amiltonian mechanics in terms of geometric algebras
Petr Vašík -- Brno University of Technology
This presentation aims to formulate Hamilton’s equations in terms of any GA
for any Hamiltonian, not just the Hamiltonian for the rigid body motion.
Actually, one generalizes this approach by replacing SE(3) by an arbitrary Lie
group G and by defining Hamiltonian by an arbitrary function on its cotangent
bundle. The form of the Hamilton’s equations in such cases is well-known, and
if G is formed by rotors of a geometric algebra, we can directly translate
this general result for dynamics on Lie groups into the GA language. The case
of the rigid body corresponds to the choice of geometric algebra PGA and G =
SE(3). Of course, we may choose SE(n) to obtain an n-dimensional version of
the rigid body motion. But not just that, we may freely choose Hamiltonian to
describe various interactions, and we may also change geometric algebra, for
example, to CGA, which then leads to Lie group G = SO(n + 1, 1). In
particular, we get a description of the rigid body motion in CGA, and we also
get its generalization to an "elastic body motion".