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Mini projects

Rules

  • Course participants, in agreement with the course instructors, are required to choose a topic related to the course. They will then work with this topic, understanding it in greater depth;
  • The aim is to demonstrate a deep understanding and ability working with the chosen material;
  • A written document detailing this work will be submitted for evaluation (or, subject to prior agreement, some equivalent product of work, for instance computer code or other).
  • Additionally the course participant will give a short presentation on their work and respond to questions;
  • The project should demonstrate approximately 20 hours of work per person on topics related to the mathematics of the course;
  • Projects are carried out in a team of 1,2 or 3 people;
  • Each person in a team must submit their own written text related to their part of the work and each must respond to questions related to the work, the work of each person in the team must be described;
  • Presentations must happen on the scheduled days or, if required to avoid scheduling conflicts, they can be done earlier with a mutually agreed choice of date.

Schedule

  • 20/12/2024: deadline for agreeing project topic and project team (this is flexible but best to aim for having a definite plan even a week before this)
  • 13/01/2025 - 17/01/2025: project presentations (this is flexible if required to avoid other commitments by arranging the presentation earlier)

Possible topics

  • The proof of Stokes' theorem
  • Manifolds, differential forms, exterior derivative (an introduction)
  • The rocket equation and hence why having multiple stages improves efficiency
  • The geometry of aerospike engines
  • Why are Lagrange points good places for spacecraft like the JW space telescope?
  • Analysis of predator-prey dynamics
  • Using Lean in order to verify proofs
  • An analysis of spacecraft station keeping using gyroscopes
  • Analysis of electro-magnetic fields
  • Derivation of orbital mechanics, period of orbits, etc. from the underlying physics
  • Didactic methods for rigorous mathematics
  • Analysis of moment of inertia, etc for different geometries of wheel design
  • Could the technology of Star Catcher be used to prolong the Voyager missions (huge mirrors to focus and transmit solar energy)?
  • Whatever else seems interesting and contains sufficient quantity of material related to MA2!
  • What's going on with the Dzhanibekov effect?
  • The history and properties of the exponential function
  • Picard-Lindelöf theorem
  • Optimizing long flight routes through regions with different prevailing winds
  • Some calculations related to flywheel electricity storage
  • Derivation and motivation of Pappus's theorem
  • The error involved in approximating celestial bodies as point masses and when this becomes significant
  • The difference between the Fourier transform and the FFT (fast Fourier transform)
  • Is it better to include 3, 4, 5, 6 or more legs in the design when one needs to maximise stability whilst minimizing weight?