Superintegrability of the Calogero–Moser system: Constants of motion, master symmetries, and time-dependent symmetries

1999 ◽  
Vol 40 (1) ◽  
pp. 236-247 ◽  
Author(s):  
Manuel F. Rañada
Keyword(s):  
Time Dependent ◽  
Constants Of Motion ◽  
Moser System

THREE ELABORATIONS ON BERRY’S CONNECTION, CURVATURE AND PHASE

1988 ◽  
Vol 03 (02) ◽  
pp. 285-297 ◽  
Author(s):  
R. JACKIW
Keyword(s):  
Conservation Laws ◽  
Quantum System ◽  
Canonical Transformation ◽  
Time Dependent ◽  
Berry's Phase ◽  
Berry’S Phase ◽  
Gauge Transformations ◽  
Constants Of Motion ◽  
Symmetry Transformations ◽  
Higher Order Corrections

We discuss how symmetries and conservation laws are affected when Berry’s phase occurs in a quantum system: symmetry transformations of coordinates have to be supplemented by gauge transformations of Berry’s connection, and consequently constants of motion acquire terms beyond the familiar kinematical ones. We show how symmetries of a problem determine Berry’s connection, curvature and, once a specific path is chosen, the phase as well. Moreover, higher order corrections are also fixed. We demonstrate that in some instances Berry’s curvature and phase can be removed by a globally well-defined, time-dependent canonical transformation. Finally, we describe how field theoretic anomalies may be viewed as manifestations of Berry’s phase.

Factorization of the constants of motion

2006 ◽  
Vol 84 (8) ◽  
pp. 717-722
Author(s):  
P L Nash ◽  
L Y Chen
Keyword(s):  
Model System ◽  
First Integrals ◽  
Time Dependent ◽  
05.70 Ln ◽  
Constant Of Motion ◽  
Sum Of Products ◽  
Constants Of Motion ◽  
Independent Constant ◽  
Complete Set

A complete set of first integrals, or constants of motion, for a model system is constructed using “factorization”, as described below. The system is described by the effective Feynman Lagrangian L = [Formula: see text], with one of the simplest, nontrivial, potentials V(x) = (1/2)m ω2x2 selected for study. Four new, explicitly time-dependent, constants of the motion ci±, i = 1, 2 are defined for this system. While [Formula: see text]ci± ≠ 0, [Formula: see text]ci± = [Formula: see text]ci± + [Formula: see text]ci± + [Formula: see text]ci± + · · · = 0 along an extremal of L. The Hamiltonian H is shown to equal a sum of products of the ci±, and verifies [Formula: see text] = 0. A second, functionally independent constant of motion is also constructed as a sum of the quadratic products of ci±. It is shown that these derived constants of motion are in involution.PACS Nos.: 02.30.Jr, 02.30.Ik, 02.60.Cb, 02.30.Hq, 05.70.Ln, 02.50.–r

Sign in / Sign up

Export Citation Format

Share Document