Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 8E (Classical Dynamics)

The phase space of an $n$-degree-of-freedom mechanical system is the $2n$-dimensional space of canonical positions and momenta $(q_1,\dots ,q_n,p_1,\dots ,p_n)$, geometrically the cotangent bundle of configuration space. A point specifies a complete instantaneous state.

For the canonical Poisson bracket,Hamilton's equations areMore generally $\dot{f}=\{f,H\}+\partial f/\partial t$.

With $H=p_j{p}_j/(2m)$, the stated brackets giveandThus $m\ddot{\mathbf{q}}=e\dot{\mathbf{q}}\times \mathbf{B}$, the Lorentz-force equation with zero electric field. The magnetic interaction has been moved from the Hamiltonian into the noncanonical symplectic structure.