Dork Scratchings

  I picked up this fantastic little book in Cambridge market a few days ago.  Published in 1908 by the Society for the Promotion of Christian Knowledge , it summarizes a lecture performed by Professor J Perry in 1890 in Leeds. The theme of the Romance of Science series is that remarkable and unexpected things can be discovered from the study of the apparently mundane.  In this lecture Professor Perry performs a series of experiments on various designs of spinning top and shows i.) What causes the precessional motion of a spinning top ii.) That a subterranean race could determine that we live on a spinning planet without ever seeing the stars iii.) Why the precessional motion of a top is in the same orientation as the spin, but the precessional motion of the Earth is in the opposite orientation iv.) Why some spinning tops stand up when you spin them on their side and even v.) A model of how a magnetic field might cause the observed rotation of plane polarized light The q...

Simple example from The Theoretical Minimum I recently re-read The Theoretical Minimum - Classical mechanics , an excellent book. The authors give an example of the use of the Hamiltonian formulation of mechanics to illustrate how much easier it makes things (compared to the Newtonian formulation) and that inspired me to have a go myself. The example they give in the book is of a negligible mass charged sphere spinning in a magnetic field pointing in the z direction. In this case the Hamiltonian $H = T + V$ is proportional to $L_z$ where $\mathbf{L}$ is the angular momentum $L_x \hat{i} + L_y \hat{j} + L_z \hat{k}$. Let's say $H=\omega^2L_z$. The book explains that for any function $G(q,p)$ of the generalized coordinates $q_i$ and the conjugate momenta $p_i$ that $$ \dot{G} = \{G,H\} $$ i.e the time derivative of $G$ is the poisson bracket of $G$ and the Hamiltonian. And indeed this does make the solution simple because the components of angular momentum satisfy $$ \begin{align} ...