| Contenu : |
Vectors, tensors, and linear transformations. Exterior algebra: determinants, oriented frames and oriented volumes. The hodge-star operator and the vector cross product. Kinematics and moving frames: from the angular velocity to gauge fields. Differentiable manifolds: the tangent and cotangent bundles. Exterior calculus: differential forms. Vector calculus by differential forms. The Stokes theorem. Cartan's method of moving frames: curvilinear coordinates in R. Mechanical constraints: the Frobenius theorem. Flows and lie derivatives. Newton's laws: inertial and non-inertial frames. Simple applications of Newton's laws. Potential theory: Newtonian gravitation. Centrifugal and Coriolis forces. Harmonic oscillators: Fourier transforms and Green's functions. Classical model of the atom: power spectra. Dynamical systems and their stabilities. Many-particle systems and the conservation principles. Rigid-body dynamics: the Euler-Poisson equations of motion. Topology and systems with holonomic constraints: homology and de Rham cohomology. Connections on vector bundles: affine connections on tangent bundles. The parallel translation of vectors: the Foucault pendulum. Geometric phases, gauge fields, and the mechanics of deformable bodies: the "falling cat" problem. Force and curvature. The Gauss-Bonnet-Chern theorem and holonomy. The curvature tensor in Riemannian geometry. Frame bundles and principal bundles, connections on principal bundles. Calculus of variations, the Euler-Lagrange equations, the first variation of arclength and geodesics. The second variation of arclength, index forms, and Jacobi Fields. The Lagrangian formulation of classical mechanics: Hamilton's principle of least action, Lagrange multipliers in constrained motion. Small oscillations and normal modes. The Hamiltonian formulation of classical mechanics: Hamilton's equations of motion. Symmetry and conservation. Symmetric tops. Canonical transformations and the symplectic group. Generating functions and the Hamilton-Jacobi equation. Integrability, invariant Tori, action-angle variables. Symplectic geometry in Hamiltonian dynamics, Hamiltonian flows, and Poincaré-Cartan integral invariants. Darboux's theorem in symplectic geometry. The Kolmogorov-Arnold-Moser (KAM) theorem. The homoclinic tangle and instability, shifts as subsystems. The restricted three-body problem |
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