- Classical mechanics overview [mln69]
- Space and time -- Galilean relativity -- Newton's laws of dynamics [mln1]
- Impact of symmetry [mln70]
- Conservation laws [mln2]

Exercises:

- Particle sliding down a sphere [mex1]
- Atwood machine [mex9]
- Water projected into air by wheel rolling on wet road [mex11]
- The shortest path is not the quickest path [mex100]
- Time of slide and time of flight [mex102]
- Design of a lawn sprinkler [mex113]
- Optimized time of travel [mex136]
- Acceleration from clocking consecutive space intervals [mex137]
- Rubber speed [mex138]
- 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]
- Minimizing time of slide when friction is present [mex154]
- On frozen pond [mex204]
- The quick, the short, and the scenic [mex205]
- When push comes to shove [mex206]
- 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]
- Elastic collision on airtrack [mex267]

- Anharmonic oscillator of sorts [mex270]

- Round and round, back and forth [mex273]

- Bouncing ball [mex277]

- Kickback [mex284]

- Newton's law of gravitation [mln3]

- 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]

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]

- 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:

- Periodic motion in 2D phase space [mex6]

- 2D phase portrait I [mex7]

- 2D phase portrait II [mex8]

- Fixed points of the plane pendulum [mex12]
- Predator and prey [mex13]
- Host and parasite [mex14]

- Hopf bifurcation [mex19]

- Continuous logistic model [mex107]
- Isoclines and fixed points [mex108]
- Fierce competition versus mild competition [mex109]
- Balancing a heavy object on a light rod [mex110]

- 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]

- 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]

- 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]

- 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]

- Motion in time on elliptic Kepler orbit [mln19]
- Orbital differential equation [mln46]
- Laplace-Runge-Lenz vector [mln45]
- Precession of the perihelion [mln21]

- 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]

- 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]

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]

- Coordinate systems used in rigid body dynamics [mln24]
- Rotational kinetic energy [mln25]
- Principal axes of inertia [mln80]
- Angular momentum [mln26]
- Euler's equations [mln27]

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]

- 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:

- Make the billiard ball roll [mex4]
- Inertia tensor of a cone [mex71]
- Cube standing on edge [mex72]
- Cone on the roll [mex74]
- Balancing act of board on cylinder [mex75]
- Rotating rectangular box [mex174]
- Steady precession of symmetric top [mex176]
- Stability of sleeping top [mex177]
- Rolling pendulum [mex178]
- Rolling inhomogeneous disk [mex179]
- From sliding to rolling motion [mex220]
- Inertia tensor of four-atomic molecule [mex255]
- Falling flat [mex256]
- Rod off balance [mex258]
- Solid sphere rolling on plane [mex260]
- Inelastic crossroad collision [mex268]

- T-bar pendulum [mex275]
- Motorcycle treadmill [mex282]

- 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]

- 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]

- 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]

- 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]

- 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]

- 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]

- 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]

- 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:

- Hello Earth [mex207]
- Who passes more quickly [mex208]
- Pion decay in accelerator [mex209]

- Interstellar travel [mex210]
- TGV [mex211]
- Optical birthday cards [mex212]
- Two views of an event [mex213]
- Hello earth again [mex214]
- Interstellar speed control [mex218]
- K meson decay [mex225]
- Time on the fly [mex236]

- Coordinate transformations [mln58]
- Relative and absolute [mln59]
- Observing transverse motion of meter stick [mln60]
- Skater's paradox [mln61]
- Mass and energy [mln62]
- Relativistic momentum [mln63]
- Relativistic energy I [mln64]
- Relativistic energy II [mln65]

Exercises:

- Lorentz transformation I [mex215]

- Lorentz transformation II [mex216]
- Skate mail fallacy [mex217]
- Momentum conservation [mex221]
- Relativistic mass [mex222]
- Photon rocket [mex223]

- Photon absorption and photon emission [mex224]

**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.

- 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.

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?

Drop a note to gmuller@uri.edu.

Last updated 11/27//22