Classical Dynamics

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1. Newtonian Mechanics

Table of contents [mtc] 
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]

2. 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]

3. 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]
Position-dependent acceleration [mex203]
Growth of falling raindrop [mex229]
Modeling attenuation [mex230]
Exponential attenuation [mex257] 

4. Fixed Points and Limit Cycles

Phase portrait: particle in double-well 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]

5. Lagrangian Mechanics I

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
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]  

6. Lagrangian Mechanics II

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]

7. Lagrangian Mechanics III

Calculus of variation [mln78]
Shortest path between two points in a plane I [mex26]
Economy plastic cup [mex27]
Variational problems with auxiliary conditions [mln16]
Isoperimetric problem

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]

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] *
Unstable circular orbit [mex51]
Orbit of the inverse-square potential at large angular momentum [mex46]
Orbit of the inverse-square 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]

9. Central Force Motion II

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 inverse-square 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]
Laplace-Runge-Lenz 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]   

10. 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 power-law potential [mex242]
Small-angle scattering [mln105]
Small-angle scattering from-power-law 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] 

11. Dynamics in Rotating Frames of Reference

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]

12. Rigid Body Dynamics I

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]
Parallel-axis theorem [mex69]
Perpendicular-axis theorem [mex73]
Inertia tensor of homogeneous cube [mex68]
Principal moment of a solid cylinder [mex252] 
Principal moments of a solid sphere [mex253] 

Principal moments of a solid ellipsoid [mex254] 
Inertia tensor of four-atomic molecule [mex255]
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]

13. Rigid Body Dynamics II

Torque-free motion of symmetric top [msl27]
Torque-free 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]
Falling flat [mex256] 
Rod off balance [mex258] 
Solid sphere rolling on plane [mln106]
Solid sphere rolling on plane [mex260] 

14. Oscillations

Damped harmonic oscillator [mln6]
Harmonic oscillator with friction [mex150]
Harmonic oscillator with attenuation [mex261]
Driven harmonic oscillator I [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]
Driven harmonic oscillator: runaway resonance [mex262] 
Driven harmonic oscillator II [mln29]
Fourier coefficients of a sawtooth force [mex184]
Fourier coefficients of periodic seuence of rectangular pulses [mex185]
Driven harmonic oscillator III [mln107] 
Driven harmonic oscillator with Coulomb damping [mex263]
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]

15. Hamiltonian Mechanics

Legendre transform [tln77]
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]
Velocity-dependent potential energy [mln85]
Charged particle in electromagnetic field [mln86]
Velocity-dependent central force [mex76] *
Charged particle in a uniform magnetic field [mex190]
Particle with position-dependent mass moving in 1D potential [mex88]
Pendulum with string of slowly increasing length [mex89]
Librations between inclines [mex259]  

16. 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]
Time-dependent generating functions [mex83]
Canonical transformation from rest frame to moving frame [mex85]
Canonical transformation applied to harmonic oscillator [mex86]  

17. Action-Angle Coordinates

Action-angle coordinates [mln92]
Actions and angles for librations [mln93]
Actions and angles for rotations [mln94]

Action-angle coordinates of the harmonic oscillator [mex91]
Action-angle 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]
Action-angle coordinates of plane pendulum: librations [mex200]

Hamiltonian system specified by noncanonical variables [mex94]
Generating a pure Galilei transformation [mex197]
Exponential potential [mex199]

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

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]
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]

21. Relativistic Mechanics II

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

Some Relevant Monographs

Advanced Course in Nonlinear Dynamics at URI: MCE663

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Last updated 11/15/16