30 research outputs found
Families of Canonical Transformations by Hamilton-Jacobi-Poincar\'e equation. Application to Rotational and Orbital Motion
The Hamilton-Jacobi equation in the sense of Poincar\'e, i.e. formulated in
the extended phase space and including regularization, is revisited building
canonical transformations with the purpose of Hamiltonian reduction. We
illustrate our approach dealing with orbital and attitude dynamics. Based on
the use of Whittaker and Andoyer symplectic charts, for which all but one
coordinates are cyclic in the Hamilton-Jacobi equation, we provide whole
families of canonical transformations, among which one recognizes the familiar
ones used in orbital and attitude dynamics. In addition, new canonical
transformations are demonstrated.Comment: 21 page
The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables
The Hamiltonian theory of Earth rotation, known as the Kinoshita-Souchay
theory, operates with nonosculating Andoyer elements. This situation parallels
a similar phenomenon that often happens (but seldom gets noticed) in orbital
dynamics, when the standard Lagrange-type or Delaunay-type planetary equations
unexpectedly render nonosculating orbital elements. In orbital mechanics,
osculation loss happens when a velocity-dependent perturbation is plugged into
the standard planetary equations. In attitude mechanics, osculation is lost
when an angular-velocity-dependent disturbance is plugged in the standard
dynamical equations for the Andoyer elements. We encounter exactly this
situation in the theory of Earth rotation, because this theory contains an
angular-velocity-dependent perturbation (the switch from an inertial frame to
that associated with the precessing ecliptic of date).
While the osculation loss does not influence the predictions for the figure
axis of the planet, it considerably alters the predictions for the
instantaneous spin-axis' orientation. We explore this issue in great detail
The Serret-Andoyer Formalism in Rigid-Body Dynamics: I. Symmetries and Perturbations
This paper reviews the Serret-Andoyer (SA) canonical formalism in rigid-body
dynamics and presents some new results. As is well known, the problem of
unsupported and unperturbed rigid rotator can be reduced. The availability of
this reduction is offered by the underlying symmetry, which stems from
conservation of the angular momentum and rotational kinetic energy. When a
perturbation is turned on, these quantities are no longer preserved.
Nonetheless, the language of reduced description remains extremely instrumental
even in the perturbed case. We describe the canonical reduction performed by
the Serret-Andoyer (SA) method, and discuss its applications to attitude
dynamics and to the theory of planetary rotation. Specifically, we consider the
case of angular-velocity-dependent torques, and discuss the
variation-of-parameters-inherent antinomy between canonicity and osculation.
Finally, we address the transformation of the Andoyer variables into the
action-angle ones, using the method of Sadov
Geographic distribution of the species assigned to the <i>Trichomycterus stawiarski</i> group: <i>T</i>. <i>crassicaudatus</i> (green symbols), <i>T</i>. <i>igobi</i> (white symbols), <i>T</i>. <i>stawiarski</i> (yellow symbols) and <i>T</i>. <i>ytororo</i> (red symbol).
<p>Stars represent the type localities. Some triangles symbols represent more than one collection locality. Numbers 1, 2, 3 indicate the Paraguay, Paraná and Iguazú Rivers, respectively.</p