Aims:

to introduce students to a range of problems in Newtonian Mechanics, including, principally, the Kepler problem.

Learning objectives:

by the end of the module students should be able to

• manipulate vectors fluently, including use of scalar and vector products and time-differentiation thereof;
• use these techniques in a range of unseen mathematical problems based on the syllabus below;
• derive results from the syllabus, including Newton’s deduction of Kepler’s laws from the inverse-square law of gravity.

Syllabus:

• Vectors (revision): Scalar, vector and triple products; time-derivatives.
• Vector mechanics: Conserved quantities (energy, momentum, angular momentum). Frames of reference and Galilean relativity.
• Rotational motion: Circular motion and angular velocity. Rotating frames and the Coriolis effect. Rigid-body rotations, inertia tensor, moment of inertia. Euler's equations of motion for a rigid body, stability of rotational motion.
• Systems of particles: Centre-of-mass of two- and many-particle systems. Virial theorem (and implications for dark matter). Reduced mass of two particles.
• Newtonian Gravity: Historical background, Kepler's laws. A geometric equation for orbits. Conic sections and relation with energy and angular momentum. Deduction of Kepler's laws. Stability of orbits.

Recommended texts:

• R Douglas Gregory, Classical Mechanics (Cambridge University Press 2006) - ISBN 0521534097
• P Smith and RC Smith, Mechanics (John Wiley and Sons 1990) - ISBN 0471927376

Teaching:

2 lectures and 1 seminar per week

Assessment:

One and a half hour closed examination taken in week 1 of Spring Term (90%), Coursework (10%)

Elective information:

Brief details: Requires fluency (at Further Mathematics A2 level) with calculus and vectors, and some background in Physics or Mechanics.

Prerequisites

• Core Mathematics, Applied Mathematics, Matrices, Introduction to Differential Equations and Fourier Series.
• Calculus, Vectors Taught in: Core Mathematics
• Elementary mechanics and modelling Taught in: Core Applied Mathematics
• Ordinary differential equations Taught in: Introduction to Differential Equations and Fourier Series