Classical Dynamics
Newtonian mechanics
Classical mechanics overview [mln69]
Space and time  Galilei's principle of
relativity  Newton's laws of dynamics [mln1]
Impact of symmetry [mln70]
Conservation laws [mln2]
The shortest path is not the quickest path [mex100]
Minimizing time of slide when
friction
is present [mex154]
Optimized time of travel [mex136]
Acceleration from clocking consecutive space intervals [mex137]
Particle sliding down a sphere [mex1]
Time of slide and time of flight [mex102]
Atwood machine [mex9]
On frozen pond [mex204]
The quick, the short, and the
scenic
[mex205]
When push comes to shove
[mex206]
Rubber speed [mex138]
Water projected into air by wheel rolling on wet
road
[mex11]
Design of a lawn sprinkler [mex113]
Longest shot from the top of a hill [mex139]
Lowest shot to target across hill [mex140]
Reel of thread I: statics [mex141]
Reel of thread II: dynamics [mex142]
Spherical pendulum of varying
length
[mex226]
Dragging block by elastic cord
[mex227]
Centripetal elevator [mex228]
Lateral force on hanging chain
[mex231]
Let's meet again... and again [mex247]
Newtonian gravitation
Newton's law of gravitation [mln3]
Gravitational potential of a homogeneous rod [mex103]
Gravitational field of a homogeneous massive sphere [mex105]
Gravitational field of an inhomogeneous massive sphere [mex106]
Gravitational self energy of a homogeneous massive sphere [mex104]
Gravitational field and and potential of interstellar dust
cloud [mex3]
Gravitational collapse of cold cloud of dust [mex2]
Gravitational potential of a homogeneous disk
[mex152]
Flat Earth versus round Earth [mex153]
Simple dynamical systems
One degree of freedom [mln71]
Solution by quadrature [mln4]
Phase portraits of conservative systems [msl5]
Periodic motion in quadratic and quartic
potentials
[mex5]
Potential
energy of
periodic motion reconstructed [mex232]
Periodic motion in 2D phase space [mex6]
Separatrix tangent lines at hyperbolic point
[mex111]
Solution by separation of variables [mln72]
Rocket launch in uniform gravitational field [mex18]
A drop of fluid disappearing [mex101]
Range and duration of attenuated motion [mex15]
Projectile in resistive medium [mex16]
Balancing the water level in a cone [mex112]
Rocket motion in resistive medium [mex17]
Positiondependent
acceleration [mex203]
Growth of falling raindrop
[mex229]
Modeling attenuation [mex230]
Damped harmonic oscillator [mln6]
Harmonic oscillator with friction [mex150]
Phase portrait: particle in doublewell potential
[msl7]
Phase portrait: plane pendulum [msl8]
Phase portrait: magnetic pendulum [msl9]
Classification of fixed points in plane [mln73]
Table of fixed points in 2D phase space [msl10]
Isoclines [mln31]
Fixed points of the plane pendulum [mex12]
2D phase portrait I [mex7]
2D phase portrait II [mex8]
Predator and prey [mex13]
Host and parasite [mex14]
Isoclines and fixed points [mex108]
Fierce competition versus mild competition [mex109]
Limit cycles [mln74]
Hopf bifurcation [mex19]
Feedback control [mln33]
Balancing a heavy object on a light rod [mex110]
Logistic model (continuous version) [mln32]
Continuous logistic model
[mex107]
Summary of properties [mln14]
Lagrangian mechanics
Challenges for Newtonian mechanics [mln75]
Holonomic constraints [mln36]
Example: disk rolling along incline [mln76]
Differential constraints [mln37]
Heading toward moving target
[mex235]
Newtonian mechanics in the presence of holonomic constraints [mln5]
Plane pendulum I [mex132]
Heavy particle sliding inside cone I [mex133]
D'Alembert's principle [mln7]
Plane pendulum II [mex134]
Heavy particle sliding inside cone II [mex135]
Plane pendulum III: librations [mex146]
Plane pendulum IV: separatrix motion and rotations [mex147]
Lagrange equations derived from D'Alembert's
principle
[mln8]
Simple applications of Lagrangian mechanics [mln77]
Invariance under point transformations of Lagrange equations
[mex79]
Gauge invariance of Lagrange equations
[mex21]
Find a simpler Lagrangian [mex22]
Lagrangian of plane double pendulum [mex20]
Parabolic slide [mex131]
Pendulum without gravity [mex25]
Disk rolling on rotating track [mex116]
Rotating and sliding [mex115]
Pendulum under forced rotation [mex23]
Pendulum with sliding pivot: Lagrange equations [mex24]
Pendulum with sliding pivot: reduction to
quadrature [mex233]
Pendulum oscillations in rotating plane [mex39]
Chain sliding off the edge of table without friction
[mex148]
Chain sliding off the edge of table with friction [mex149]
Plane pendulum with periodically driven pivot I [mex248]
Plane pendulum with periodically
driven pivot II [mex249]
Plane pendulum with
periodically driven pivot III [mex250]
Restoring force of elastic
string
[mex251]
Constants of the motion [mln10]
Conservation laws and symmetry [mln11]
Kinetic energy in Lagrangian mechanics [mex155]
Spherical pendulum: reduction to quadrature [mex156]
Routhian function [mln39]
Routhian function for heavy particle sliding inside cone [mex157]
Routhian function of 2D harmonic oscillator [mex121]
Noether's theorem I [mln12]
Noether's theorem: translation in space
[mex35]
Noether's theorem: rotation in space [mex36]
Noether's theorem II [mln13]
Noether's theorem: pure Galilei
transformation
[mex37]
Noether's theorem III [mln42]
Dissipative forces in Lagrangian mechanics [mln9]
Motion with friction on inclined plane [mex151]
Linearly damped spherical pendulum [mex158]
Generalized forces of constraint in Lagrangian mechanics [mln15]
Particle sliding down sphere (revisited) [mex34]
Static frictional force of constraint [mex32]
Normal force of constraint [mex33]
Particle sliding inside cone: normal force of
constraint
[mex159]
Calculus of variation [mln78]
Shortest path between two points in a plane I
[mex26]
Economy paper cup [mex27]
Variational problems with auxiliary conditions [mln16]
Isoperimetric problem [mex28]
Catenary problem [mex38]
Athletic refraction [mex29]
Brachistochrone problem I [mex30]
Brachistochrone problem II [mex31]
Isochronous potential well [mex144]
Geodesics [mln38]
Shortest path between two point in a plane II [mex117]
Geodesics on a sphere [mex118]
Dynamical trap without potential energy [mex119]
Vertical range of particle sliding inside cone [mex120]
Extremum principles [msl20]
Generalized forces of constraint and Hamilton's
principle
[mln17]
Bead sliding down cylindrical spiral [mex160]
Massive dimer on skates [mex122]
Massive dimer skating on incline [mex161]
Wave equation from Hamilton's principle [mex162]
Central force motion
Central force motion: twobody problem [mln66]
Central force motion: onebody problem [mln67]
Central force problem: formal solution [mln18]
Orbits of powerlaw potentials [msl21]
Unstable circular orbit [mex51]
Orbit of the inversesquare potential at
large
angular momentum [mex46]
Orbit of the inversesquare potential at
small
angular momentum [mex47]
In search of some hyperbolic orbit [mex41]
Virial theorem [mln68]
Changing orbit by brief rocket
boost
[mex163]
Discounted gravity: 50% off [mex40]
Bounded orbits open or closed [mln79]
Bertrand's theorem [mln44]
Stability of circular orbits
[mex53]
Small oscillations of radial
coordinate
about circular orbit [mex125]
Angle between apsidal vectors
for
nearly
circular orbits [mex126]
Robustness of apsidal angles
[mex127]
Apsidal angle reinterpreted
[mex128]
Apsidal angle at very high
energies
[mex129]
Apsidal angle at very low
energies
[mex130]
Kepler's laws of planetary motion [msl22]
Orbits of Kepler problem [msl23]
Motion in time on elliptic orbit [mln19]
Cometary motion on parabolic orbit [mex44]
Cometary motion on hyperbolic orbit [mex234]
Close encounter of
the
first kind [mex145]
Kepler's second and third laws
[mex43]
Circular and radial motion in
inversesquare
law potential [mex164]
Circular orbit of the Yukawa potential [mex54]
Orbital differential equation [mln46]
Exponential spiral orbit [mex49]
Orbital differential equation applied to
the
Kepler
problem [mex48]
Linear spiral orbit [mex52]
Crash course on circular orbit [mex50]
LaplaceRungeLenz vector [mln45]
Precession of the perihelion [mln21]
Precession of the
perihelion:
orbital integral [mex165]
Precession of the perihelion:
orbital
differential
equation [mex166]
The comet
and
the
planet [mex45]
Free fall with or without angular momentum
[mex42]
Elliptic and hyperbolic
orbits
[mex169]
Scattering
from
central
force potential
Scattering from stationary central force potential
[msl2]
Determination of the scattering angle [mln20]
Total cross section for shower of
meteorites
[mex 58]
Rutherford scattering formula [mex56]
Scattering from hard spheres [mex55]
Elastic scattering from hard ellipsoids
[mex60]
Scattering cross section for inverse
square
potential
[mex59]
Particle experiencing soft Coulomb kick [mex10]
Scattering angle in the laboratory frame [msl3]
Loss
of
kinetic energy in elastic collision [mex57]
Elastic collision: angle between
scattered particles [mex240]
Elastic collision: velocities of
scattered particles [mex241]
Mechanical refraction [mex167]
Scattering from a spherical
potential
well
[mex168]
Grazing collision between flat
surfaces [mex219]
Absorption cross section of
powerlaw
potential [mex242]
Smallangle scattering [mln105]
Smallangle scattering
frompowerlaw
potential [mex246]
Classical inverse scattering [mln104]
Classical inverse scattering
problem I
[mex243]
Classical
inverse scattering problem II [mex244]
Classical inverse
scattering problem III [mex245]
Decay of particle I [mln102]
Decay of particle II [mln103]
Decay of particle: maximum
kinetic
energy [mex237]
Decay of particle: directions in
lab
frame I [mex238]
Decay of particle: directions in
lab
frame II [mex239]
Dynamics in rotating frames of reference
Noninertial reference frames
Motion in rotating frame of reference [mln22]
Effect of Coriolis force on falling object
[mex61]
Effects of Coriolis force on an object projected
vertically
up [mex62]
Foucault pendulum [mex64]
Effects of Coriolis force and centrifugal force
on
falling
object [mex63]
Lateral deflection of projectile due to Coriolis
force
[mex65]
Effect of Coriolis force on range of projectile
[mex66]
What is vertical? [mex170]
Lagrange equations in
rotating
frame
[mex171]
Holonomic constraints in rotating frame [mln23]
Parabolic slide on rotating
Earth
[mex172]
Rigid body dynamics
Coordinate systems used in rigid body dynamics [mln24]
Rotational kinetic energy [mln25]
Translational and rotational kinetic energies
[mex67]
Kinetic energy of rolling
cylinder
[mex173]
Principal
axes
of
inertia
[mln80]
Parallelaxis theorem [mex69]
Perpendicularaxis theorem [mex73]
Inertia tensor of homogeneous cube [mex68]
Inertia tensor of a cone [mex71]
Simulating a stick by three
point
masses
[mex143]
Angular momentum [mln26]
Eulerian angles of rotation [msl25]
Eulerian angular velocities [msl26]
Rotating rectangular
box
[mex174]
Euler's equations [mln27]
Heavy wheels [mex175]
Torquefree motion of symmetric top [msl27]
Torquefree motion of asymmetric top [msl28]
Stability of rigid body rotations about principal
axes
[mex70]
Steady precession of
symmetric
top [mex176]
Heavy symmetric top: general solution [mln47]
Heavy symmetric top: steady precession [mln81]
Heavy symmetric top: precession and nutation [msl49]
Stability of sleeping top
[mex177]
Cube standing on edge [mex72]
Rolling
pendulum
[mex178]
Cone on the roll [mex74]
Make the billiard ball roll [mex4]
From sliding to rolling motion
[mex220]
Rolling inhomogeneous
disk
[mex179]
Balancing act of board on cylinder [mex75]
Oscillations
Driven harmonic oscillator [mln28]
Amplitude resonance and phase angle [msl48]
Driven harmonic oscillator:
steady
state
solution [mex180]
Driven harmonic oscillator:
kinetic
and
potential energy [mex181]
Driven harmonic oscillator:
power
input
[mex182]
Quality factor of damped
harmonic
oscillator
[mex183]
Anharmonically driven harmonic oscillator [mln29]
Fourier coefficients of a
sawtooth
force
[mex184]
Fourier coefficients of periodic
seuence
of rectangular pulses [mex185]
Small oscillations [mln43]
Transformation to principal axes [mln30]
Elastic chain [mln48]
Blocks and springs in series
[mex123]
Two coupled oscillators
[mex186]
Three coupled oscillators
[mex187]
What is the physical nature of
these
modes?
[mex114]
Small oscillations of the double
pendulum
[mex124]
Hamiltonian mechanics
Hamiltonian formulation of classical mechanics
Legendre transform
Hamiltonian and canonical equations [mln82]
Lagrangian from Hamiltonian via Legendre transform [mex188]
Can you find the Hamiltonian of this system? [mex189]
Variational principle in phase space [mln83]
Properties of the Hamiltonian [mln87]
When does the Hamiltonian represent the total
energy?
[mex81]
Hamiltonian: conserved quantity or total energy?
[mex77]
Bead
sliding on rotating rod in vertical plane
[mex78]
Use of cyclic coordinates in Lagrangian and
Hamiltonian
mechanics [mln84]
Velocitydependent
potential energy [mln85]
Charged particle in electromagnetic field [mln86]
Velocitydependent central force [mex76]
Charged particle in a uniform magnetic field [mex190]
Particle with positiondependent mass
moving
in
1D potential [mex88]
Pendulum with string of slowly increasing length
[mex89]
Canonical transformations
Point
transformations (of coordinates in configuration
space) [mln88]
Effect of point transformation on Hamiltonian
[mex80]
Effect of point transformation on canonical
equations
[mex82]
Hamiltonian of free particle in rotating frame
[mex193]
Canonical transformations (of coordinates in phase
space) [mln89]
Canonicity and volume preservation [mln90]
Determine canonicity and generating function I [mex87]
Determine canonicity and generating function II
[mex90]
Determine canonicity and generating function III [mex194]
Determine
canonicity
and generating function IV [mex198]
Infinitesimal canonical transformations [mln91]
Canonicity of time evolution: Liouville theorem [tln45], [tln46]
Canonicity of gauge transformation [mex195]
Electromagnetic gauge transformation [mex196]
Check the canonicity of coordinate
transformations
[mex84]
Timedependent generating functions [mex83]
Canonical transformation from rest frame to
moving
frame
[mex85]
Canonical transformation applied to harmonic
oscillator
[mex86]
Actionangle coordinates
Actionangle coordinates [mln92]
Actions and angles for librations [mln93]
Actions and angles for rotations [mln94]
Actionangle coordinates of the harmonic
oscillator
[mex91]
Actionangle coordinates of an anharmonic
oscillator
[mex92]
Unbounded motion in piecewise constant periodic
potential
[mex96]
Unbounded motion in piecewise linear periodic
potential
[mex93]
Bounded motion in piecewise constant periodic
potential
[mex95]
Poisson brackets [msl30]
Specifications of Hamiltonian system [mln95]
Poisson's theorem [mex191]
Poisson brackets of angular momentum variables [mex192]
Actionangle coordinates of plane pendulum: librations [mex200]
Hamiltonian system specified by noncanonical
variables
[mex94]
Generating a pure Galilei transformation [mex197]
Exponential potential [mex199]
HamiltonJacobi theory
Hamilton's principal
function [mln96]
Hamilton's characteristic
function [mln97]
HamiltonJacobi equation for the harmonic
oscillator
[mex97]
Hamilton's principal function for central force
problem
[mex98]
Hamilton's characteristic function for central
force
problem
[mex99]
Particle in timedependent field [mex201]
HamiltonJacobi theory for projectile motion [mex202]
Hamiltonian chaos
Integrability as a universal property [mln98]
Integrability as a contingent property [mln99]
Poincaré surface of section [mln100]
Summary of properties [msl15]
Toda system (integrable) [msl12]
HenonHeiles
system
(nonintegrable) [msl13]
Dissipative chaos
Dissipative dynamical systems [mln101]
Fixed points in 3D phase flow [msl16]
Limit cycles in 3D phase flow [msl17]
Toroidal attractor in 3D phase flow [msl18]
Strange attractor in 3D phase flow: Roessler band
[msl19]
Relativistic mechanics (largely following Kogut 2001)
Relativistic versus Newtonian mechanics [mln49]
Relativity of space and time [mln50]
Relativity of simultaneity [mln51]
Time dilation paradox [mln52]
Length contraction paradox [mln53]
Hello Earth [mex207]
Who passes more quickly?
[mex208]
Time on the fly [mex236]
Pion decay in accelerator
[mex209]
Interstellar travel [mex210]
TGV [mex211]
Minkowski diagram I: relativity of simultaneity [mln54]
Minkowski diagram II: length contraction and time dilation [mln55]
Twin paradox [mln56]
Longitudinal Doppler effect [mln57]
Optical birthday cards [mex212]
Two views of an event [mex213]
Hello Earth again [mex214]
Coordinate transformations [mln58]
Relative and absolute [mln59]
Lorentz transformation I
[mex215]
Lorentz transformation II
[mex216]
Observing transverse motion of meter stick [mln60]
Skater's paradox [mln61]
Skate mail fallacy [mex217]
Interstellar speed control
[mex218]
Mass and energy [mln62]
Relativistic momentum [mln63]
Momentum conservation [mex221]
Relativistic mass [mex222]
Relativistic energy I [mln64]
Relativistic energy II [mln65]
Photon rocket [mex223]
Photon absorption and photon
emission
[mex224]
K meson decay [mex225]
Some Relevant Textbooks
 H. Goldstein: Classical Mechanics. Addison Wesley,
1981.
 I. Percival and D. Richards: Introduction to Dynamics.
Cambridge
University Press, 1982.
 L. D. Landau and E. M. Lifshitz: Mechanics. Pergamon
Press,
1976.
 J. V. José and E. J. Saletan: Classical Dynamics: A
Contemporary
Approach. Cambridge University Press, 1998.
 J. L. McCauley: Classical Mechanics: Transformation,
Flows,
Integrable
and Chaotic Dynamics. Cambridge University Press, 1997.
 D. T. Greenwood: Classical Dynamics. Dover
Publications
1997.
 J. B. Kogut: Introduction
to
Relativity. Harcourt/Academic Press 2001.
 Friedhelm Kuypers: Klassische
Mechanik. WileyVCH 1997.
Some Relevant Monographs
 V. I. Arnold: Mathematical Methods of Classical Mechanics.
SpringerVerlag,
1978.
 A. J. Lichtenberg and M. A. Lieberman: Regular and
Stochastic
Motion.
SpringerVerlag, 1983.
 R. C. Hilborn: Chaos and Nonlinear Dynamics. An
Introduction
for
Scientists and Engineers. 2nd edition. Oxford University
Press 2000.
 R. C. Hilborn and N. B. Tufillaro (Eds.): Chaos and
Nonlinear
Dynamics.
(collection of reprinted articles) AAPT Publication, 1999.
 M. Tabor: Chaos and Integrability in Nonlinear Dynamics 
An
Introduction.
Wiley, 1989.
 R. H. Abraham and C. D. Shaw: Dynamics

The Geometry of Behavior. Aerial Press, Santa Cruz
1984.
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[mex]?
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Last updated 10/19/12