Null lifts and projective dynamics

Abstract

We describe natural Hamiltonian systems using projective geometry. The null lift procedure endows the tangent bundle with a projective structure where the null Hamiltonian is identified with a projective conic and induces a Weyl geometry. Projective transformations generate a set of known and new dualities between Hamiltonian systems, as for example the phenomenon of coupling-constant metamorphosis. We conclude outlining how this construction can be extended to the quantum case for Eisenhart-Duval lifts.Comment: 11 pages, no figures. Some minor changes, typos amende

Jacobi-Maupertuis-Eisenhart metric and geodesic flows

“…respectively. This shows that the Jacobi metric in the non-relativistic limit can be deduced from projective transformations of time-dependent systems, just as [27] demonstrates it for time-independent systems.…”

“…Projective geometry can be used to describe natural Hamiltonian systems and generate the dualities between them. The Jacobi metric can be alternatively formulated from a projective transformation in the phase space as described in [27]. This is described by the null Hamiltonian, for which the curve is parametrised by the arc length.…”

“Kepler Harmonies” and conformal symmetries

“…It states that the motion may be regarded as the projection of the motion of light rays moving in a fivedimensional extended spacetime and obtain for the first time Kepler For further details and applications of conformal symmetries for gravitational waves, see [15,16]. Other examples of Chrono-Projective transformations include the Schrödinger-Newton equations [17], hydrodynamics [18], Schrödinger operators [19] and projective dynamics [20]. where l the total angular momentum and n is the principal quantum number, respectively.…”

Cosmological aspects of the Eisenhart–Duval lift

“…which is a relationship between dynamical variables of order 2 in the momenta. Then an important step is to transform this relation into a homogeneous one by introducing a conserved momentum, p v -which is the physical equivalent of constructing a projective conic from a standard one, as described in [64]. In our case the process transforms (IV.27) into…”