville 1809 – 1882 cf. section 9.9 1900: Albert Einstein 1879 – 1955 cf. section 7.1 ∼ Emmy Amalie Noether 1882 – 1935 cf. section 8.2 1950: Vladimir Igorevich Arnold 1937 – 2010 cf. section 11.2 ≥ Alexandre Aleksandrovich Kirillov 1936 – BertramKostant 1928– Jean-MarieSouriau 1922–2012 JerroldEldonMarsden 1942–2010 Alan David Weinstein 1943 –

FYGB08 – HT14 1 2014-11-24 1.1 Mechanics of a particle

1.1 Concepts: space, time kinematics, dynamics, statics coordinate system, reference frame, inertial frame, Galilean frame position, velocity, acceleration mass point, point mass inertial mass, gravitational mass, rest mass momentum, angular momentum force, torque, force ﬁeld work, kinetic energy, conservative force, friction simply connected region, curl-free ﬁeld potential energy, potential, total energy conservation law, conserved quantity, conserved charge Results: Newton’s second law conservation of momentum conservation of angular momentum conservation of total energy Formulas: (1.3) F~ = ~p˙ (1.12) W (P)= F~ d~s 12 P · (1.16) F~ (~r)= ~R V (~r) −∇ Elementary fact: Physics is independent of the choice of coordinate system.

Warnings: j Do not mix up the notions ‘frame’ and ‘coordinate system’. j There are systems with F~ = ~ V but V time-dependent. −∇

FYGB08 – HT14 2 2014-11-24 1.2 Mechanics of a system of particles

1.2 Concepts: internal and external forces distance vector center of mass strong law of action and reaction mechanical equilibrium Results: Newton’s third law (weak law of action and reaction) for conservative forces depending only on distance center of mass motion conservation of total momentum conservation of total angular momentum conservation of total energy angular momentum as sum of c.m. term and term for relative motion kinetic energy as sum of c.m. term and term for relative motion Formulas: ˙ ~ ext ~ (1.19) ~pi = Fi + j Fji 1 ~ ~ (1.20 2 ) Fij = Fji P 2 − d ~ ~ ext (1.22) M dt2 R = F d ~ ~ ext (1.26) dt L = N ′ ~ (1.27) ~ri = ~ri + R (1.28) L~ = R~ M ~v + ~r′ ~p′ × i i × i 1 2 1 ′2 (1.31) T = 2 M v + 2 P i mi vi P

FYGB08 – HT14 3 2014-11-24 1.3 Constraints

1.3 Concepts: dynamical and non-dynamical parts of a system constraint holonomic and non-holonomic constraints, semi-holonomic constraints skleronomic and rheonomic constraints independent dynamical variables, generalized coordinate, degree of freedom constraint force, applied force Results: presence of constraints = particle positions no longer independent ⇒ constraint forces are usually not known explicitly Formulas : (1.37) fκ = fκ(~r1, ~r2, ... ; t)=0

(1.38) ~ri = ~ri(q1, q2, ... , qN◦ ; t) q = q (~r , ~r , ... , ~r ; t) (holonomic constraints) − j j 1 2 N Examples: rigid body bead sliding on a wire disk rolling on a plane

Warnings: j A generalized coordinate need not have dimension of length.

FYGB08 – HT14 4 2014-11-24 1.4 D’Alembert’s principle and Euler--Lagrange equations

1.4 Concepts: virtual displacement, virtual work eﬀective force generalized velocity generalized force Lagrangian total derivative Results: principle of virtual work, d’Alemberts principle ‘cancellation of dots’ rule for holonomic constraints (Euler-)Lagrange equations of motion

Formulas: (1.43) F~ appl δ~r =0 i · i i X (1.45) (F~ appl ~p˙ ) δ~r =0 i − i · i i X ∂~r (1.49) Q = F~ i j i · ∂q i j X ∂~v ∂~r (1.51) i = i ∂q˙j ∂qj d ∂T ∂T (1.52) [ Qj] δqj =0 dt ∂q˙ − ∂q − j j j X d ∂T ∂T (1.53) = Qj dt ∂q˙j − ∂qj ∂V (1.54) Qj = ∂qj (1.56) L = T V − d ∂L ∂L (1.57) =0 dt ∂q˙j − ∂qj

Warnings: j The notation F~ is often used for the applied rather than the total force. j Generically the kinetic energy depends also on the generalized coordinates, not only on the generalized velocities.

FYGB08 – HT14 5 2014-11-24 1.5 Velocity dependent potentials and the dissipation function

1.5 Concepts: generalized potential, velocity dependent potential monogenic system Rayleigh’s dissipation function Results:— Formulas: ∂U d ∂U (1.58) Qj = + −∂qj dt ∂q˙j (1.62) U = eφ e A~ ~v e.m. − · (1.68) F~ fric = ~ −∇~vF ∂ (1.69) Qj = F −∂q˙j d ∂L ∂L ∂ (1.70) + F =0 dt ∂q˙j − ∂qj ∂q˙j

Examples: charged particle in an electric and magnetic ﬁeld frictional drag force on a sphere (Stokes’s law)

1.6 Simple applications of the Lagrangian formulation

1.6 Examples: particle in free space particle in free space, in cylindrical coordinates Atwood’s machine bead sliding on a uniformly rotating wire

FYGB08 – HT14 6 2014-11-24 2 Variational Principles and Lagrange’s Equations

2.1 Hamilton’s principle

2.1 Concepts: integral principle functional, extremum of a functional conﬁguration space, path in conﬁguration space action, action integral Hamilton’s principle Results:— Formulas: t2 (2.1) S = S(t1, t2)= L dt t1 t2 Z (2.2) δS[q] δ L(q, q,˙ t) dt =0 ≡ Zt1

Warnings: j Two distinct meanings of “q”: as a coordinate of conﬁguration space as a path q = q(t) in conﬁguration space

FYGB08 – HT14 7 2014-11-24 2.2 Calculus of variations

2.2 Concepts: functional, stationarity condition calculus of variations, independent and dependent variables path, neighboring paths, one-parameter family of paths boundary condition catenary, brachistochrone problem Results:— Formulas: x2 (2.3) J[y]= f(y, y,˙ x) dx Zx1 x2 ∂f d ∂f ∂y (2.9) ( ) dx =0 ∂y − dx ∂y˙ ∂α Zx1 α=0

Examples : shortest path between two points in a plane minimum area of a surface of revolution brachistochrone problem

2.3 Derivation of Lagrange’s equations from Hamilton’s principle

2.3 Results: Euler--Lagrange equations of motion derived from Hamilton’s principle – for monogenic systems with holonomic constraints

FYGB08 – HT14 8 2014-11-24 2.4 Lagrange multipliers

2.4 Concepts: Lagrange multiplier semi-holonomic constraints Results: Formulas: t2 m (2.21) δ t1 (L + s=1 λs fs) dt =0

dR ∂L ∂LP m ∂fs (2.23) = Q˜j := λs dt ∂q˙j − ∂qj − s=1 ∂qj P Examples: Hoop rolling down an inclined plane

Warnings: j Lagrange multipliers are directly applicable only in the case of holonomic systems

Warning This section has been rewritten completely in newer versions of the book

2.5 Advantages of a variational principle formulation

2.5 Concepts: resistor, inductor, capacitor battery, electromotive force Results: electric-circuit analogues of mechanical quantities Formulas 1 : (2.42) j q¨j + jk q¨k + Rj q˙j + qj = j(t) L L Cj V k Xj=6 k Examples: battery in series with a resistance and an inductance inductance in series with a capacitance, as analogue of the simple harmonic oscillator

FYGB08 – HT14 9 2014-11-24 2.6 Conservation laws and symmetry properties

2.6 Concepts: initial conditions ﬁrst integral of equations of motion generalized momentum, canonical momentum, conjugate momentum cyclic coordinate, ignorable coordinate translational symmetry, rotational symmetry Results: conservation of the generalized momentum for a cyclic coordinate Formulas: ∂L (2.44) pj = ∂q˙j

Examples: system of charged particles in an electromagnetic ﬁeld uniform translation of a system uniform rotation of a system about a prescribed axis

Warnings: j even when pj has dimension of momentum, it does not necessarily coincide with ordinary mechanical momentum

FYGB08 – HT14 10 2014-11-24 3 The central force problem

3.1 Reduction to the equivalent one-body problem

3.1 Concepts: two-body problem interaction potential reduced mass Results: conservation of total momentum reduction of a two-body to a one-body problem Formulas: ′ m2 ′ m1 (3.2) ~r1 = ~r, ~r2 = ~r −m1 + m2 m1 + m2 ˙ (3.3) L = 1 M R~ 2 + 1 µ ~r˙ 2 U(~r, ~r,˙ ...) 2 2 − 1 1 1 (3.5) = + µ m1 m2

FYGB08 – HT14 11 2014-11-24 3.2 The equations of motion and ﬁrst integrals

3.2 Concepts: central potential, central force spherical symmetry areal velocity planar polar coordinates quadrature Results: conservation of angular momentum, implying motion in a plane Kepler’s second law, resulting from angular momentum conservation energy conservation, yielding t = t(r) reduction of the central force problem to two quadratures Formulas: (3.8) ℓ = mr2 θ˙ dV ℓ2 (3.12) f(r) = m r¨ ≡ − dr − mr3 dr (3.17) dt = 2 ℓ2 (E V 2 ) m − − 2mr q ℓ dt (3.19) dθ = mr2

FYGB08 – HT14 12 2014-11-24 3.4 The virial theorem

3.4 Concepts: virial ideal gas, equipartition, pressure, Boltzmann constant Results: virial theorem for periodic motion ideal gas law Formulas: d (3.24) ~p ~r =2 T + F~ ~r dt i · i i · i i i X X 1 (3.26) T = F~ ~r −2 i · i 1 XdV (3.28) T = r 2 dr n +1 (3.29) T = V 2

FYGB08 – HT14 13 2014-11-24 3.3 The equivalent one-dimensional problem, and classiﬁcation of orbits

3.3 Concepts: eﬀective potential, angular momentum barrier bounded and unbounded motion turning point

Results: qualitative form of possible orbits from graph for Veﬀ circular orbits when the energy is minimal Formulas: ℓ2 (3.22) V = V + eﬀ 2mr2

3.5 The diﬀerential equation for the orbit, and integrable power-law potentials

3.5 Concepts: orbit, orbit equation turning point elliptic functions Results: mirror symmetry of an orbit with at least one turning point orbits given by elementary functions for some power-law potentials Formulas: d ℓ d (3.32) = dt mr2 dθ d2u m d (3.34) + u = V ( 1 ) dθ2 −ℓ2 du u du (3.37) dθ = − 2mE 2mV 2 2 2 u ℓ − ℓ − q

FYGB08 – HT14 14 2014-11-24 3.6 Conditions for closed orbits (Bertrand’s theorem)

3.6 Concepts: closed orbit, periodic motion stability against small perturbations Results: Bertrand’s theorem: all bounded orbits closed 1/r or r2 potential ⇐⇒ Formulas: r df (3.47′) β2 =3+ f dr

3.7 The Kepler problem: Inverse-square law of force

3.7 Concepts: conic sections: ellipse, parabola, hyperbola eccentricity, focal point semiminor and semimajor axes, turning points, apsidal distances Results: orbits in the Kepler problem Kepler’s ﬁrst law

2 Formulas: 1 mk 2Eℓ ′ (3.55) u = = (1+ 1+ 2 cos(θ θ )) r ℓ2 mk − ◦ q 2Eℓ2 (3.57) e = 1+ 2 r mk k (3.61) a = −2E a (1 e2) (3.64) r = − 1+ e cos(θ θ˜ ) − ◦

FYGB08 – HT14 15 2014-11-24 3.8 The motion in time in the Kepler problem

3.8 Concepts: eccentric anomaly period Results: Kepler’s third law Kepler equation Formulas: ℓ3 θ dθ′ (3.66) t = 2 2 ′ mk θ0 [1 + e cos(θ θ˜ )] Z − ◦ 2π a3/2 (3.74) = T G (m1 + m2) (3.76) ω t = ψ e sin ψ p −

FYGB08 – HT14 16 2014-11-24 3.10 Scattering in a central force ﬁeld

3.10 Concepts: scattering, scattering angle, trajectory beam, ﬂux density solid angle, cross section, impact parameter, periapsis total cross section, long and short range potentials Rutherford scattering long range potential spiraling, rainbows, glory scattering Results: Rutherford cross section

Formulas n : (3.88) σ(Ω)~ dΩ = | | I (3.89) dΩ=2π sin ϑ dϑ (3.90) ℓ = s √2mE s ∂s (3.93) σ(ϑ)= sin ϑ ∂ϑ

∞ s dr (3.96) ϑ = π 2 − V (r) Zrmin r r2 s2 − E − k ϑq (3.101) s(ϑ)= 2E cot 2 k2 (3.102) σ(ϑ)= sin−4 ϑ , k = ZZ′e2 16 E2 2

FYGB08 – HT14 17 2014-11-24 3.11 Transformation of the scattering problem to laboratory coordinates

3.11 Concepts: recoil, laboratory frame scattering angles in laboratory and center of mass frames elastic and inelastic scattering, excitation energy Results: Formulas: µ v0 m1 v0 (3.108) ρ = ′ = m2 v1 m2 v cos ϑ + ρ (3.110) cos θ = 1+2ρ cos ϑ + ρ2

p 1+2ρ cos ϑ + ρ2 3/2 (3.116) σlab(θ)= σc.m.(ϑ) 1+ ρ cos ϑ θ = ϑ and θ = π for ρ =1 − 2 max 2 σ (θ) = 4 cos θ σ (2θ) for ρ =1 − lab c.m.

3.9 The Laplace-Runge-Lentz vector

3.9 Concepts: (Laplace-) Runge-Lentz vector Results: Conservation of the RL vector purely algebraic solution of the orbit equation Formulas: ~r (3.82) A~ =~p L~ mk × − r

FYGB08 – HT14 18 2014-11-24 4 The kinematics of rigid body motion

4.1 Degrees of freedom of a rigid body

4.1 Concepts: rigid body space system, body system direction cosines

Kronecker symbol δi,j

Results: A rigid body has N◦ =6 ′ Formulas: (4.2) cos θ = ~e ~e ij i · j

4.2 Orthogonal transformations

4.2 Concepts: orthogonal transformation, rotation, reﬂection matrix notation active and passive transformations Results: Formulas : (4.11) aij = cos θij 3 ′ (4.12) xi = aij xj j=1 3 X

(4.15) aij aik = δj,k i=1 X (4.19) ~r′ = A ~r

Warnings: j Do not mix up active and passive transformations. j The bracket notaion “ (~r)′ ” is not established practice.

FYGB08 – HT14 19 2014-11-24 4.3 Properties of the transformation matrix

4.3 Concepts: composition of transformations commutativity, associativity, distributivity transpose matrix, symmetric matrix, antisymmetric matrix shape of a matrix, row and column vectors product of matrices, unit matrix, inverse matrix, determinant orthogonal matrices similarity transformation Results: the determinant is invariant under similarity transformations Formulas: (4.35) A−1 = At (4.41) B′ = A B A−1 (4.42) det(A)= 1 ±

Warnings: j The sign indicating the transpose is usually omitted on row vectors. j In [Goldstein--Poole--Safko] the role of A and B in the description of similarity transformations can be a bit confusing.

FYGB08 – HT14 20 2014-11-24 4.4 The Euler angles

4.4 Concepts: rotation, reﬂection Euler angles (φ,θ,ψ) zxz-convention line of nodes Results:—

Formulas: A = Aψ Aθ Aφ

4.5 The Cayley-Klein parameters and related quantities

4.5 Concepts: Cayley-Klein parameters α, β, γ, δ unitary matrix special unitary group SU(2), special orthogonal group SO(3) spin Results:—

Formulas: α = ei(ψ+φ)/2 cos θ , β = i ei(ψ−φ)/2 sin θ , γ = β∗ , δ = α∗ − 2 2 − U = αδ βγ = α 2 + β 2 =1 − − | | | | U − 1 = U † −

FYGB08 – HT14 21 2014-11-24 4.6 Euler’s theorem on the motion of a rigid body

4.6 Concepts: axis of rotation eigenvector, eigenvalue, eigenvalue problem characteristic equation, secular equation diagonalization of a matrix, similarity transformation trace of a matrix Results: a rotation matrix has (at least) one eigenvalue 1 eigenvalues of an orthogonal matrix are real or form complex pairs invariance of the trace under a similarity transformation Euler’s theorem Chasles’ theorem Formulas: (4.52) A λ11 =0 − (4.61) tr( A) = tr(λ)=1+2 cosΦ

Φ φ + ψ θ (4.63) cos 2 = cos 2 cos 2

4.7 Finite rotations

4.7 Concepts: ﬁnite rotation Results: rotation formula Formulas: (4.62) ~r′ = cos Φ ~r + (1 cos Φ) (~n ~r) ~n + sin Φ ~r ~n − · ×

FYGB08 – HT14 22 2014-11-24 4.8 Inﬁnitesimal rotations

4.8 Concepts: infinitesimal rotation pseudovector, axial vector, polar vector generators of infinitesimal rotations Levi-Civita symbol Results: rotational part of the motion of a rigid body described by time dependence of Ω~ infinitesimal rotations commute Formulas: (4.70) d~r = ǫ ~r (4.72) d~r = ~r dΩ~ × (4.74) dΩ~ B det(B) B dΩ~ 7−→ (4.76) dΩ~ = ~n dΦ 1 ǫ 3 (4.78 2 ) = α=1 Mα dΩα

(4.80) [Mα,P Mβ]= α εαβγ Mγ P1 for (α,β,γ) (1, 2, 3) , (2, 3, 1) , (3, 1, 2) ∈{ } ε = 1 for (α,β,γ) (2, 1, 3) , (3, 2, 1) , (1, 3, 2) − αβγ  − ∈{ }  0 else  Warnings: j choosing inﬁnitesimal Euler angles does not give the most general inﬁnitesimal rotation j switch from passive to active rotations after formula (4.77)

FYGB08 – HT14 23 2014-11-24 4.9 Rate of change of a vector

4.9 Concepts: rate of change of a vector in the space and body systems instantaneous angular velocity operator equality Results: Formulas: dΩ~ (4.83) ~ω = dt d d (4.86) = + ~ω dt space dt body ×

˙ ˙ ′ (4.87) ~ω =( φ sin θ sin ψ + θ cos ψ) ~ex +(φ˙ sin θ cos ψ θ˙ sin ψ) ~e ′ +(φ˙ cos θ + ψ˙) ~e ′ (body s.) − y z 1 ˙ ˙ (4.87 2 ) ~ω =(ψ sin θ sin φ + θ cos φ) ~ex +( ψ˙ sin θ cos φ + θ˙ sin φ) ~e +(ψ˙ cos θ + φ˙) ~e (space s.) − y z

4.10 The Coriolis eﬀect

4.10 Concepts: ﬁctitious force, eﬀective force centrifugal force, Coriolis force geoid, cyclone patterns, Foucault pendulum freely falling particle Results: Formulas: (4.88) ~v = ~v + ~ω ~r s r × (4.89) ~a = d + ~ω (~v + ~ω ~r) s dt s × r × = ~a +2(~ω ~r)+ ~ω (~ω ~r)+ ~ω˙ ~r r × × × ×

Warnings: j The centrifugal ‘force’ and Coriolis ‘force’ are ﬁctitious. j A ﬁctitious force is not a force.

FYGB08 – HT14 24 2014-11-24 FYGB08 – HT14 25 2014-11-24 5 The dynamics of rigid body motion

5.1 Angular momentum of motion about a point

5.1 Concepts: moment of inertia tensor continuous mass distribution, mass density bac-cab rule for double cross products Results: linear relation between vectors described by a tensor Formulas: (5.2) ~v = ~ω ~r i × i (5.3) L~ = m [r2 ~ω (~r ~ω) ~r ] i i i − i · i (5.9) L~ = PI ~ω (5.8′) I = m (δ r2 r r ) αβ i i αβ i − iα iβ P 2 (5.8) Iαβ = ρ(~r)(δαβ ~r rα rβ) dV V − 3 Z ε ε = δ δ δ δ − αβγ αµν β,µ γ,ν − γ,µ β,ν α=1 X

FYGB08 – HT14 26 2014-11-24 5.2 Tensors

5.2 Concepts: tensor, rank of a tensor, pseudotensor tensor product of two vectors covariant and contravariant tensors contraction, matrix product, dot product self-contraction, trace Results:

Formulas: (5.10) T ′ = A A A T α1α2...αm ··· α1β1 α2β2 ··· αmβm β1β2...βm Xβ1 Xβ2 Xβm

Warnings: j Do not mix up the geometric object T with the collection T of numbers. { α1α2...αm }

5.3 The inertia tensor and the moment of inertia

5.3 Concepts: moment of inertia tensor moment of inertia perpendicular distance from rotation axis Results: displaced axis theorem Formulas: (5.16) T = 1 ~ω L~ = 1 ~ω I ~ω 2 · 2 1 2 I 1 2 (5.17) T = 2 ω ~n ~n = 2 Iω (5.18) I I = ~n I ~n = m [ r2 (~r ~n)2 ] ≡ ~n i i i − i · (5.21) I = I + M (R~ P~n)2 ◦ ×

Warnings: j In the notation I for the moment of inertia, the dependence on the rotation axis ~n (as well as the dependence on the choice of origin) is suppressed.

FYGB08 – HT14 27 2014-11-24 5.4 The eigenvalues of the inertia tensor and the principal axis transformation

5.4 Concepts: diagonal matrix principal moment principal axis, principal axes system inertia ellipsoid ellipsoid of revolution Results: the principal moments are positive Formulas: (5.24) It = I I1 0 0 Idiag Idiag (5.29) ( )αβ = Iα δαβ i.e. =  0 I2 0   0 0 I3    (5.25) Lα = Iα ωα in principal axes system

1 3 2 (5.26) T = 2 α=1 Iα ωα in principal axes system (5.31) det(I Pλ 11)=0 − (5.33) ~ρ = ~n/√I~n

2 (5.35) 1= α Iα ρα

(5.36) R◦ =P I/M p

FYGB08 – HT14 28 2014-11-24 5.5 Solving rigid body problems and the Euler equations of motion

5.5 Concepts: Results: Euler equations of motion Formulas: T = 1 M v2 + 1 Iω2 − 2 c.m. 2 ~ (5.37) ∂L + ~ω L~ = N~ ∂t × d (5.39) I ω + ε ω ω I = N α dt α αβγ β γ γ α Xβ,γ

Warnings: j b ody subscript suppressed

FYGB08 – HT14 29 2014-11-24 5.6 Torque-free motion of a rigid body

5.6 Concepts: angular velocity space, angular momentum space Poinsot construction, invariable plane, polhode, herpolhode body cone, space cone Binet ellipsoid symmetric top precession oblate top, prolate top, spherical top Chandler wobble Results: rolling of the inertia ellipsoid at ﬁxed height on the invariable plane rolling of the body cone on the space cone (symmetric top) steady motion only when ~ω is along one of the principal axes stable steady motion only for axis with smallest or largest moment Formulas : (5.40) Iα ω˙ α = εαβγ Iβ ωβ ωγ Xβ,γ (5.43) F = √2 T −1 L~ ∇~ρ I3 I1 (5.49) Ω= − ω3 I1

FYGB08 – HT14 30 2014-11-24 5.7 The heavy symmetric top with one point ﬁxed

5.7 Concepts: symmetric top, heavy top ﬁgure axis rotation, precession, nutation turning angle falling top, fast top, slow and fast regular precession sleeping top, tippie-top Results: φ and ψ are cyclic coordinates Formulas: (5.52) L = 1 I (θ˙2 + φ˙2 sin2 θ)+ 1 I (ψ˙ + φ˙ cos θ)2 Mgl cos θ 2 1 2 3 − (b a cos θ)2 (5.59) E′ = 1 I θ˙2 + 1 I − + Mgl cos θ 2 1 2 1 sin2 θ u = cos θ − (5.62′)u ˙ 2 = (1 u2)(α βu) (b au)2 f(u) − − − − ≡

Warnings: j (x, y, z) used for the body system

5.8 Precession of the equinoxes and of satellite orbits

5.8 Concepts: precession of the equinoxes Poisson equation, Legendre polynomial Results: full vs torque-free motion of Earth’s axis Formulas: G M m G M (5.86) V = + (3 I tr(I)) − r 2 r3 r −

FYGB08 – HT14 31 2014-11-24 6 Oscillations

6.1 Formulation of the problem

6.1 Concepts: simple harmonic oscillator (free, damped, forced / driven) mechanical equilibrium, equilibrium configuration stable and unstable equilibrium, indifferent / neutral equilibrium small perturbation Taylor expansion Results: stable (unstable) equilibrium at minimum (maximum) of the potential indifferent equilibrium implies that V is degenerate Formulas: ∂V (6.1) Qi =0 ≡ ∂qi 0

1 ∂2V (6.4) V = V η η , V = 2 ij i j ij ∂q ∂q i,j i j η=0 X 1 1 (6.5) T = m q˙ q˙ = m η˙ η˙ 2 ij i j 2 ij i j i,j i,j X X 1 ∂2T (6.6) T = T η˙ η˙ , T = 2 ij i j ij ∂q˙ ∂q˙ i,j i j η˙=0 X

(6.8) (Tij η¨j + Vij ηj) =0 j X L = 1 ~η˙ T ~η˙ ~η V ~η , T ~η¨ + V ~η =0 − 2 −

FYGB08 – HT14 32 2014-11-24 6.2 The eigenvalue equation and the principal axis transformation

6.2 Concepts: harmonic motion generalized eigenvalue problem Gram-Schmidt orthogonalization hermitian adjoint of a matrix congruence transformation Results: T and V can be diagonalized simultaneously

λk > 0 for stable equilibrium −iωt Formulas: (6.11) ηi = C ai e (6.13) det(V ω2 T)=0 − (6.14) V~a = λ T~a † V † T (6.21) λk = ~ak ~ak/~ak ~ak (6.23) At T A = 11 A =(A )= (~a ) ) − ij j i (6.26) At V A = Λ Vdiag ≡

FYGB08 – HT14 33 2014-11-24 6.3 Frequences of free vibration, and normal coordinates

6.3 Concepts: frequency of free vibration, resonant frequency normal coordinate, normal mode commensurable frequencies Results: separation of equation of motions in normal coordinates small vibrations about stable equilibrium give superposition of harmonic oscillators periodic motion for commensurable frequencies

Formulas −iωkt : (6.35) ηi = k Ck Aik e 1 t T (6.38 2 ) Re CP= A ~η(0) (6.41) ~η = A ζ~ 1 ~ ~ 1 2 2 (6.43) V = 2 ζ Λ ζ = 2 i ωi ζi ˙ ˙ (6.44) T = 1 ζ~ ζ~ = 1 Pζ˙2 2 · 2 i i ¨ 2 (6.46) ζk + ωk ζk =0 P

6.4 Free vibrations of a linear triatomic molecule

6.4 Examples: linear triatomic molecule

FYGB08 – HT14 34 2014-11-24 6.5 Forced vibrations and the eﬀect of dissipative forces

6.5 Concepts: driving force transient solution harmonic driving force dissipation, damping, pure damping resonance steady state, transient solution Results: general solution to inhomogeneous diﬀerential equation as general solution to homogeneous equation plus a particular solution no decoupling (in general) in the presence of dissipation Formulas : (6.60) Qi = j Aji Fi ¨ 2 (6.61) ζk + ωPk ζk = Qk Q cos(ωt + δ ) (6.66) η = A ζ = A 0,i i j ji i ji ω2 ω2 i i i − (6.68) TXη¨ + η˙X+ V η =0 j ij j Fij j ij j (6.76) PV~a + γ F~a + γ2 T~a =0

1 ij (αiαj + βiβj) (6.79) κ = 1 (γ + γ∗)= i,j F 2 2 T (α α + β β ) − Pi,j ij i j i j

(6.82) Aj = Dj(ω)/D(ω) P

FYGB08 – HT14 35 2014-11-24 6.6 Beyond small oscillations: the damped driven pendulum and the Josephson junction

6.6 Concepts: static and dynamic steady states quasi-static motion hysteresis Results: critical value of the torque Formulas: N = m g R − c 1 1 N (6.92) 2 φ¨ + φ˙ + sin φ = ω◦ ωc Nc

N (6.93) ω =0 , φ◦ = arcsin Nc Examples: damped driven oscillator Josephson junction

FYGB08 – HT14 36 2014-11-24 7 The special theory of relativity

7.1 Basic postulates

7.1 Concepts: Galilei transformation space-time, event space-like, light-like, and time-like curves laboratory frame, rest frame laboratory time, proper time signal / information transmission, limiting velocity causal past and future, causally disconnected regions forward / backward light cone Results: non-invariance of electrodynamics under Galilei transformations (constancy of the speed of light) observer dependence of distinction between space and time speed of light as limiting velocity Formulas: (7.4) ds2 = c2 dt2 d~x 2 − (7.5) ds′ 2 = ds2

v2 (7.6) dτ = 1 2 dt − c q Warnings: j Deﬁnition of ds2 with reversed signs is also in use

FYGB08 – HT14 37 2014-11-24 7.2 Lorentz transformations

7.2 Concepts: Lorentz transformation, boost, pseudo-rotation metric rotation group, Lorentz group Poincar´etransformation SO(3), SO(3, 1), SO(m, n) Results: Formulas: v 1 (7.7) β = , γ = c 1 β2 − (7.8) x′ = γ (x β c t)p, c t′ = γ (c t β x) − − (7.9) ct′ = γ (ct β~ ~x) , ~x ′ = ~x + β−2 (γ 1)(β~ ~x) β~ γ β~ ct − · − · −

7.3 Velocity addition and Thomas precession

7.3 Concepts: velocity addition Thomas rotation, electron spin, Thomas precession Results: c as limiting velocity ′ Formulas: ′′ β+β (7.15) β = 1+ββ′ β′′ (7.22) ∆Ω=(γ 1) y − β 1 (7.25) ~ω = 2 ~a ~v 2c ×

FYGB08 – HT14 38 2014-11-24 7.4 Vectors and the metric tensor

7.4 Concepts: four-vector, four-velocity, four-momentum relativistic kinetic energy, rest mass metric, metric tensor one-form, covariant and contravariant tensors electromagnetic ﬁeld tensor Results: Formulas: c (7.27) u = γ ~v (7.37) p 2 = m2 c2 k k (7.39) T = E mc2 =(γ 1) mc2 = (mc2)2 +~p 2 mc2 − − − (7.33) u v = 3 g uµ vν q · µ,ν=0 µν P

FYGB08 – HT14 39 2014-11-24 8 The Hamilton Equations of Motion

8.1 Legendre transformations and the Hamilton equations of motion

8.1 Concepts: conﬁguration space, phase space canonical(ly conjugate) variables Hamiltonian, canonical (= Hamilton) formalism, Dirac formalism Legendre transformation total diﬀerential of a function symplectic formulation of Hamiltonian mechanics

Results: Hamilton equations of motion (2N◦ ﬁrst order diﬀerential equations) Formulas: (8.15) H(q,p; t)= q˙ p L(q, q˙; t) i i i − ∂H P ∂H (8.18)q ˙i = , p˙i = ∂pi − ∂qi ∂H ∂L (8.19) = ∂t − ∂t q for i =1, 2, ... , N (8.36) η = i i p for i = N+1, N+2, ... , 2N i−N ∂H ∂H ∂H ∂H (8.37) = , = (i =1, 2, ... , N) ∂ηi ∂qi ∂ηN+i ∂pi

0N×N 11N×N (8.38) J = 11 0 − N×N N×N ! ∂H (8.39) ~η˙ = J J ~ H ∂~η ≡ ∇~η

Warnings: j In full generality, a Legendre transformation need not exist

FYGB08 – HT14 40 2014-11-24 8.2 Cyclic coordinates and conservation theorems

8.2 Concepts: cyclic coordinate comoving frame Results: cyclic coordinates do not appear in the Hamiltonian all time dependence of H is explicit H = T + V vs. conservation of H (coordinate dependent) Formulas: dH ∂H (8.41) = dt ∂t

Examples: a harmonic oscillator attached to a uniformly moving vehicle

8.3 Routh’s procedure

8.3 Concepts: Routhian Results: conserved generalized momentum parametrizing motions in phase space Formulas: (8.48) R(q;q ˙ , ..., q˙ ,p , ..., p ; t)= N p q˙ L(q, q˙; t) 1 s s+1 N i=s+1 i i −

(8.49) R(q;q ˙1, ..., q˙s,ps+1, ..., pN ; t) P = H (p , ..., p ) L (q , ..., q ;q ˙ , ..., q˙ ; t) cycl. s+1 N − noncycl. 1 s 1 s

FYGB08 – HT14 41 2014-11-24 8.5 Derivation of Hamilton’s equations from a variational principle

8.5

Concepts: modiﬁed Hamilton principle, with or without δpi freely varying Results: derivation of Hamilton equations without knowledge of a Lagrangian Formulas: t2 (8.65) δ [ pi q˙i H] dt =0 − t1 i Z X t2 (8.71) δ [ p˙i qi + H] dt =0 t1 i Z X

8.6 The principle of least action

8.6 Concepts: variation of boundary conditions principle of least action Jacobi’s form of the least action principle metric, geodesic abbreviated action Results: Formulas: t2 t2+∆t2 t2 (8.73) ∆ L dt = L(α) dt L(α=0)dt − Zt1 Zt1+∆t1 Zt1 2 t t2 (8.77) ∆ L dt = [ pi ∆qi H ∆t] − t1 t1 i Z X t2 (8.80) ∆ pi q˙i dt =0 t1 i Z X s2 (8.89) ∆ H V (q) ds =0 s1 − Z p

FYGB08 – HT14 42 2014-11-24 9 Canonical Transformations

9.1 The equations of canonical transformation

9.1 Concepts: canonical variables point transformation, scale transformation canonical transformation, extended canonical transformation restricted canonical transformation generating function Results: various speciﬁc choices of generating function

F2 = i qi Pi gives the identity transformation

F1 = Pi qi Qi and F4 = i pi Pi exchange, up to signs, the coordinates and momenta P P Formulas ∂F1 ∂F1 ∂F1 : (9.12) & (9.14) F = F1(q, Q; t) K = H + pi = Pi = ∂t ∂qi − ∂Qi

∂F2 ∂F2 (9.15) & (9.17) F = F2(q,P ; t) Qi Pi pi = Qi = − i ∂qi ∂Pi P ∂F3 ∂F3 F = F3(p, Q; t)+ qi pi qi = Pi = − i − ∂pi − ∂Qi P ∂F4 ∂F4 F = F4(p,P ; t)+ (qi pi QiPi) qi = Qi = − i − − ∂pi ∂Pi P Warnings: j the choices F1, F2, F3, F4 do not describe all possible canonical transformations

FYGB08 – HT14 43 2014-11-24 9.2 Examples of canonical transformations

9.2 Concepts: identity transformation Results: exchange of coordinates and (minus) momenta by a canonical transf. every point transformation is canonical Formulas : (9.28) F2 = fi(q; t) Pi + g(q; t) (point transformation) i X

9.4 The symplectic approach to canonical transformations

9.4 Concepts: inﬁnitesimal canonical transformation symplectic matrix Jacobian Results:

Formulas: ∂Qi ∂pj ∂Qi ∂qj ∂q = ∂P ∂p = ∂P j q,p i Q,P j q,p − i Q,P (9.48) ∂Pi ∂pj ∂Pi ∂qj ∂q = ∂Q ∂p = ∂Q j q,p − i Q,P j q,p i Q,P ~˙ M ˙ ∂ζi (9.50) & (9.51) ζ = ~η, Mij = ∂ηj (9.55) M J Mt = J

det(M)= 1 − ±

(9.62) F2 = i qi Pi + ǫG(q,P ; t) ∂G (9.63) δ~η = ǫPJ ∂~η

FYGB08 – HT14 44 2014-11-24 9.5 Poisson brackets and other canonical invariants

9.5 Concepts: Poisson bracket fundamental Poisson brackets canonical invariants Lagrange bracket integral invariants of Poincar´e correspondence principle (quantum mechanics) Jacobi identity, Lie algebra, Leibniz property Lagrange brackets, integral invariants of Poincar´e Results: Poisson brackets are canonical invariants the functions on phase space form a Lie algebra with respect to the Poisson bracket phase space volumes are canonical invariants Formulas: ∂u ∂v ∂u ∂v (9.67) u, v = { }q,p ∂q ∂p − ∂p ∂q i i i i i X ∂u ∂v (9.68) u, v = J { }~η ∂~η ∂~η (9.69) q ,p = δ p ,p =0= q , q { i j}q,p i,j { i j}q,p { i j}q,p (9.70) ~η, ~η = J { }~η

FYGB08 – HT14 45 2014-11-24 9.6 Equations of motion, inﬁnitesimal canonical transformations, and conservation theorems

9.6 Concepts: total time derivative Results: Poisson’s theorem: u and v conserved = u, v conserved ⇒ { } time evolution as continuous sequence of canonical transformations existence of canonical transformation to constant canonical variables constant of motion as generating function of canonical transformation Formulas: du ∂u (9.94) = u,H + dt { } ∂t (9.95) ~η˙ = ~η, H { } ∂u (9.97) H,u = (u conserved) { } ∂t (9.100) δ~η = ε ~η, G { }

9.7 The angular momentum Poisson bracket relations

9.7 Concepts: system quantity, system vector Lie algebra so(3), vector representation of so(3) Results: any set of canonical variables can contain at most one component of L~ Formulas: (9.123) F,~ L~ ~n = ~n F~ { · } × (9.128) L , L = 3 ǫ L { α β} γ=1 αβγ γ (9.129) L2, L =0P { α}

FYGB08 – HT14 46 2014-11-24 9.8 Symmetry groups of mechanical systems

9.8 Concepts: Lie group SU(2), Lie algebra su(2) Pauli matrix, spinor representation Lie algebras so(4), so(3, 1), iso(3) Lie algebras su(n) Results: description of rotations in terms of su(2) extended Lie algebra of symmetries of the Kepler problem Lie algebra of symmetries of the isotropic harmonic oscillator Formulas: 3 (9.134) D , L = ǫ D { α β} αβγ γ γ=1 X 3

ǫαβγ Lγ for E< 0 ,  γ=1 (9.135) Dα,Dβ =  X { }  3  ǫ L for E> 0 − αβγ γ γ=1  X  

9.9 Liouville’s theorem

9.9 Concepts: identical systems, ensemble isolated system statistically distributed initial conditions density in phase space time reversal symmetry Results: trajectories in phase space do not cross the density in phase space is constant (Liouville’s theorem) D,H is zero in statistical equilibrium { } Formulas: dD ∂D (9.149) = D,H + dt { } ∂t ∂D (9.150) = D,H ∂t −{ }

FYGB08 – HT14 47 2014-11-24 FYGB08 – HT14 48 2014-11-24 10 Hamilton-Jacobi theory and action angle variables

10.1 The Hamilton-Jacobi equation for Hamilton’s principal function

10.1 Concepts: Hamilton’s principal function S Hamilton-Jacobi equation Results: new Hamiltonian zero, new coordinates constant new momenta as integration constants S generates canonical transformation to these variables S is the action Formulas: ∂F2 ∂F2 ∂F2 (10.3) H(q1, ..., qN , , ..., ; t)+ =0 ∂q1 ∂qN ∂t (10.13) S = L dt + const R

10.3 The Hamilton-Jacobi equation for Hamilton’s characteristic function

10.3 Concepts: Hamilton’s characteristic function W restricted Hamilton-Jacobi equation orbit equations Results: W is the abbreviated action Hamiltonian as one of the new momenta Formulas: (10.14) S(q; α; t)= W (q; α) a t − ∂W (10.43) H(qi, )= a ∂qi

FYGB08 – HT14 49 2014-11-24 10.2 The harmonic oscillator problem as an example of the Hamilton- Jacobi method

10.2 Concepts:— Results:— Formulas: 1 ∂S 2 ∂S (10.20) + m2ω2 q2 + =0 2m ∂q ∂t h i 1 ∂W 2 (10.21) + m2ω2 q2 = a 2m ∂q h i mω2 q2 (10.23) S = √2m a dq 1 a t − 2a − Z r

FYGB08 – HT14 50 2014-11-24 10.4 Separation of variables in the Hamilton-Jacobi equation

10.4 Concepts: separation of variables separable coordinate completely separable system Results: a completely separable system can be solved by quadratures Formulas: N N (10.49) S = i=1 Si , Si = Si(qi; α1, ... , αN ; t) , H = i=1 Hi

P∂Si ∂Si P (10.50) Hi(qi, ,α1, ... , αN ; t)+ =0 ∂qi ∂t

∂Wi (10.52) Hi(qi, ,α1, ... , αN )= αi ∂qi

10.5 Ignorable coordinates and the Kepler problem

10.5 Concepts: natural orthogonal form of a Hamiltonian elliptic, parabolic, spheroconical coordinates Results: a cyclic coordinate is separable St¨ackel theorem Formulas ∂W ∂W : (10.53) H(q2, ... , qN , γ, , ... , )= α1 ( q1 cyclic ) ∂q2 ∂qN ′ (10.56) W = W + γ q1

N N U = U (q ) , w = w (q ) , g U = δ , g w = V − ij ij i i i i j jk 1,k j j j=1 j=1 X X 1 1 (10.78) V (r, ϑ, ϕ)= V (r)+ 2 V (ϑ)+ 2 V (ϕ) r r ϑ r2 sin ϑ ϕ

FYGB08 – HT14 51 2014-11-24 10.6 Action-angle variables in systems of one degree of freedom

10.6 Concepts: action-angle variables libration, rotation unbounded angular variables bifurcation separatrix Results: J has dimension of action, w is dimensionless v obtainable without solving the motion in the original variables parameter regions of libration / rotation for the simple pendulum Formulas : (10.79) p = p(q,α1) (10.81) p = 2mℓ2 (E + mgℓ cos θ) ( pendulum ) θ ± p (10.82) J = p dq ∂W (10.85) w = H ∂J (10.87) w(t)= v t + β ∂w (10.90) ∆w = dq =1 ∂q I (10.91) v = −1 T 2α 2π 2πα (10.93) J = cos2 φ dφ = ( pendulum ) ω ω Z0

FYGB08 – HT14 52 2014-11-24 10.7 Action-angle variables for completely separable systems

10.7 Concepts: multiply periodic system commensurability Results: existence of action-angle variables libration or rotation for projection to each q -p -plane ⇐⇒ i i (simply) periodic motion commensurability of frequencies for all N projected motions ⇐⇒ Formulas: (10.100) Ji = pi dqi I (10.101) Ji =2πpi ( cyclic coordinate )

W = W (q ,J , ... , J ) − i i i 1 N ∂W ∂W (q ,J , ... , J ) (10.103) w = P = j j 1 N i ∂J ∂J i j i X (10.106) ~w(t)= ~vt + β~

~ (10.111) q (t)=Re a(k)e2πi~j·(~vt+β) ( libration ) k ~j X~j ~ (10.114) q (t)= q (v t + β )+Re a(k)e2πi~j·(~vt+β) k 0,k k k ~j X~j

FYGB08 – HT14 53 2014-11-24 10.8 The Kepler problem and action-angle variables

10.8 Concepts: classical astronomical parameters line of nodes, ascending node, inclination Results: close relationship between action-angle variables and astronomical parameters Formulas: ∂W (10.129) J = dϕ = α dϕ = α dϕ =2πα ϕ ∂ϕ ϕ ϕ ϕ I I I ∂W α2 J = dϑ = α2 ϕ dϑ ϑ ∂ϑ ϑ − sin2 ϑ I I s ∂W 2mk α2 J = dr = 2m E + ϑ r ∂r r − r2 I I s 2π2 mk2 (10.140) H = E = 2 −(Jr + Jϑ + Jϕ) (10.144) w = w w , w = w w , w = w 1 ϕ − ϑ 2 ϑ − r 3 r

(10.145) J1 = Jϕ , J2 = Jϑ + Jϕ J3 = Jϑ + Jϕ + Jr

(10.146) H = 2π2 mk2/J 2 − 3

FYGB08 – HT14 54 2014-11-24 5.9 Precession of systems of charges in a magnetic ﬁeld

5.9 Concepts: magnetic moment gyromagnetic ratio magnetic dipole Larmor frequency Results: precession of angular momentum in a constant magnetic ﬁeld Larmor’s theorem Formulas: (5.99) M~ = γ L~ (5.100) γ = q/2m (5.101) V = M~ B~ − · ~ (5.103) dL = γ L~ B~ dt × q (5.104) ~ω = B~ L − 2m (5.110) L = 1 m v′2 V (r ) 1 I ω2 2 i i i − jk − 2 L L P

FYGB08 – HT14 55 2014-11-24 10 Classical Chaos

10.1 Periodic motion

10.1 Concepts: chaotic system deterministic chaos submanifold, N-dimensional torus Results: multiply periodic motion as a system of uncoupled harmonic oscillators Formulas:

10.2 Perturbations and the Kolmogorov-Arnold-Moser theorem

10.2 Concepts: canonical perturbation theory iteration procedure set of measure zero Results: KAM theorem Formulas: ∂ ∂ (11.10) ∆Ki(Qi,Pi)= Q˙ i+1 , ∆Ki(Qi,Pi)= P˙i+1 ∂Pi ∂Qi −

FYGB08 – HT14 56 2014-11-24 10.3 Attractors

10.3 Concepts: attraction towards stable N-torus regular attractor, ﬁxed point, limit cycle strange attractor, fractal dimension van der Pol equation Results: Formulas: (11.11) m x¨ ǫ (1 x2)x ˙ + mω2 x − − o

10.4 Chaotic trajectories and Lyapunov exponents

10.4 Concepts: mixing, quasi-periodicity, sensitivity to initial conditions domain of motion butterﬂy eﬀect Lyapunov exponent Results: positive Lyapunov exponents 10−8 ... 10−10 in the solar system ≃ Formulas: (11.12) s(t) s eλt ∼ 0

10.5 Poincar´emaps (sections)

10.5 Concepts: Poincar´esection, Poincar´emap

FYGB08 – HT14 57 2014-11-24 10.6 H´enon-Heiles Hamiltonian

10.6 Concepts: H´enon-Heiles potential island of integrability

FYGB08 – HT14 58 2014-11-24