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Geometry and Differential Equations (22/23)
| Faculteit | Science and Engineering |
| Jaar | 2022/23 |
| Vakcode | WMMA017-05 |
| Vaknaam | Geometry and Differential Equations (22/23) |
| Niveau(s) | master |
| Voertaal | Engels |
| Periode | semester II a |
| ECTS | 5 |
| Rooster | tweejaarlijks, niet in 2019/2020 |
| Uitgebreide vaknaam | Geometry and Differential Equations (tweejaarlijks 2022/2023) | ||||||||||||||||||||
| Leerdoelen | At the end of the course, the student is able to: 1. derive the Euler-Lagrange equations of motion and obtain the conserved currents from a given Noether symmetry of the action; 2. construct and classify exact solutions of Kepler's problem, derive and verify Kepler's three laws of orbital motion, and indicate the location of Lagrange's parking points; 3. justify the qualitative behaviour of orbital motion (precession of orbits, their eccentricity and its drift, as well as the linear velocity portrait for stars in galaxy's spiral arms); 4. average the oscillations by introducing the slow-fast variables according to the Krylov-Bogolyubov method, and explain the mathematics of parametric resonance and of the amplitude breakdown in anharmonic oscillators. |
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| Omschrijving | In this course about Lagrangian celestial mechanics we study: * the origins and implications of the least action principle; * the superintegrability of Kepler's problem, classes of its exact solutions, and their use for space parking at the Lagrange points; * the virial theorem and evidence of the dark matter presence in the spiral galaxy structure; * the averaging of (non)linear oscillations by using the Krylov-Bogolyubov method of slow-fast variables and power series expansions, and the parametric resonance and amplitude breakdown in anharmonic oscillations. The course is equally oriented towards mathematicians, physicists, and astronomers. The course is aimed to help student develop a coherent, synthesizing view of Nature around us by using neat mathematical elements and language. |
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| Uren per week | |||||||||||||||||||||
| Onderwijsvorm | Hoorcollege (LC), Opdracht (ASM), Werkcollege (T) | ||||||||||||||||||||
| Toetsvorm |
Schriftelijk tentamen (WE)
(If Exam grade is at least 4.5, Homeworks count; otherwise not (so that final grade = Exam grade). Max (100% final exam, 60% final exam + 40% homeworks)) |
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| Vaksoort | master | ||||||||||||||||||||
| Coördinator | A.V. Kiselev | ||||||||||||||||||||
| Docent(en) | A.V. Kiselev | ||||||||||||||||||||
| Verplichte literatuur |
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| Entreevoorwaarden | Knowledge assumed: Linear algebra, multivariate calculus, and ODE. Helpful but not compulsory is the knowledge of Newtonian mechanics (Newton's three laws of motion, linear harmonic oscillator) and elementary geometry of curves and surfaces. This course is independent from (hence both prequel and sequel to) any course about Hamiltonian mechanics, symplectic geometry, and Poisson dynamical systems. This course revisits the Math of analytic mechanics. | ||||||||||||||||||||
| Opmerkingen | This course was registered last year with course code WMMA14002 | ||||||||||||||||||||
| Opgenomen in |