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Rotation

Dive into the world of rotational dynamics with detailed lecture notes on astrometry topics like Finite Rotation, Euler's Theorem, and Euler Angles. Explore matrix operations, basic rotations, and angular velocity concepts. Understand different sequences and conventions for describing rotational dynamics effectively. Includes examples and solutions for practical applications in astrometry studies.

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Rotation

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  1. Rotation • (Euclidean) Distance-Invariant • Finite Rotation: Matrix representation • Orthogonality Lecture Notes on Astrometry

  2. Infinitesimal Rotational Displacement • AntisymmetricMatrix • Vector Product Lecture Notes on Astrometry

  3. Finite Rotation • Expressions: Matrix, Spinol, Quarternion • Rotation = Matrix Operation • Rot. Matrix = Set of Basis Vectors (= Triad) Z Y X Lecture Notes on Astrometry

  4. Euler’s Theorem • Any Finite Rotation = 3 Basic Rotation • Euler angles: 3 Angles of Basic Rotations Lecture Notes on Astrometry

  5. Y y P x q X Basic Rotation • Rotation around z-axis by angle q Lecture Notes on Astrometry

  6. Basic Rotation (contd.) • Rotation around j-axis by angle q • Inverse Rotation Lecture Notes on Astrometry

  7. Basic Rotation Matrix • Example: Equatorial – Ecliptic • Obliquity of Ecliptic Lecture Notes on Astrometry

  8. Basic Rotation Matrix (contd.) • Small Angle Approximation Lecture Notes on Astrometry

  9. Angular Velocity Lecture Notes on Astrometry

  10. Euler Rotation • 3x2x2 = 12 different combinations • 3-1-3 Sequence (= x-convention) • Most popular (Euler angles) • Used to describe rotational dynamics Lecture Notes on Astrometry

  11. Euler Angles (3-1-3) Lecture Notes on Astrometry

  12. Euler Angles Z P f q X y Y N Lecture Notes on Astrometry

  13. Demerit of 3-1-3 Sequence • Degeneration in case of small angles • Solution: 3-2-1-like Sequences Lecture Notes on Astrometry

  14. 3-2-3 Sequence • y-convention: precession • Conic Rotation • Rotation around a fixed direction Lecture Notes on Astrometry

  15. Other Sequences • 1-3-1: Nutation • 2-1-3: Polar Motion + Sidereal Rotation • 1-2-3: Aerodynamics, Attitude Control • Best Recommended Lecture Notes on Astrometry

  16. Small Angle Rotation Lecture Notes on Astrometry

  17. Rotational Velocity Lecture Notes on Astrometry

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