Classical Mechanics II
Analytical mechanics, the reformulation of Newton's laws through action principles that underlies all of modern theoretical physics, developed with a runnable Python check in every lesson: the action and Hamilton's principle of stationary action, tested by minimizing a discretized action and by finding paths that are saddles rather than minima, the calculus of variations with the Euler–Lagrange equation, cyclic coordinates, the Beltrami identity and Fermat's principle of least time leading to Snell's law, classic variational problems including the brachistochrone and tautochrone cycloid, the catenary found with a Lagrange multiplier, minimal surfaces and geodesics on the sphere, Lagrange's equations derived from d'Alembert's principle with generalized coordinates, holonomic and rheonomic constraints, the bead on a spinning hoop and its pitchfork bifurcation and the double pendulum, constraint forces from Lagrange multipliers, loss of contact, rolling and nonholonomic constraints with the Lie bracket that lets a wheel move sideways, Noether's theorem linking translation, rotation and time-shift symmetry to momentum, angular momentum and energy, central forces, the spherical pendulum and the heavy symmetric top with Euler angles, precession and nutation, small oscillations as a generalized eigenvalue problem with mass and stiffness matrices, normal modes of CO2, chains and the triple pendulum, the Legendre transform and Hamilton's equations, phase space, charged particles with the vector potential and symplectic integrators, Poisson brackets, Liouville's theorem and its consequences for statistical mechanics, beams and optics, canonical transformations, generating functions, the Hamilton–Jacobi equation, action-angle variables and adiabatic invariants, and a capstone on integrability, the KAM theorem and chaos in the standard map, and the passage from classical mechanics to quantum mechanics and field theory.
PHYS-344 Year 3 3 credits 12 lessons · 5 h 17 min ▶ Video lessons coming soon
Offered by HSM University · Physics
What you'll learn
- Explain the principle of stationary action, compute the action of trial paths, and distinguish a stationary path from a minimum.
- Derive the Euler–Lagrange equation for a functional, use its first integrals including the Beltrami identity, and apply Fermat's principle to refraction.
- Solve classic variational problems, including the brachistochrone, the tautochrone, the catenary with a length constraint and geodesics on a sphere.
- Derive Lagrange's equations from d'Alembert's principle and use them, with suitable generalized coordinates, to obtain equations of motion for constrained systems.
- Use Lagrange multipliers to find constraint forces and loss of contact, and recognize and treat nonholonomic constraints such as rolling.
- State and prove Noether's theorem, and identify the conserved quantities that follow from translation, rotation and time-shift symmetries.
- Analyse central-force motion, the spherical pendulum and the heavy symmetric top from their Lagrangians, including effective potentials, precession and nutation.
- Find the normal modes and frequencies of systems near stable equilibrium by solving the generalized eigenvalue problem for the mass and stiffness matrices.
- Construct Hamiltonians by Legendre transformation, write and solve Hamilton's equations, and explain why symplectic integrators conserve energy over long runs.
- Use Poisson brackets to compute time derivatives and test for conserved quantities, and apply Liouville's theorem to ensembles of states.
- Test and construct canonical transformations, solve problems with the Hamilton–Jacobi equation, and use action-angle variables and adiabatic invariants.
- Explain integrability, the KAM theorem and the onset of chaos in area-preserving maps, and describe how action, brackets and phase space carry over to quantum mechanics and field theory.
Course content
12 lessons · 5 h 17 min total · final exam
- 1 Why analytical mechanics: from forces to principles 29 min · quiz
- 2 The calculus of variations and the Euler–Lagrange equation 27 min · quiz
- 3 Classic variational problems: brachistochrone, catenary and geodesics 28 min · quiz
- 4 Lagrangian mechanics: generalized coordinates and constraints 25 min · quiz
- 5 Constraint forces, Lagrange multipliers and nonholonomic systems 26 min · quiz
- 6 Symmetry and conservation: Noether's theorem 26 min · quiz
- 7 Central forces and rigid bodies revisited with Lagrangians 24 min · quiz
- 8 Small oscillations and normal modes: the eigenvalue method 27 min · quiz
- 9 Hamiltonian mechanics: Legendre transform and Hamilton's equations 27 min · quiz
- 10 Phase space, Poisson brackets and Liouville's theorem 25 min · quiz
- 11 Canonical transformations and Hamilton–Jacobi theory 27 min · quiz
- 12 Capstone: from classical to modern physics — action, integrability, chaos and the road to quantum mechanics and field theory 26 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
Analytical mechanics, the reformulation of Newton's laws through action principles that underlies all of modern theoretical physics, developed with a runnable Python check in every lesson: the action and Hamilton's principle of stationary action, tested by minimizing a discretized action and by finding paths that are saddles rather than minima, the calculus of variations with the Euler–Lagrange equation, cyclic coordinates, the Beltrami identity and Fermat's principle of least time leading to Snell's law, classic variational problems including the brachistochrone and tautochrone cycloid, the catenary found with a Lagrange multiplier, minimal surfaces and geodesics on the sphere, Lagrange's equations derived from d'Alembert's principle with generalized coordinates, holonomic and rheonomic constraints, the bead on a spinning hoop and its pitchfork bifurcation and the double pendulum, constraint forces from Lagrange multipliers, loss of contact, rolling and nonholonomic constraints with the Lie bracket that lets a wheel move sideways, Noether's theorem linking translation, rotation and time-shift symmetry to momentum, angular momentum and energy, central forces, the spherical pendulum and the heavy symmetric top with Euler angles, precession and nutation, small oscillations as a generalized eigenvalue problem with mass and stiffness matrices, normal modes of CO2, chains and the triple pendulum, the Legendre transform and Hamilton's equations, phase space, charged particles with the vector potential and symplectic integrators, Poisson brackets, Liouville's theorem and its consequences for statistical mechanics, beams and optics, canonical transformations, generating functions, the Hamilton–Jacobi equation, action-angle variables and adiabatic invariants, and a capstone on integrability, the KAM theorem and chaos in the standard map, and the passage from classical mechanics to quantum mechanics and field theory.
Part of the Science stream, typically taken in year 3.