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: Isaac Newton 1643 – 1727 cf. section 1.1

∼ 1750 – 1800: Leonhard Euler 1707 – 1783 cf. section 1.4

Jean Le Rond d’Alembert 1717 – 1783 cf. section 1.4

Joseph-Louis Lagrange 1736 – 1813 cf. section 1.4, 2.3

∼ 1850: Carl Gustav Jacob Jacobi 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 –Bertram Kostant 1928 –Jean-Marie Souriau 1922 – 2012Jerrold Eldon Marsden 1942 – 2010Alan David Weinstein 1943 –

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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) W12(P) =∫

P~F · d~s

(1.16) ~F (~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.

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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: (1.19) ~pi = ~F exti +

∑

j~Fji

(1.2012) ~Fij = −~Fji

(1.22) M d2

dt2~R = ~F ext

(1.26) ddt~L = ~N ext

(1.27) ~ri = ~r′i + ~R

(1.28) ~L = ~R×M ~v +∑

i ~r′i×~p′i

(1.31) T = 12 M v2 + 1

2

∑

imi v′2i

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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)

− qj = qj(~r1, ~r2, ... , ~rN ; t) (holonomic constraints)

Examples: rigid body

bead sliding on a wire

disk rolling on a plane

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

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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)∑

i

~F appli · δ~ri = 0

(1.45)∑

i

(~F appli − ~pi) · δ~ri = 0

(1.49) Qj =∑

i

~Fi ·∂~ri∂qj

(1.51)∂~vi∂qj

=∂~ri∂qj

(1.52)∑

j

[ ddt

∂T

∂qj− ∂T

∂qj−Qj] δqj = 0

(1.53)d

dt

∂T

∂qj− ∂T

∂qj= Qj

(1.54) Qj =∂V

∂qj

(1.56) L = T − V

(1.57)d

dt

∂L

∂qj− ∂L

∂qj= 0

Warnings: j The notation ~F is often used for the applied rather than thetotal force.

j Generically the kinetic energy depends also on thegeneralized coordinates, not only on the generalized velocities.

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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:(1.58) Qj = −∂U

∂qj+

d

dt

∂U

∂qj

(1.62) Ue.m. = e φ− e ~A ·~v(1.68) ~F fric = −~∇~vF(1.69) Qj = −∂F

∂qj

(1.70)d

dt

∂L

∂qj− ∂L

∂qj+∂F∂qj

= 0

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

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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:(2.1) S = S(t1, t2) =

∫ t2

t1

L dt

(2.2) δS[q] ≡ δ

∫ t2

t1

L(q, q, t) dt = 0

Warnings: j Two distinct meanings of “q”:

as a coordinate of configuration space

as a path q= q(t) in configuration space

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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:(2.3) J [y] =

∫ x2

x1

f(y, y, x) dx

(2.9)

∫ x2

x1

(∂f

∂y− d

dx

∂f

∂y)∂y

∂α

∣

∣

∣

∣

α=0

dx = 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

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2.4 Lagrange multipliers

2.4

Concepts: Lagrange multiplier

semi-holonomic constraints

Results:

Formulas: (2.21) δ∫ t2t1(L+

∑ms=1 λs fs) dt = 0

(2.23) ddt

∂L∂qj

− ∂L∂qj

= Qj := −∑m

s=1 λs∂fs∂qj

Examples: Hoop rolling down an inclined plane

Warnings: j Lagrange multipliers are directly applicable onlyin 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: (2.42) Lj qj +∑

kj 6=k

Ljk qk +Rj qj +1Cjqj = Vj(t)

Examples: battery in series with a resistance and an inductance

inductance in series with a capacitance,as analogue of the simple harmonic oscillator

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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:(2.44) pj =

∂L

∂qj

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 necessarilycoincide with ordinary mechanical momentum

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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: (3.2) ~r′1 = − m2

m1 +m2~r , ~r′2 =

m1

m1 +m2~r

(3.3) L = 12 M

~R2 + 12 µ ~r

2 − U(~r, ~r, ...)

(3.5)1

µ=

1

m1+

1

m2

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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

(3.12) f(r) ≡ −dV

dr= m r − ℓ2

mr3

(3.17) dt =dr

√

2m(E − V − ℓ2

2mr2)

(3.19) dθ =ℓ dt

m r2

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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:(3.24)

d

dt

(

∑

i

~pi · ~ri)

= 2 T +∑

i

~Fi · ~ri

(3.26) T = −1

2

∑

~Fi · ~ri

(3.28) T =1

2rdV

dr

(3.29) T =n+ 1

2V

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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 Veffcircular orbits when the energy is minimal

Formulas:(3.22) Veff = V +

ℓ2

2mr2

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

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:(3.32)

d

dt=

ℓ

mr2d

dθ

(3.34)d2u

dθ2+ u = −m

ℓ2d

duV ( 1

u)

(3.37) dθ = − du√

2mEℓ2

− 2mVℓ2

− u2

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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:(3.47′) β2 = 3 +

r

f

df

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

Formulas:(3.55) u =

1

r=mk

ℓ2(1 +

√

1 + 2Eℓ2

mk2cos(θ− θ′))

(3.57) e =

√

1 +2Eℓ2

mk2

(3.61) a = − k

2E

(3.64) r =a (1− e2)

1+ e cos(θ− θ)

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3.8 The motion in time in the Kepler problem

3.8

Concepts: eccentric anomaly

period

Results: Kepler’s third law

Kepler equation

Formulas:(3.66) t =

ℓ3

mk2

∫ θ

θ0

dθ′

[1 + e cos(θ′ − θ)]2

(3.74) T =2π a3/2

√

G (m1+m2)

(3.76) ω t = ψ − e sinψ

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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: (3.88) σ(~Ω) |dΩ| = n

I

(3.89) dΩ = 2π sin ϑ dϑ

(3.90) ℓ = s√2mE

(3.93) σ(ϑ) =s

sinϑ

∣

∣

∣

∂s

∂ϑ

∣

∣

∣

(3.96) ϑ = π − 2

∫ ∞

rmin

s dr

r

√

r2 − V (r)E

− s2

(3.101) s(ϑ) = k2E

cot(ϑ2

)

(3.102) σ(ϑ) =k2

16E2sin−4

(ϑ2

)

, k = ZZ ′e2

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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: (3.108) ρ =µ

m2

v0v′1

=m1

m2

v0v

(3.110) cos θ =cosϑ+ ρ

√

1 + 2ρ cosϑ+ ρ2

(3.116) σlab(θ) =

(

1 + 2ρ cosϑ+ ρ2)3/2

1 + ρ cosϑσc.m.(ϑ)

− θ = ϑ2 and θmax =

π2 for ρ=1

− σlab(θ) = 4 cos θ σc.m.(2θ) for ρ=1

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:(3.82) ~A =~p× ~L−mk

~r

r

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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 θij = ~ei · ~e′

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

(4.12) x′i =

3∑

j=1

aij xj

(4.15)3∑

i=1

aij aik = δj,k

(4.19) ~r′ = A~r

Warnings: j Do not mix up active and passive transformations.

j The bracket notaion “ (~r)′ ” is not established practice.

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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′ = ABA−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 thedescription 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 θ2 , β = i ei(ψ−φ)/2 sin θ2 , γ = −β∗ , δ = α∗

−∣

∣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 cos Φ

(4.63) cos Φ2 = cos

φ+ψ2 cos θ2

4.7 Finite rotations

4.7

Concepts: finite rotation

Results: rotation formula

Formulas: (4.62) ~r′ = cos Φ ~r + (1− cosΦ) (~n · ~r)~n+ sinΦ ~r×~n

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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) ~dΩB7−→ det(B)B ~dΩ

(4.76) ~dΩ = ~n dΦ

(4.7812) ǫ =

∑3α=1Mα dΩα

(4.80) [Mα,Mβ] =∑

α εαβγMγ

− εαβγ =

1 for (α, β, γ) ∈ (1, 2, 3) , (2, 3, 1) , (3, 1, 2)−1 for (α, β, γ) ∈ (2, 1, 3) , (3, 2, 1) , (1, 3, 2)0 else

Warnings: j choosing infinitesimal Euler angles does not givethe 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:(4.83) ~ω =

~dΩ

dt

(4.86)d

dt

∣

∣

∣

space=

d

dt

∣

∣

∣

body+ ~ω×

(4.87) ~ω = (φ sin θ sinψ+ θ cosψ)~ex′

+ (φ sin θ cosψ− θ sinψ)~ey′ + (φ cos θ+ ψ)~ez′ (body s.)

(4.8712) ~ω = (ψ sin θ sinφ+ θ cosφ)~ex

+ (−ψ sin θ cosφ+ θ sin φ)~ey + (ψ cos θ+ φ)~ez (space s.)

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) ~vs = ~vr + ~ω× ~r

(4.89) ~as =(

ddt

∣

∣

s+ ~ω×

)

(~vr+ ~ω× ~r)

= ~ar + 2 (~ω× ~r) + ~ω× (~ω× ~r) + ~ω× ~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

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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) ~vi = ~ω× ~ri

(5.3) ~L =∑

imi [r2i ~ω − (~ri · ~ω) ~ri]

(5.9) ~L = I ~ω

(5.8′) Iαβ =∑

imi (δαβ r2i − riα riβ)

(5.8) Iαβ =

∫

V

ρ(~r) (δαβ ~r2 − rα rβ) dV

−3∑

α=1

εαβγ εαµν = δβ,µ δγ,ν − δγ,µ δβ,ν

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 ′α1α2...αm

=∑

β1

∑

β2

· · ·∑

βm

Aα1β1 Aα2β2 · · ·Aαmβm Tβ1β2...βm

Warnings: j Do not mix up the geometric object Twith the collection Tα1α2...αm

of numbers.

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 = 12 ~ω · ~L = 1

2 ~ω I ~ω

(5.17) T = 12 ω

2 ~n I~n = 12 I ω

2

(5.18) I ≡ I~n = ~n I~n =∑

imi [ r2i − (~ri ·~n)2 ]

(5.21) I = I +M (~R×~n)2

Warnings: j In the notation I for the moment of inertia, the dependence on therotation 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 axistransformation

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

(5.29) (Idiag)αβ = Iα δαβ i.e. Idiag =

I1 0 0

0 I2 0

0 0 I3

(5.25) Lα = Iα ωα in principal axes system

(5.26) T = 12

∑3α=1 Iα ω

2α in principal axes system

(5.31) det(I−λ 11) = 0

(5.33) ~ρ = ~n/√I~n

(5.35) 1 =∑

α Iα ρ2α

(5.36) R =√

I/M

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 = 12 M v2c.m. +

12 I ω

2

(5.37) ∂~L∂t

+ ~ω× ~L = ~N

(5.39) Iαd

dtωα +

∑

β,γ

εαβγ ωβ ωγ Iγ = Nα

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 ωα =∑

β,γ

εαβγ Iβ ωβ ωγ

(5.43) ∇~ρF =√2 T−1 ~L

(5.49) Ω =I3− I1I1

ω3

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 = 12I1 (θ

2+ φ2 sin2θ) + 12I3 (ψ+ φ cos θ)2 −M g l cos θ

(5.59) E ′ = 12I1 θ

2 + 12I1

(b− a cos θ)2

sin2θ+M g l cos θ

− u = cos θ

(5.62′) u2 = (1−u2) (α− β u)− (b− a u)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:(5.86) V = − GM m

r+GM

2 r3(3 Ir− tr(I))

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:(6.1) Qi ≡

∂V

∂qi

∣

∣

∣

0= 0

(6.4) V =1

2

∑

i,j

Vij ηi ηj , Vij =∂2V

∂qi∂qj

∣

∣

∣

∣

η=0

(6.5) T =1

2

∑

i,j

mij qi qj =1

2

∑

i,j

mij ηi ηj

(6.6) T =1

2

∑

i,j

Tij ηi ηj , Tij =∂2T

∂qi∂qj

∣

∣

∣

∣

η=0

(6.8)∑

j

(Tij ηj + Vij ηj) = 0

− L = 12

(

~η T ~η − ~ηV ~η)

, T ~η + V ~η = 0

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

Formulas: (6.11) ηi = C ai e−iωt

(6.13) det(V−ω2T) = 0

(6.14) V~a = λT~a

(6.21) λk = ~a †k V~ak/~a

†k T~ak

(6.23) At TA = 11

− A = (Aij) =(

(~aj)i )

(6.26) AtVA = 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 equilibriumgive superposition of harmonic oscillators

periodic motion for commensurable frequencies

Formulas: (6.35) ηi =∑

k Ck Aik e−iωkt

(6.3812) ReC = At T ~η(0)

(6.41) ~η = A~ζ

(6.43) V = 12~ζΛ ~ζ = 1

2

∑

i ω2i ζ

2i

(6.44) T = 12~ζ · ~ζ = 1

2

∑

i ζ2i

(6.46) ζk + ω2k ζk = 0

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 generalsolution to homogeneous equation plus a particular solution

no decoupling (in general) in the presence of dissipation

Formulas: (6.60) Qi =∑

j Aji Fi

(6.61) ζk + ω2k ζk = Qk

(6.66) ηj =∑

i

Aji ζi =∑

i

AjiQ0,i cos(ωt+ δi)

ω2i − ω2

(6.68)∑

j

(

Tij ηj + Fij ηj + Vij ηj)

= 0

(6.76) V~a+ γ F~a+ γ2 T~a = 0

(6.79) κ = −12 (γ+ γ∗) =

1

2

∑

i,j Fij (αiαj + βiβj)∑

i,j Tij (αiαj + βiβj)

(6.82) Aj = Dj(ω)/D(ω)

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: − Nc = mgR

(6.92) 1ω 2

φ+1

ωcφ+ sinφ =

N

Nc

(6.93) ω = 0 , φ = arcsin(N

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

(7.6) dτ =

√

1− v2

c2dt

Warnings: j Definition of ds2 with reversed signs is also in use

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7.2 Lorentz transformations

7.2

Concepts: Lorentz transformation, boost, pseudo-rotation

metric

rotation group, Lorentz group

Poincare transformation

SO(3), SO(3, 1), SO(m,n)

Results:

Formulas:(7.7) β =

v

c, γ =

1√

1−β2

(7.8) x′ = γ (x− β c t) , 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: (7.15) β ′′ =β+β′

1+ββ′

(7.22) ∆Ω = (γ− 1)β′′

y

β

(7.25) ~ω = 12c2 ~a×~v

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:(7.27) u = γ

(

c~v

)

(7.37) ‖p‖2 = m2 c2

(7.39) T = E −mc2 = (γ− 1)mc2 =√

(mc2)2+~p 2 −mc2

(7.33) u · v =∑3

µ,ν=0 gµν uµ vν

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) =∑

i qi pi − L(q, q; t)

(8.18) qi =∂H

∂pi, pi = −∂H

∂qi

(8.19)∂H

∂t= −∂L

∂t

(8.36) ηi =

qi for i=1, 2, ... , N

pi−N for i=N+1, N+2, ... , 2N

(8.37)∂H

∂ηi=∂H

∂qi,

∂H

∂ηN+i=∂H

∂pi(i=1, 2, ... , N)

(8.38) J =

(

0N×N 11N×N

−11N×N 0N×N

)

(8.39) ~η = J∂H

∂~η≡ J ~∇~ηH

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:(8.41)

dH

dt=∂H

∂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; q1, ..., qs, ps+1, ..., pN ; t) =∑N

i=s+1 pi qi − L(q, q; t)

(8.49) R(q; q1, ..., qs, ps+1, ..., pN ; t)

= Hcycl.(ps+1, ..., pN)− Lnoncycl.(q1, ..., qs; q1, ..., qs; t)

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:(8.65) δ

∫ t2

t1

[∑

i

pi qi −H]dt = 0

(8.71) δ

∫ t2

t1

[∑

i

pi qi +H] dt = 0

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:(8.73) ∆

∫ t2

t1

L dt =

∫ t2+∆t2

t1+∆t1

L(α) dt−∫ t2

t1

L(α=0) dt

(8.77) ∆

∫ t2

t1

L dt = [∑

i

pi∆qi −H∆t]t2

t1

(8.80) ∆

∫ t2

t1

∑

i

pi qi dt = 0

(8.89) ∆

∫ s2

s1

√

H −V (q) ds = 0

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=∑

i qiQi and F4=∑

i pi Pi exchange, up to signs,the coordinates and momenta

Formulas: (9.12) & (9.14) F = F1(q, Q; t) K = H +∂F1

∂tpi =

∂F1

∂qiPi = −∂F1

∂Qi

(9.15) & (9.17) F = F2(q, P ; t)−∑

iQi Pi pi =∂F2

∂qiQi =

∂F2

∂Pi

− F = F3(p,Q; t) +∑

i qi pi qi = −∂F3

∂piPi = −∂F3

∂Qi

− F = F4(p, P ; t) +∑

i(qi pi−QiPi) qi = −∂F4

∂piQi =

∂F4

∂Pi

Warnings: j the choices F1, F2, F3, F4 do not describeall 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 =∑

i

fi(q; t)Pi + g(q; t) (point transformation)

9.4 The symplectic approach to canonical transformations

9.4

Concepts: infinitesimal canonical transformation

symplectic matrix

Jacobian

Results:

Formulas:

(9.48)

(

∂Qi

∂qj

)

q,p=(

∂pj∂Pi

)

Q,P

(

∂Qi

∂pj

)

q,p= −

(

∂qj∂Pi

)

Q,P(

∂Pi

∂qj

)

q,p= −

(

∂pj∂Qi

)

Q,P

(

∂Pi

∂pj

)

q,p=(

∂qj∂Qi

)

Q,P

(9.50) & (9.51) ~ζ = M ~η , Mij =∂ζi∂ηj

(9.55) MJMt= J

− det(M) = ± 1

(9.62) F2 =∑

i qi Pi + ǫG(q, P ; t)

(9.63) δ~η = ǫ J∂G

∂~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 Poincare

correspondence principle (quantum mechanics)

Jacobi identity, Lie algebra, Leibniz property

Lagrange brackets, integral invariants of Poincare

Results: Poisson brackets are canonical invariants

the functions on phase space form a Lie algebrawith respect to the Poisson bracket

phase space volumes are canonical invariants

Formulas:(9.67) u, vq,p =

∑

i

( ∂u

∂qi

∂v

∂pi− ∂u

∂pi

∂v

∂qi

)

(9.68) u, v~η =∂u

∂~ηJ∂v

∂~η

(9.69) qi, pjq,p = δi,j pi, pjq,p = 0 = qi, qjq,p

(9.70) ~η, ~η~η = J

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:(9.94)

du

dt= u,H+ ∂u

∂t

(9.95) ~η = ~η,H

(9.97) H, u =∂u

∂t(u conserved)

(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) Lα, Lβ =∑3

γ=1 ǫαβγ Lγ

(9.129) L2, Lα = 0

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:(9.134) Dα, Lβ =

3∑

γ=1

ǫαβγ Dγ

(9.135) Dα, Dβ =

3∑

γ=1

ǫαβγ Lγ for E < 0 ,

−3∑

γ=1

ǫαβγ Lγ for E > 0

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:(9.149)

dD

dt= D,H+ ∂D

∂t

(9.150)∂D

∂t= −D,H

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 func-

tion

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: (10.3) H(q1, ..., qN ,∂F2

∂q1, ...,

∂F2

∂qN; t) +

∂F2

∂t= 0

(10.13) S =∫

L dt + const

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

(10.43) H(qi,∂W∂qi

) = a

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:(10.20)

1

2m

[(∂S

∂q

)2

+m2ω2 q2]

+∂S

∂t= 0

(10.21)1

2m

[(∂W

∂q

)2

+m2ω2 q2]

= a

(10.23) S =√2ma

∫

dq

√

1− mω2 q2

2a − a t

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: (10.49) S =∑N

i=1 Si , Si = Si(qi;α1, ... , αN ; t) , H =∑N

i=1Hi

(10.50) Hi(qi,∂Si

∂qi, α1, ... , αN ; t) +

∂Si∂t

= 0

(10.52) Hi(qi,∂Wi

∂qi, α1, ... , αN) = αi

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

Stackel theorem

Formulas: (10.53) H(q2, ... , qN , γ,∂W∂q2

, ... , ∂W∂qN

) = α1 ( q1 cyclic )

(10.56) W =W ′ + γ q1

− Uij = Uij(qi) , wi = wi(qi) ,N∑

j=1

gj Ujk = δ1,k ,N∑

j=1

gj wj = V

(10.78) V (r, ϑ, ϕ) = Vr(r) +1r2Vϑ(ϑ) +

1r2 sin2 ϑ

Vϕ(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) pθ = ±√

2mℓ2 (E +mg ℓ cos θ) ( pendulum )

(10.82) J =∮

p dq

(10.85) w =∂W

∂J

(10.87) w(t) = v t+ β

(10.90) ∆w =

∮

∂w

∂qdq = 1

(10.91) v = T −1

(10.93) J =2α

ω

∫ 2π

0

cos2φ dφ =2π α

ω( pendulum )

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 qi-pi-plane

(simply) periodic motion⇐⇒ commensurability of frequencies for all N projected motions

Formulas:(10.100) Ji =

∮

pi dqi

(10.101) Ji = 2π pi ( cyclic coordinate )

− W =∑

iWi(qi, J1, ... , JN)

(10.103) wi =∂W

∂Ji=∑

j

∂Wj(qj , J1, ... , JN)

∂Ji

(10.106) ~w(t) = ~v t + ~β

(10.111) qk(t) = Re(

∑

~j

a(k)~j

e2πi~j·(~v t+~β)

)

( libration )

(10.114) qk(t) = q0,k (vk t+ βk) + Re(

∑

~j

a(k)~j

e2πi~j·(~v t+~β)

)

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 variablesand astronomical parameters

Formulas:(10.129) Jϕ =

∮

∂W

∂ϕdϕ =

∮

αϕ dϕ = αϕ

∮

dϕ = 2π αϕ

Jϑ =

∮

∂W

∂ϑdϑ =

∮

√

α2ϑ −

α2ϕ

sin2 ϑdϑ

Jr =

∮

∂W

∂rdr =

∮

√

2mE +2mk

r− α2

ϑ

r2

(10.140) H = E = − 2π2mk2

(Jr + Jϑ+ Jϕ)2

(10.144) w1 = wϕ − wϑ , w2 = wϑ − wr , w3 = wr

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

(10.146) H = − 2π2mk2/J 23

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) d~Ldt = γ ~L× ~B

(5.104) ~ωL = − q2m

~B

(5.110) L = 12

∑

imi v′i2 − V (rjk)− 1

2 IL ω2L

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)

∂

∂Pi∆Ki(Qi, Pi) = Qi+1 ,

∂

∂Qi∆Ki(Qi, Pi) = − Pi+1

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) mx− ǫ (1−x2) x+mω2o x

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) ∼ s0 eλt

10.5 Poincare maps (sections)

10.5

Concepts: Poincare section, Poincare map

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10.6 Henon-Heiles Hamiltonian

10.6

Concepts: Henon-Heiles potential

island of integrability

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