Classical Mechanics I
A calculus-based second course in Newtonian mechanics that pairs every derivation with runnable Python: Newton's laws as differential equations in state space, the domain where classical mechanics holds and the RK4 integrator, inertial frames, Galilean relativity, systems of particles with centre-of-mass motion, momentum and angular momentum conservation and the limits of the third law, linear and quadratic drag with the Reynolds number, terminal velocity and projectile range, and variable-mass motion with the rocket equation, staging and gravity losses, energy methods in one dimension with turning points, stability, the period integral and the exact pendulum period from the arithmetic-geometric mean, anharmonic and nonlinear oscillators including Lindstedt-Poincaré perturbation theory, the driven Duffing oscillator and its hysteresis, parametric resonance and limit cycles, polar and spherical coordinates, gradient and curl tests for conservative forces and charged particles in magnetic and crossed fields, the two-body problem with reduced mass, the effective potential, the Binet equation, stability of circular orbits, precession and Bertrand's theorem, Kepler's laws derived from the inverse-square law with vis-viva, Kepler's equation, Hohmann transfers and gravity assists, scattering with impact parameters, differential cross-sections and the Rutherford formula tested by Monte Carlo, rotating frames with centrifugal and Coriolis forces, effective gravity and the Foucault pendulum, rigid bodies with the inertia tensor, principal axes, Euler's equations, the intermediate axis theorem and gyroscopic precession, and a capstone on chaos through Lyapunov exponents, the logistic map, the driven and double pendulums and the three-body problem.
PHYS-244 Year 2 3 credits 12 lessons · 4 h 45 min ▶ Video lessons coming soon
Offered by HSM University · Physics
This is the Canadian version of this course. See the American version.
What you'll learn
- Write Newton's second law as a first-order system in state space, judge when classical mechanics applies, and integrate equations of motion with the RK4 method.
- Apply Newton's laws to systems of particles, using the centre of mass, momentum and angular momentum conservation, and recognize when the third law needs field momentum.
- Solve motion with linear and quadratic drag, choose between them using the Reynolds number, and apply the rocket equation, staging and gravity losses.
- Use potential energy curves to find turning points, equilibria and their stability, and compute periods of one-dimensional oscillations, including the exact pendulum period.
- Analyse anharmonic and nonlinear oscillators with Lindstedt–Poincaré perturbation theory, and explain hysteresis, parametric resonance and limit cycles.
- Use polar, cylindrical and spherical coordinates, test forces for conservativeness with the curl, and describe charged-particle motion in magnetic and crossed fields.
- Reduce the two-body problem to one body with reduced mass, use the effective potential and the Binet equation, and assess the stability and precession of orbits.
- Derive Kepler's laws from the inverse-square force and apply vis-viva, Kepler's equation and Hohmann transfers to orbital problems.
- Define and compute differential and total cross-sections, derive the Rutherford formula, and convert between laboratory and centre-of-mass frames.
- Derive and apply the fictitious forces in accelerating and rotating frames, including effective gravity, Coriolis deflection and the Foucault pendulum.
- Compute inertia tensors and principal axes, use Euler's equations to analyse rotational stability, and calculate free and torque-induced precession.
- Identify chaotic motion through Lyapunov exponents, bifurcation diagrams and Poincaré sections, and explain how chaos limits prediction in mechanical systems.
Course content
12 lessons · 4 h 45 min total · final exam
- 1 Why classical mechanics: from Newton to the edge of chaos 25 min · quiz
- 2 Newton's laws revisited: frames, forces and momentum 24 min · quiz
- 3 Motion with drag and variable mass 26 min · quiz
- 4 Energy, potentials and stability in one dimension 22 min · quiz
- 5 Oscillators revisited: anharmonic and nonlinear motion 22 min · quiz
- 6 Motion in three dimensions and vector calculus tools 26 min · quiz
- 7 Central forces and the two-body problem 23 min · quiz
- 8 Kepler orbits and orbital mechanics 22 min · quiz
- 9 Scattering and the Rutherford cross-section 23 min · quiz
- 10 Non-inertial frames: centrifugal, Coriolis and the Foucault pendulum 23 min · quiz
- 11 Rigid bodies: inertia tensor, Euler's equations and gyroscopes 25 min · quiz
- 12 Capstone: chaos and the limits of prediction 24 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A calculus-based second course in Newtonian mechanics that pairs every derivation with runnable Python: Newton's laws as differential equations in state space, the domain where classical mechanics holds and the RK4 integrator, inertial frames, Galilean relativity, systems of particles with centre-of-mass motion, momentum and angular momentum conservation and the limits of the third law, linear and quadratic drag with the Reynolds number, terminal velocity and projectile range, and variable-mass motion with the rocket equation, staging and gravity losses, energy methods in one dimension with turning points, stability, the period integral and the exact pendulum period from the arithmetic-geometric mean, anharmonic and nonlinear oscillators including Lindstedt-Poincaré perturbation theory, the driven Duffing oscillator and its hysteresis, parametric resonance and limit cycles, polar and spherical coordinates, gradient and curl tests for conservative forces and charged particles in magnetic and crossed fields, the two-body problem with reduced mass, the effective potential, the Binet equation, stability of circular orbits, precession and Bertrand's theorem, Kepler's laws derived from the inverse-square law with vis-viva, Kepler's equation, Hohmann transfers and gravity assists, scattering with impact parameters, differential cross-sections and the Rutherford formula tested by Monte Carlo, rotating frames with centrifugal and Coriolis forces, effective gravity and the Foucault pendulum, rigid bodies with the inertia tensor, principal axes, Euler's equations, the intermediate axis theorem and gyroscopic precession, and a capstone on chaos through Lyapunov exponents, the logistic map, the driven and double pendulums and the three-body problem.
Part of the Science stream, typically taken in year 2.