Analytical Mechanics and Field Theory
Content:
• This course covers an introduction to variational calculus and the Lagrange multiplier method, along with Hamilton’s variational principle and the Lagrangian formulation of mechanics. Topics include symmetries and conservation laws, with applications to motion in central fields, the dynamics of rigid bodies, oscillations, and the Lagrangian formulation of special relativity. It also explores Hamilton’s formulation of mechanics.
• This course explores continuous systems and fields, focusing on the Lagrangian and Hamiltonian formulations of the mechanics of continuous systems. It covers conservation laws for fields, including the energy-momentum tensor, and provides an overview of key classical field theories, with Maxwell’s electrodynamics discussed as a relativistic field theory.
Learning outcome (Course Objectives):
Upon completion of the course, students should be able to:
• Possess knowledge of the Lagrangian and Hamiltonian formulations of classical mechanics, along with key applications of these formalisms.
• Understand relativistic mechanics and its range of validity.
• Possess knowledge of classical field theory, including the specific field theories that are central to our understanding of nature.
• Be capable of applying the Lagrangian and Hamiltonian formalisms to moderately advanced mechanical systems, deriving and solving the equations of motion.
• Be capable of analyzing and solving classical mechanics problems involving rotating rigid bodies.
• Be able to analyze key central applications in detail using the Lagrangian formalism.
• Develop an understanding of how the concepts covered in the course (Lagrangian, Hamiltonian, relativistic, and field theory mechanics) relate to and interconnect with various areas of physics, including Newtonian mechanics, electromagnetism, quantum mechanics, and field theory applications.
• This course covers an introduction to variational calculus and the Lagrange multiplier method, along with Hamilton’s variational principle and the Lagrangian formulation of mechanics. Topics include symmetries and conservation laws, with applications to motion in central fields, the dynamics of rigid bodies, oscillations, and the Lagrangian formulation of special relativity. It also explores Hamilton’s formulation of mechanics.
• This course explores continuous systems and fields, focusing on the Lagrangian and Hamiltonian formulations of the mechanics of continuous systems. It covers conservation laws for fields, including the energy-momentum tensor, and provides an overview of key classical field theories, with Maxwell’s electrodynamics discussed as a relativistic field theory.
Learning outcome (Course Objectives):
Upon completion of the course, students should be able to:
• Possess knowledge of the Lagrangian and Hamiltonian formulations of classical mechanics, along with key applications of these formalisms.
• Understand relativistic mechanics and its range of validity.
• Possess knowledge of classical field theory, including the specific field theories that are central to our understanding of nature.
• Be capable of applying the Lagrangian and Hamiltonian formalisms to moderately advanced mechanical systems, deriving and solving the equations of motion.
• Be capable of analyzing and solving classical mechanics problems involving rotating rigid bodies.
• Be able to analyze key central applications in detail using the Lagrangian formalism.
• Develop an understanding of how the concepts covered in the course (Lagrangian, Hamiltonian, relativistic, and field theory mechanics) relate to and interconnect with various areas of physics, including Newtonian mechanics, electromagnetism, quantum mechanics, and field theory applications.
Lecturer
Date
17th February ~ 12th May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Tuesday | 13:30 - 15:05 | A14-202 | ZOOM 2 | 638 227 8222 | BIMSA |
Prerequisite
Required prerequisite knowledge: None, Recommended prerequisites: Mechanics, Electromagnetism and Special Relativity, Mathematical Methods in Physics
Syllabus
Week 1: Mathematical introduction
Keywords: Functional, functional derivative, Euler-Lagrange equation, first integral, Lagrange multiplier.
Background reading: Sec. 2.2 of Goldstein
Week 2: Lagrangian mechanics
Keywords: Hamilton’s principle, generalized coordinates, Lagrangian, action, Lagrange equations, systems with (holonomic) constraints, constants (integrals) of motion, conservation laws.
Background reading: Chaps. 1 and 2 of Goldstein
Week 3: Application: motion in central fields
Keywords: Two-body problem, effective potential, Binet’s equation, Kepler’s laws, Laplace-Runge-Lenz vector, differential cross-section, Rutherford scattering.
Background reading: Chap. 3 of Goldstein.
Week 4: Application: dynamics of rigid bodies
Keywords: Rigid body, finite and infinitesimal rotation, fictitious forces in a rotating frame, tensor of inertia, principal axes of inertia, Euler equations, precession.
Background reading: Chaps. 4 and 5 of Goldstein
Week 5: Application: oscillations of mechanical systems
Keywords: Damped driven harmonic oscillator, normal modes and normal frequencies.
Background reading: Chap. 6 of Goldstein.
Week 6: Hamiltonian mechanics
Keywords: Generalized momenta, Hamiltonian, Hamilton equations, Legendre transform, phase space, phase portrait, cyclic coordinates, Routhian.
Background reading: Chap. 8 of Goldstein.
Week 7: Selected further aspects of theoretical mechanics
Keywords: Geometry of configuration and phase space, symplectic form, Darboux coordinates, canonical transformations, Poisson bracket, Hamiltonian symmetries and conservation laws.
Background reading: Lecture of week 7
Week 8: Lagrangian and Hamiltonian mechanics of continuous systems
Keywords: Continuum limit of discrete systems, Lagrangian density, action for fields, Klein-Gordon theory, Schr¨odinger theory, Hamiltonian density.
Background reading: Secs. 13.1, 13.2 and 13.4 of Goldstein, Lecture of week 8
Week 9: Symmetries and conservation laws
Keywords: Local conservation law, continuous symmetry, Noether’s theorem, Noether current, stress-energy tensor.
Background reading: Secs. 13.3 and 13.7 of Goldstein, Lecture of week 9
Week 10: Application: fluid dynamics
Keywords: Continuity equation, Euler’s equation, momentum flux tensor, irrotational flow, velocity potential, Bernoulli’s equation, gravity waves, sound waves, viscous stress tensor, shear and bulk viscosity, Navier-Stokes equation.
Background reading: Lecture of week 10
Week 11: Relativistic mechanics and field theory
Keywords: Minkowski metric and spacetime, Lorentz transformations, rapidity, covariant and contravariant four-vectors, proper time, four-velocity, four-momentum, Lorentz-scalar and vector fields, energy-momentum and angular-momentum tensor.
Background reading: Chap. 7 and Secs. 13.5 and 13.6 of Goldstein, Lecture of week 11
Week 12: Application: electrodynamics
Keywords: Four-current, four-potential, electromagnetic field-strength tensor, gauge invariance, retarded potentials, electric dipole radiation, Rayleigh scattering, improved stress-energy tensor, magneto-electric coupling, Chern-Simons term.
Background reading: Lecture of week 12
Keywords: Functional, functional derivative, Euler-Lagrange equation, first integral, Lagrange multiplier.
Background reading: Sec. 2.2 of Goldstein
Week 2: Lagrangian mechanics
Keywords: Hamilton’s principle, generalized coordinates, Lagrangian, action, Lagrange equations, systems with (holonomic) constraints, constants (integrals) of motion, conservation laws.
Background reading: Chaps. 1 and 2 of Goldstein
Week 3: Application: motion in central fields
Keywords: Two-body problem, effective potential, Binet’s equation, Kepler’s laws, Laplace-Runge-Lenz vector, differential cross-section, Rutherford scattering.
Background reading: Chap. 3 of Goldstein.
Week 4: Application: dynamics of rigid bodies
Keywords: Rigid body, finite and infinitesimal rotation, fictitious forces in a rotating frame, tensor of inertia, principal axes of inertia, Euler equations, precession.
Background reading: Chaps. 4 and 5 of Goldstein
Week 5: Application: oscillations of mechanical systems
Keywords: Damped driven harmonic oscillator, normal modes and normal frequencies.
Background reading: Chap. 6 of Goldstein.
Week 6: Hamiltonian mechanics
Keywords: Generalized momenta, Hamiltonian, Hamilton equations, Legendre transform, phase space, phase portrait, cyclic coordinates, Routhian.
Background reading: Chap. 8 of Goldstein.
Week 7: Selected further aspects of theoretical mechanics
Keywords: Geometry of configuration and phase space, symplectic form, Darboux coordinates, canonical transformations, Poisson bracket, Hamiltonian symmetries and conservation laws.
Background reading: Lecture of week 7
Week 8: Lagrangian and Hamiltonian mechanics of continuous systems
Keywords: Continuum limit of discrete systems, Lagrangian density, action for fields, Klein-Gordon theory, Schr¨odinger theory, Hamiltonian density.
Background reading: Secs. 13.1, 13.2 and 13.4 of Goldstein, Lecture of week 8
Week 9: Symmetries and conservation laws
Keywords: Local conservation law, continuous symmetry, Noether’s theorem, Noether current, stress-energy tensor.
Background reading: Secs. 13.3 and 13.7 of Goldstein, Lecture of week 9
Week 10: Application: fluid dynamics
Keywords: Continuity equation, Euler’s equation, momentum flux tensor, irrotational flow, velocity potential, Bernoulli’s equation, gravity waves, sound waves, viscous stress tensor, shear and bulk viscosity, Navier-Stokes equation.
Background reading: Lecture of week 10
Week 11: Relativistic mechanics and field theory
Keywords: Minkowski metric and spacetime, Lorentz transformations, rapidity, covariant and contravariant four-vectors, proper time, four-velocity, four-momentum, Lorentz-scalar and vector fields, energy-momentum and angular-momentum tensor.
Background reading: Chap. 7 and Secs. 13.5 and 13.6 of Goldstein, Lecture of week 11
Week 12: Application: electrodynamics
Keywords: Four-current, four-potential, electromagnetic field-strength tensor, gauge invariance, retarded potentials, electric dipole radiation, Rayleigh scattering, improved stress-energy tensor, magneto-electric coupling, Chern-Simons term.
Background reading: Lecture of week 12
Reference
1. Classical mechanics by Goldstein, Safko and Poole.
2. Classical mechanics by Taylor.
3. Introduction to Lagrangian & Hamiltonian mechanics by Brevik and Linder (requires a free account
to access), along with a YouTube play list Links to an external site. with recordings of the lectures
by Jacob Linder.
4. Analytical Mechanics and Field Theory (FYS500) lecture notes by Tomas Brauner
2. Classical mechanics by Taylor.
3. Introduction to Lagrangian & Hamiltonian mechanics by Brevik and Linder (requires a free account
to access), along with a YouTube play list Links to an external site. with recordings of the lectures
by Jacob Linder.
4. Analytical Mechanics and Field Theory (FYS500) lecture notes by Tomas Brauner
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
Video Public
No
Notes Public
No
Language
English
Lecturer Intro
Jahed Abedi is a black hole physicist with a broad interest in gravitational physics, bridging both observational and theoretical domains. On the observational side, his work focuses on the search for gravitational wave (GW) echoes and Quasi-Normal Modes (QNMs) in LIGO/Virgo data, while his theoretical research delves into black hole perturbations, QNMs, and Quantum Field Theory (QFT) in curved space-time. Jahed was awarded the 2019 Buchalter Cosmology First Prize for one of his publications where he served as the lead author, reflecting the high impact of his research. He holds a Bachelor's degree in Electrical Engineering, as well as a Master's and PhD in Physics. His research seeks to answer several critical questions: How can a better pipeline be developed to test the Kerr nature of observed Binary Black Hole Mergers through black hole spectroscopy? With improved methods, can additional subdominant Quasi-Normal Modes (QNMs) be detected? Can these results validate previous searches or reveal deviations from General Relativity in GW data? What quantum effects might be expected from black holes, and if they exist, how significant are they? Can such effects be observed? Lastly, how can gravitational wave data confirm or rule out alternatives to classical black holes or their mimickers? Jahed's work continues to push the frontiers of black hole physics, and he remains open to collaborations and inquiries from those interested in his research.