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Canonical transformation
About: Canonical transformation is a research topic. Over the lifetime, 1854 publications have been published within this topic receiving 38019 citations.
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TL;DR: In this paper, a mathematical technique for deriving dynamical invariants (i.e., constants of motion) in time-dependent gravitational potentials is proposed. But the method relies on the construction of a canonical transformation that removes the explicit time-dependence from the Hamiltonian of the system by referring the phase-space locations of particles to a coordinate frame in which the potential remains'static' and the dynamical effects introduced by the time evolution vanish.
Abstract: This paper explores a mathematical technique for deriving dynamical invariants (ie constants of motion) in time-dependent gravitational potentials The method relies on the construction of a canonical transformation that removes the explicit time-dependence from the Hamiltonian of the system By referring the phase-space locations of particles to a coordinate frame in which the potential remains `static' the dynamical effects introduced by the time evolution vanish It follows that dynamical invariants correspond to the integrals of motion for the static potential expressed in the transformed coordinates The main difficulty of the method reduces to solving the differential equations that define the canonical transformation, which are typically coupled with the equations of motion We discuss a few examples where both sets of equations can be exactly de-coupled, and cases that require approximations The construction of dynamical invariants has far-reaching applications These quantities allow us, for example, to describe the evolution of (statistical) microcanonical ensembles in time-dependent gravitational potentials without relying on ergodicity or probability assumptions As an illustration, we follow the evolution of dynamical fossils in galaxies that build up mass hierarchically It is shown that the growth of the host potential tends to efface tidal substructures in the integral-of-motion space through an orbital diffusion process The inexorable cycle of deposition, and progressive dissolution, of tidal clumps naturally leads to the formation of a `smooth' stellar halo
16 citations
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TL;DR: In this article, a detailed investigation of the action for pure Yang-Mills theory which L. Mason formulated in twistor space is carried out, and explicit solutions are found in each case and connections with earlier work are examined.
Abstract: This thesis carries out a detailed investigation of the action for pure Yang- Mills theory which L. Mason formulated in twistor space. The rich structure of twistor space results in greater gauge freedom compared to the theory in ordinary space-time. One particular gauge choice, the CSW gauge, allows simplifications to be made at both the classical and quantum level.
The equations of motion have an interesting form in the CSW gauge, which suggests a possible solution procedure. This is explored in three special cases. Explicit solutions are found in each case and connections with earlier work are examined. The equations are then reformulated in Minkowski space, in order to deal with an initial-value, rather than boundary-value, problem. An interesting form of the Yang-Mills equation is obtained, for which we propose an iteration procedure.
The quantum theory is also simplified by adopting the CSW gauge. The Feynman rules are derived and are shown to reproduce the MHV diagram formalism straightforwardly, once LSZ reduction is taken into account. The three-point amplitude missing in the MHV formalism can be recovered in our theory. Finally, relations to Mansfield's canonical transformation approach are elucidated.
16 citations
01 Jan 2016
TL;DR: In this paper, the authors introduce the class of linear canonical transformations, which includes as particular cases the Fourier transformation, the Fresnel transformation, and magnifier, rotation and shearing operations.
Abstract: In this chapter we introduce the class of linear canonical transformations, which includes as particular cases the Fourier transformation (and its generalization: the fractional Fourier transformation), the Fresnel transformation, and magnifier, rotation and shearing operations. The basic properties of these transformations—such as cascadability, scaling, shift, phase modulation, coordinate multiplication and differentiation—are considered. We demonstrate that any linear canonical transformation is associated with affine transformations in phase space, defined by time-frequency or position-momentum coordinates. The affine transformation is described by a symplectic matrix, which defines the parameters of the transformation kernel. This alternative matrix description of linear canonical transformations is widely used along the chapter and allows simplifying the classification of such transformations, their eigenfunction identification, the interpretation of the related Wigner distribution and ambiguity function transformations, among many other tasks. Special attention is paid to the consideration of one- and two-dimensional linear canonical transformations, which are more often used in signal processing, optics and mechanics. Analytic expressions for the transforms of some selected functions are provided.
16 citations
TL;DR: The canonical transformation which Scheifele proposed to make a coordinate of the true anomaly is the product of a Whittaker transformation by an extension to space-time of the one-parameter family of canonical transformations that Hill (1913) defined for the same purpose as mentioned in this paper.
Abstract: The canonical transformation which Scheifele (1970) proposes to make a coordinate of the true anomaly is the product of a Whittaker transformation by an extension to space-time of the one-parameter family of canonical transformations that Hill (1913) defined for the same purpose.
16 citations
TL;DR: In this article, the authors generalize the Serret-andoyer transformation to the case of Hamiltonian systems on T*SO(3) with left-invariant, hyperregular Hamiltonian functions.
Abstract: The Serret-Andoyer transformation is a classical method for reducing the free rigid body dynamics, expressed in Eulerian coordinates, to a 2-dimensional Hamiltonian flow. First, we show that this transformation is the computation, in 3-1-3 Eulerian coordinates, of the symplectic (Marsden-Weinstein) reduction associated with the lifted left-action of SO(3) on T*SO(3)—a generalization and extension of Noether's theorem for Hamiltonian systems with symmetry. In fact, we go on to generalize the Serret-Andoyer transformation to the case of Hamiltonian systems on T*SO(3) with left-invariant, hyperregular Hamiltonian functions. Interpretations of the Serret-Andoyer variables, both as Eulerian coordinates and as canonical coordinates of the co-adjoint orbit, are given. Next, we apply the result obtained to the controlled rigid body with momentum wheels. For the class of Hamiltonian controls that preserve the symmetry on T*SO(3), the closed-loop motion of the main body can again be reduced to canonical form. This simplifies the stability proof for relative equilibria , which then amounts to verifying the classical Lagrange-Dirichlet criterion. Additionally, issues regarding numerical integration of closed-loop dynamics are also discussed. Part of this work has been presented in LumBloch:97a.
16 citations