| PHYS 498GMP Spring 99 |
| PHYS 498GMP Spring 99 |
| =======> |
GEOMETRIC METHODS IN PHYSICS dynamical systems |
Note: The first 10 lectures form a
"crush course in differential geometry."
Also, see the list of handouts.
Lecture 1:
General overview (see map)
Lecture 2:
Linear algebra: space and the dual space
Lecture 3:
Exterior forms and multivectors. Extra: What is the Hodge star?
(reading)
Lecture 4:
Manifolds. The concept of vector after 2000 years.
Lecture 5:
Tangent and cotangent space. Vector fields, differential forms.
Lecture 6:
Exterior derivative, induced maps.
Lecture 7:
Cohomology (de Rham), Stokes' theorem.
Lecture 8:
Example 1: Maxwell's equations --- from 3D to 4D. And back!
Lecture 9:
Dynamical systems. Lie derivative.
(reading)
Lecture 10:
Newton equations.
(Remark for programmers).
Lecture 11:
Symplectic geometry and Hamiltonian mechanics 1.
Lecture 12:
Symplectic geometry and Hamiltonian mechanics 2.
Lecture 13:
Special topics in symplectic geometry.
Lecture 14:
What are the Lagrange/Hamilton equations, really ?
Lecture 15:
Digression 1: Thermodynamics and symplectic geometry (the magic cube)
Lecture 16:
Digression 2: Symplectic geometry and geometric optics.
Digression 3: Schroedinger equations and symplectic geometry
Lecture 17:
Poisson manifolds and the Schouten-Nijenhuis bracket ---
a big erratum to Hamiltonian description.
Lecture 18:
Beyond the symplectic paradigm (1) (Poincare-Cartan description revised).
Lecture 19:
Beyond the symplectic paradigm (2) (Myths concerning CM).
Lecture 20:
Beyond the symplectic paradigm (3) (Nambu mechanics, bi-Hamiltonian systems).
Lecture 21:
Lie groups and algebras via differential geometry.
Lecture 22:
Quaternions, SO(3), and SU(2).
Lecture 23:
Clifford algebras.
Lecture 24:
Geometry of Lie algebras and co-algebras.
Lecture 25:
Geometry of Lie algebras and co-algebras -- cont.
Lecture 26:
What are Lax equations.
Lecture 27:
Fiber bundles, connections, and curvature -- 1.
Lecture 28:
Fiber bundles, connections, and curvature -- 2.
Lecture 29:
What is quantization: geometric approaches.
NEAR FUTURE:
Lecture 30:
Everything you wanted to know about n-spheres
(Hopf fibrations, spin and polarization, Utnapishtim, etc)
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