arXiv subjects

Anthony M. Bloch

Publications and source records attributed to Anthony M. Bloch.

58 records · Page 4Linked to original sources

arXiv2005

Orbits in Extended Mass Distributions: General Results and the Spirographic Approximation

This paper explores orbits in extended mass distributions and develops an analytic approximation scheme based on epicycloids (spirograph patterns). We focus on the Hernquist potential which provides a good model for many astrophysical systems, including elliptical galaxies, dark matter halos, and young embedded star clusters. For a given potential, one can readily calculate orbital solutions as a function of energy and angular momentum using numerical methods. In contrast, this paper presents a number of analytic results for the Hernquist potential and proves a series of general constraints showing that orbits have similar properties for any extended mass distribution (including, e.g., the NFW profile). We discuss circular orbits, radial orbits, zero energy orbits, different definitions of eccentricity, analogs of Kepler's law, the definition of orbital elements, and the relation of these orbits to spirograph patterns (epicycloids). Over much of parameter space the orbits can be adequately described (with accuracy better than 10%) using the parametric equations of epicycloids, thereby providing an analytic description of the orbits. As an application of this formal development, we find a solution for the orbit of the Large Magellanic Cloud in the potential of our Galaxy.

astro-ph↗

arXiv2004

Discrete variational principles and Hamilton-Jacobi theory for mechanical systems and optimal control problems

In this paper we present a general framework that allows one to study discretization of certain dynamical systems. This generalizes earlier work on discretization of Lagrangian and Hamiltonian systems on tangent bundles and cotangent bundles respectively. In particular we show how to obtain a large class of discrete algorithms using this geometric approach. We give new geometric insight into the Newmark model for example and we give a direct discrete formulation of the Hamilton-Jacobi method. Moreover we extend these ideas to deriving a discrete version of the maximum principle for smooth optimal control problems. We define discrete variational principles that are the discrete counterpart of known variational principles. For dynamical systems, we introduce principles of critical action on both the tangent bundle and the cotangent bundle. These two principles are equivalent and allow one to recover most of the classical symplectic algorithms. We also identify a class of coordinate transformations that leave the variational principles presented in this paper invariant and develop a discrete Hamilton-Jacobi theory. This theory allows us to show that the energy error in the (symplectic) integration of a dynamical system is invariant under discrete canonical transformations. Finally, for optimal control problems we develop a discrete maximum principle that yields discrete necessary conditions for optimality. These conditions are in agreement with the usual conditions obtained from Pontryagin maximum principle. We illustrate our approach with an example of a sub-Riemannian optimal control.

math.DS↗

arXiv2001

Gyroscopic Classical and Quantum Oscillators interacting with Heat Baths

We analyze the stability of a gyroscopic oscillator interacting with a finite- and infinite-dimensional heat bath in both the classical and quantum cases. We consider a finite gyroscopic oscillator model of a particle in a magnetic field and examine the stability before and after coupling to a heat bath. It is shown that if the oscillator is gyroscopically stable, coupling to a sufficiently massive heat bath induces instability. The meaning of these ideas in the quantum context is discussed. The model extends the exact diagonalization analysis of an oscillator and field of Ford, Lewis, and O'Connell to the gyroscopic setting.

cond-mat.stat-mech↗

arXiv2000

Control of Squeezed States

In this paper we consider the classical and quantum control of squeezed states of harmonic oscillators. This provides a method for reducing noise below the quantum limit and provides an example of the control of under-actuated systems in the stochastic and quantum context. We consider also the interaction of a squeezed quantum oscillator with an external heat bath.

cond-mat.stat-mech↗