Four Anharmonic Oscillators on a Circle

Four identical, uniformly separated particles interconnected by ideal anharmonic springs are constrained to move on a fixed, frictionless circular track. The Lagrangian for the system is written and then transformed by matrix operations suggested by the symmetry of the arrangement of springs and particles. The equations of motion derived from the transformed Lagrangian yield four natural frequencies of motion

Breaking the Symmetry of a Circular System of Coupled Harmonic Oscillators

First we compute the natural frequencies of vibration of four identical particles coupled by ideal, massless harmonic springs. The four particles are constrained to move on a fixed circle. The initial computations are simplified by a transformation to symmetry coordinates. Then the symmetry of the vibrating system is broken by changing the mass of a single particle by a very small amount. We observe the effect of applying the symmetry transformation to the now slightly nonsymmetric system. We compute the new frequencies and compare them with the frequencies of the original symmetric system of oscillators. Results of similar calculations for 2,3,5, and 6particles are given

Breaking the symmetry of a circular system of coupled harmonic oscillators

First we compute the natural frequencies of vibration of four identical particles coupled by ideal, massless harmonic springs. The four particles are constrained to move on a fixed circle. The initial computations are simplified by a transformation to symmetry coordinates. Then the symmetry of the vibrating system is broken by changing the mass of a single particle by a very small amount. We observe the effect of applying the symmetry transformation to the now slightly nonsymmetric system. We compute the new frequencies and compare them with the frequencies of the original symmetric system of oscillators. Results of similar calculations for 2,3,5, and 6 particles are given

Experimental Transport Benchmarks for Physical Dosimetry to Support Development of Fast-Neutron Therapy with Neutron Capture Augmentation

The Idaho National Laboratory (INL), the University of Washington (UW) Neutron Therapy Center, the University of Essen (Germany) Neutron Therapy Clinic, and the Northern Illinois University(NIU) Institute for Neutron Therapy at Fermilab have been collaborating in the development of fast-neutron therapy (FNT) with concurrent neutron capture (NCT) augmentation [1,2]. As part of this effort, we have conducted measurements to produce suitable benchmark data as an aid in validation of advanced three-dimensional treatment planning methodologies required for successful administration of FNT/NCT. Free-beam spectral measurements as well as phantom measurements with Lucite{trademark} cylinders using thermal, resonance, and threshold activation foil techniques have now been completed at all three clinical accelerator facilities. The same protocol was used for all measurements to facilitate intercomparison of data. The results will be useful for further detailed characterization of the neutron beams of interest as well as for validation of various charged particle and neutron transport codes and methodologies for FNT/NCT computational dosimetry, such as MCNP [3], LAHET [4], and MINERVA [5]