Lagrangian Mechanics and Rigid Body Motion

This is a group research project I did in my second year at Imperial College London as a math undergraduate student. I cooperated with the team in explaining and proving basic Lagrangian Mechanics and the investigation into the heavy-top problems.

Here is an abstract of what this project is about. The full report reveals more about the proof, examples and Matlab code.

In our report we will discuss Lagrangian Mechanics and the Motion of Rigid Bodies. La- grangian Mechanics is a reformulation of Classical Mechanics, first introduced by the famous mathematician Joseph-Louis Lagrange, in 1788. We shall discuss the uses of Lagrangian Me- chanics and include two examples - the Spherical Pendulum and the Double Pendulum. In each case we will derive the equations of motion, and then try to solve these numerically and/or analytically. We will investigate the effect of removing the gravitational field (in the case of the Spherical Pendulum) and discuss any links between the two, as well as any implications of the solutions.

A rigid body is a collection of N points such that the distance between any two of them is fixed regardless of any external forces they are subject to. We shall look at the kinematics, the Inertia Tensor and Euler’s Equation and use this to explain about the dynamical stability of rigid bodies. Symmetric tops are the main example that we will investigate and discuss. We will look into the precession rate and the spinning rate and discuss two examples, Feynman’s wobbling plate and the hula hoop. A more complicated rigid body we shall then explore is the heavy symmetric top, in which we take into account the forces exerted by a gravitational field.

Full report is here.