Search results for "Action-angle coordinates"
showing 8 items of 8 documents
Action-Angle Variables
2001
In the following we will assume that the Hamiltonian does not depend explicitly on time; ∂H/∂t = 0. Then we know that the characteristic function W(q i , P i ) is the generator of a canonical transformation to new constant momenta P i , (all Q i , are ignorable), and the new Hamiltonian depends only on the P i ,: H = K = K(P i ). Besides, the following canonical equations are valid: $$ \dot Q_i = \frac{{\partial K}} {{\partial P_i }} = v_i = const. $$ (1) $$ \dot P_i = \frac{{\partial K}} {{\partial Q_i }} = 0. $$ (2)
Canonical Adiabatic Theory
2001
In the present chapter we are concerned with systems, the change of which—with the exception of a single degree of freedom—should proceed slowly. (Compare the pertinent remarks about \(\varepsilon\) as slow parameter in Chap. 7) Accordingly, the Hamiltonian reads: $$\displaystyle{ H = H_{0}{\bigl (J,\varepsilon p_{i},\varepsilon q_{i};\varepsilon t\bigr )} +\varepsilon H_{1}{\bigl (J,\theta,\varepsilon p_{i},\varepsilon q_{i};\varepsilon t\bigr )}\;. }$$ (12.1) Here, \((J,\theta )\) designates the “fast” action-angle variables for the unperturbed, solved problem \(H_{0}(\varepsilon = 0),\) and the (p i , q i ) represent the remaining “slow” canonical variables, which do not necessarily have…
Emission and null coordinates: geometrical properties and physical construction
2011
A Relativistic Positioning System is defined by four clocks (emitters) broadcasting their proper time. Then, every event reached by the signals is naturally labeled by these four times which are the emission coordinates of this event. The coordinate hypersurfaces of the emission coordinates are the future light cones based on the emitter trajectories. For this reason the emission coordinates have been also named null coordinates or light coordinates. Nevertheless, other coordinate systems used in different relativistic contexts have the own right to be named null or light coordinates. Here we analyze when one can say that a coordinate is a null coordinate and when one can say that a coordin…
Comparison between the fCCZ4 and BSSN formulations of Einstein equations in spherical polar coordinates
2015
Recently, we generalized a covariant and conformal version of the Z4 system of the Einstein equations using a reference metric approach, that we denote as fCCZ4. We successfully implemented and tested this approach in a 1D code that uses spherical coordinates and assumes spherical symmetry, obtaining from one to three orders of magnitude reduction of the Hamiltonian constraint violations with respect to the BSSN formulation in tests involving neutron star spacetimes. In this work, we show preliminary results obtained with the 3D implementation of the fCCZ4 formulation in a fully 3D code using spherical polar coordinates.
Positioning systems in Minkowski space-time: Bifurcation problem and observational data
2012
In the framework of relativistic positioning systems in Minkowski space-time, the determination of the inertial coordinates of a user involves the {\em bifurcation problem} (which is the indeterminate location of a pair of different events receiving the same emission coordinates). To solve it, in addition to the user emission coordinates and the emitter positions in inertial coordinates, it may happen that the user needs to know {\em independently} the orientation of its emission coordinates. Assuming that the user may observe the relative positions of the four emitters on its celestial sphere, an observational rule to determine this orientation is presented. The bifurcation problem is thus…
Newtonian and relativistic emission coordinates
2009
Emission coordinates are those generated by positioning systems. Positioning systems are physical systems constituted by four emitters broadcasting their respective times by means of sound or light signals. We analyze the incidence of the space-time causal structure on the construction of emission coordinates. The Newtonian case of four emitters at rest is analyzed and contrasted with the corresponding situation in special relativity.
Time-Independent Canonical Perturbation Theory
2001
First we consider the perturbation calculation only to first order, limiting ourselves to only one degree of freedom. Furthermore, the system is to be conservative, ∂ H∕∂ t = 0, and periodic in both the unperturbed and perturbed case. In addition to periodicity, we shall require the Hamilton–Jacobi equation to be separable for the unperturbed situation. The unperturbed problem H0(J0) which is described by the action-angle variables J0 and w0 will be assumed to be solved. Thus we have, for the unperturbed frequency: $$\displaystyle{ \nu _{0} = \frac{\partial H_{0}} {\partial J_{0}} }$$ (10.1) and $$\displaystyle{ w_{0} =\nu _{0}t +\beta _{0}\;. }$$ (10.2) Then the new Hamiltonian reads, up t…
Coordinates for quasi-Fuchsian punctured torus space
1998
We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely the classical Fuchsian space with Fenchel-Nielsen coordinates and the Maskit embedding. We also show how they relate to the pleating coordinates of Keen and Series.