A formulation of Hamiltonian mechanics using geometric algebra (Q5947318)

The author uses the axioms of geometric algebra to give a compact formulation of the equation of motion for linear Hamiltonian systems. The basic concepts of Hamiltonian mechanics are derived using the geometric (or Clifford) algebra. Standard Poisson bracket and standard Lagrange bracket are introduced in terms of geometric algebra, which allows to describe the configuration \(q\) of the system and its momentum \(p\) together with the symplectic form \(J\) in the same space. Thus the Hamiltonian equations take the form \(\dot\theta= -\partial_{\overline\theta}H\), where \(\theta\) is a point in phase space which regroups both \(q\) and \(p\) (we have \(\theta=q + \overline p)\), and \(\overline\theta =\theta J\). At the end, the author writes out the equation for the standard Dirac quantization map.