Classical Dynamics

Free and easy access to complete set of presentable lecture notes and exercises is available on URI Digital Commons (downloadable pdf files covering entire sections). The most recent updates are available only here (slides marked *).

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
[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]  

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
[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]

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



Do you have a question about any of the problems [mex]?
Do you need a hint?
Do you wish to suggest additional problems?
Do you have a question about the lecture notes [mln,msl]?
Did you find any mistakes?
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Last updated 11/15/16