Classical Dynamics
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1. Newtonian Mechanics
- Classical mechanics overview [mln69]
- Space and time -- Galilean
relativity -- Newton's laws of
dynamics [mln1]
- Impact of symmetry [mln70]
- Conservation laws [mln2]
Exercises:
2. Newtonian Gravitation
- Newton's law of gravitation [mln3]
Exercises:
- Gravitational collapse of cold cloud of dust [mex2]
- Gravitational field and and potential of
interstellar dust cloud [mex3]
- Gravitational potential of a homogeneous rod [mex103]
- Gravitational self energy of a homogeneous
massive sphere [mex104]
- Gravitational field of a homogeneous massive
sphere [mex105]
- Gravitational field of an inhomogeneous
massive sphere [mex106]
- Gravitational potential of a
homogeneous disk [mex152]
- Flat Earth versus round Earth [mex153]
3. Simple Dynamical Systems
- Solution by quadrature [mln4]
- Solution by separation of variables [mln72]
Exercises:
- Periodic motion in quadratic and quartic
potentials [mex5]
- Range and duration of attenuated motion [mex15]
- Projectile in resistive medium [mex16]
- Rocket motion in resistive medium [mex17]
- Rocket launch in uniform gravitational field [mex18]
- A drop of fluid disappearing [mex101]
- Balancing the water level in a cone [mex112]
- Position-dependent acceleration [mex203]
- Growth of falling raindrop [mex229]
- Modeling attenuation [mex230]
- Potential energy of periodic motion
reconstructed [mex232]
- Exponential attenuation [mex257]
4. Fixed Points and Limit Cycles
- One degree of freedom [mln71]
- Phase portrait of conservative systems [msl5]
- Phase portrait: particle in double-well
potential [msl7]
- Phase portrait: plane pendulum [msl8]
- Velocity vector field in phase plane [mln109]
- Phase portrait: magnetic pendulum [msl9]
- Classification of fixed points in plane [mln73]
- Table of fixed points in 2D phase space [msl10]
- Isoclines [mln31]
- Limit cycles [mln74]
- Feedback control [mln33]
- Logistic model (continuous version) [mln32]
- Summary of properties [mln14]
Exercises:
5. Lagrangian Mechanics I
- Challenges for Newtonian mechanics [mln75]
- Holonomic constraints [mln36]
- Example: disk rolling along incline [mln76]
- Newtonian mechanics in the presence of holonomic
constraints [mln5]
- D'Alembert's principle [mln7]
- Lagrange equations derived from D'Alembert's
principle [mln8]
- Simple applications of Lagrangian mechanics [mln77]
Exercises:
- Lagrangian of plane double pendulum [mex20]
- Gauge invariance of Lagrange equations [mex21]
- Find a simpler Lagrangian [mex22]
- Pendulum under forced rotation [mex23]
- Pendulum with sliding pivot: Lagrange
equations [mex24]
- Pendulum without gravity [mex25]
- Pendulum oscillations in rotating plane [mex39]
- Invariance under point transformations of Lagrange
equations [mex79]
- Disk rolling on rotating track [mex116]
- Parabolic slide [mex131]
- Plane pendulum I [mex132]
- Heavy particle sliding inside cone I [mex133]
- Plane pendulum II [mex134]
- Heavy particle sliding inside cone II [mex135]
- Plane pendulum III: librations [mex146]
- Plane pendulum IV: separatrix motion and rotations [mex147]
- Chain sliding off the edge of table without friction [mex148]
- Chain sliding off the edge of table with friction [mex149]
- Pendulum with sliding pivot: reduction to
quadrature [mex233]
- Heading toward moving target [mex235]
- 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]
6. Lagrangian Mechanics II
- Constants of the motion [mln10]
- Conservation laws and symmetry [mln11]
- Routhian function [mln39]
- Noether's theorem I [mln12]
- Noether's theorem II [mln13]
- Noether's theorem III [mln42]
- Dissipative forces in Lagrangian mechanics [mln9]
- Differential constraints [mln37]
- Disk rolling down along incline [mln76]
- Generalized forces of constraint in Lagrangian mechanics
[mln15]
Exercises:
- Static frictional force of constraint [mex32]
- Normal force of constraint [mex33]
- Particle sliding down sphere (revisited) [mex34]
- Noether's theorem: translation in space [mex35]
- Noether's theorem: rotation in space [mex36]
- Noether's theorem: pure Galilei transformation
[mex37]
- Rotating and sliding [mex115]
- Routhian function of 2D harmonic oscillator [mex121]
- Motion with friction on inclined plane [mex151]
- Kinetic energy in Lagrangian mechanics [mex155]
- Spherical pendulum: reduction to quadrature [mex156]
- Routhian function for heavy particle sliding inside cone
[mex157]
- Linearly damped spherical pendulum [mex158]
- Particle sliding inside cone: normal force of constraint
[mex159]
7. Lagrangian Mechanics III
- Calculus of variation [mln78]
- Variational problems with auxiliary
conditions [mln16]
- Extremum principles [msl20]
- Geodesics [mln38]
- Generalized forces of constraint
and Hamilton's principle [mln17]
Exercises:
- Shortest path between two points in a plane
I [mex26]
- Economy plastic cup [mex27]
- Isoperimetric problem [mex28]
- Athletic refraction [mex29]
- Brachistochrone problem I [mex30]
- Brachistochrone problem II [mex31]
- Catenary problem II [mex38]
- 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]
- Massive dimer on skates [mex122]
- Isochronous potential well [mex144]
- Bead sliding down cylindrical spiral [mex160]
- Massive dimer skating on incline [mex161]
- Wave equation from Hamilton's principle [mex162]
- Catenary problem I [mex278]
- Soap film between parallel circles [mex279]
8. Central Force Motion I
- Central force motion: two-body problem [mln66]
- Central force motion: one-body problem [mln67]
- Central force problem: formal solution [mln18]
- Orbits of power-law potentials [msl21]
- Kepler's laws of planetary motion [msl22]
- Orbits of Kepler problem [msl23]
- Virial theorem [mln68]
- Bounded orbits open or closed [mln79]
- Bertrand's theorem [mln44]
Exercises:
- Discounted gravity: 50% off [mex40]
- In search of some hyperbolic orbit [mex41]
- Kepler's second and third laws [mex43]
- Orbit of the inverse-square potential at large
angular momentum [mex46]
- Orbit of the inverse-square potential at small
angular momentum [mex47]
- Unstable circular orbit [mex51]
- 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]
- Close encounter of the first kind [mex145]
- Changing orbit by brief rocket boost [mex163]
- Logarithmic central-force potential [mex265]
- Linear central-force potential [mex271]
- Kepler orbital equation [mex274]
- Sudden turn [mex281]
9. Central Force Motion II
- Motion in time on elliptic Kepler orbit [mln19]
- Orbital differential equation [mln46]
- Laplace-Runge-Lenz vector [mln45]
- Precession of the perihelion [mln21]
Exercises:
- Free fall with or without angular momentum [mex42]
- Cometary motion on parabolic orbit [mex44]
- The comet and the planet [mex45]
- Orbital differential equation applied to the
Kepler problem [mex48]
- Exponential spiral orbit [mex49]
- Crash course on circular orbit [mex50]
- Linear spiral orbit [mex52]
- Circular orbit of the Yukawa potential [mex54]
- Circular and radial motion in inverse-square
law potential [mex164]
- Precession of the perihelion: orbital integral [mex165]
- Precession of the perihelion: orbital differential
equation [mex166]
- Elliptic and hyperbolic orbits [mex169]
- Cometary motion on hyperbolic orbit [mex234]
10. Scattering from Central Force Potential
- Scattering from stationary central force
potential [msl2]
- Determination of the scattering angle [mln20]
- Scattering angle in the laboratory frame [msl3]
- Small-angle scattering [mln105]
- Classical inverse scattering [mln104]
- Decay of particle I [mln102]
- Decay of particle II [mln103]
Exercises:
- Particle experiencing soft Coulomb kick [mex10]
- Scattering from hard spheres [mex55]
- Rutherford scattering formula [mex56]
- Loss of kinetic energy in elastic collision
[mex57]
- Total cross section for shower of meteorites [mex 58]
- Scattering cross section for inverse square
potential [mex59]
- Elastic scattering from hard ellipsoids [mex60]
- Mechanical refraction [mex167]
- Scattering from a spherical potential well [mex168]
- Grazing collision between flat surfaces [mex219]
- 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]
- Elastic collision: angle between scattered
particles [mex240]
- Elastic collision: velocities of scattered
particles [mex241]
- Absorption cross section of power-law
potential [mex242]
- Classical inverse
scattering problem I [mex243]
- Classical inverse scattering problem II [mex244]
- Classical inverse scattering problem III [mex245]
- Small-angle scattering from-power-law
potential [mex246]
11. Dynamics in Rotating Frames of Reference
- Motion in rotating frame of reference [mln22]
- Holonomic constraints in rotating frame [mln23]
Exercises:
- Effect of Coriolis force on falling object [mex61]
- Effects of Coriolis force on an object
projected vertically up [mex62]
- Effects of Coriolis force and centrifugal
force on falling object [mex63]
- Foucault pendulum [mex64]
- 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]
- Parabolic slide on rotating Earth [mex172]
12. Rigid Body Dynamics I
Exercises:
- Translational and rotational kinetic
energies [mex67]
- Inertia tensor of homogeneous cube [mex68]
- Parallel-axis theorem [mex69]
- Stability of rigid body rotations about
principal axes [mex70]
- Perpendicular-axis theorem [mex73]
- Simulating a stick by three point masses [mex143]
- Kinetic energy of rolling cylinder [mex173]
- Heavy wheels [mex175]
- Principal moment of a solid cylinder [mex252]
- Principal moments of a solid sphere [mex253]
- Principal moments of a solid ellipsoid [mex254]
- Principal moments of square-shaped
tiles [mex266]
- Runaway dumbbell [mex272]
13. Rigid Body Dynamics II
- Torque-free motion of symmetric top [msl27]
- Torque-free motion of asymmetric top [msl28]
- Eulerian angles of rotation [msl25]
- Eulerian angular velocities [msl26]
- Heavy symmetric top: precession and nutation [msl49]
- Heavy symmetric top: general solution [mln47]
- Heavy symmetric top: steady precession [mln81]
- Solid sphere rolling on plane [mln106]
Exercises:
14. Oscillations
- Linearly damped harmonic oscillator [mln6]
- Driven harmonic oscillator I [mln28]
- Amplitude resonance and phase angle [msl48]
- Driven harmonic oscillator II [mln29]
- Driven harmonic oscillator III [mln107]
- Small oscillations [mln43]
- Transformation to principal axes [mln30]
- Elastic chain [mln48]
Exercises:
- What is the physical nature of these modes? [mex114]
- Blocks and springs in series [mex123]
- Small oscillations of the double pendulum [mex124]
- Harmonic oscillator with friction [mex150]
- 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]
- Fourier coefficients of a sawtooth force [mex184]
- Fourier coefficients of periodic seuence of
rectangular pulses [mex185]
- Two coupled oscillators [mex186]
- Three coupled oscillators [mex187]
- Harmonic oscillator with attenuation [mex261]
- Driven harmonic oscillator: runaway resonance [mex262]
- Driven harmonic oscillator with Coulomb
damping [mex263]
15. Hamiltonian Mechanics
- Legendre transform [tln77]
- Hamiltonian
and canonical equations [mln82]
- Variational principle in phase space [mln83]
- Properties of the Hamiltonian
[mln87]
- Use of cyclic
coordinates in Lagrangian and Hamiltonian mechanics [mln84]
- Velocity-dependent potential energy [mln85]
- Charged particle in electromagnetic field [mln86]
Exercises:
- Velocity-dependent central force [mex76]
- Hamiltonian:conserved quantity or total enrgy? [mex77]
- Bead sliding on rotating rod in vertical plane [mex78]
- When does the Hamiltonian represent the total energy? [mex81]
- Particles with position-dependent mass moving in
potential [mex88]
- Pendulum with stringnof slowly increasing length [mex89]
- Lagrangian from Hamiltonian via Legendre transform [mex188]
- Can you find the Hamiltonian of this system? [mex189]
- Charged particle in a uniform magnetic field [mex190]
- Libration between inclines [mex259]
- T-pendulum [mex264]
- Wiggling cylinder [mex269]
- Parabolic slide II [mex276]
16. Canonical Transformations
- Point transformations (in configuration space) [mln88]
- Canonicity and volume preservation [mln90]
- Canonical transformations (in phase space) [mln89]
- Infinitesimal canonical transformations [mln91]
- Classical Hamiltonian (many-body) system [tln45]
- Classical Liouville operator [tln46]
Exercises:
- Effect of point transformation on Hamiltonian
[mex80]
- Effect of point transformation on canonical
equations [mex82]
- Time-dependent generating
functions [mex83]
- Check the canonicity of coordinate
transformations [mex84]
- Canonical transformation from rest frame to
moving frame [mex85]
- Canonical transformation applied to harmonic
oscillator [mex86]
- Determine canonicity and generating function I [mex87]
- Determine canonicity and generating function
II [mex90]
- Hamiltonian of free particle in rotating frame
[mex193]
- Determine canonicity and generating function III [mex194]
- Canonicity of gauge transformation [mex195]
- Electromagnetic gauge transformation [mex196]
- Determine canonicity and generating function IV [mex198]
- Canonicity and generating function V [mex283]
17. Action-Angle Coordinates
- Action-angle coordinates [mln92]
- Actions and angles for librations [mln93]
- Actions and angles for rotations [mln94]
- Poisson brackets [msl30]
- Specifications of Hamiltonian system [mln95]
Exercises:
- Action-angle coordinates of the harmonic
oscillator [mex91]
- Action-angle coordinates of an anharmonic
oscillator [mex92]
- Unbounded motion in piecewise linear
periodic potential [mex93]
- Hamiltonian system specified by noncanonical
variables [mex94]
- Bounded motion in piecewise constant
periodic potential [mex95]
- Unbounded motion in piecewise constant
periodic potential [mex96]
- Poisson's theorem [mex191]
- Poisson brackets of angular momentum
variables [mex192]
- Generating a pure Galilei transformation [mex197]
- Exponential
potential [mex199]
- Action-angle coordinates of plane pendulum: librations [mex200]
18. Hamilton-Jacobi Theory
- Hamilton's principal function [mln96]
- Hamilton's characteristic function [mln97]
- Hamilton-Jacobi 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 time-dependent field [mex201]
- Hamilton-Jacobi theory for projectile motion
[mex202]
19. Deterministic Chaos
- Dynamical systems with one degree of freedom [mln14]
- 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]
- 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]
- Henon-Heiles system (nonintegrable) [msl13]
- Introduction to Hamiltonian chaos [mln108]
20. Relativistic Mechanics I
- Relativistic versus Newtonian mechanics [mln49]
- Relativity of space and time [mln50]
- Relativity of simultaneity [mln51]
- Time dilation paradox [mln52]
- Length contraction paradox [mln53]
- Minkowski diagram I: relativity of simultaneity [mln54]
- Minkowski diagram II: length contraction and time
dilation [mln55]
- Relative and absolute [mln59]
- Twin paradox [mln56]
- Longitudinal Doppler effect [mln57]
Exercises:
21. Relativistic Mechanics II
Exercises:
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. Wiley-VCH 1997.
- T. M. Helliwell and V. V. Sahakian: Modern Classical
Mechanics. Cambridge University Press 2021.
Some Relevant Monographs
- V. I. Arnold: Mathematical Methods of Classical Mechanics.
Springer-Verlag, 1978.
- A. J. Lichtenberg and M. A. Lieberman: Regular and
Stochastic Motion. Springer-Verlag, 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.
Advanced Course in Nonlinear Dynamics at URI: MCE663
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[mex]?
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Last updated 11/27//22