Open Subsystems of Conservative Systems

TL;DR

This paper investigates how an open subsystem of a larger conservative system can be analyzed and reconstructed, revealing that only a small part of the entire system's oscillatory motion is coupled to the subsystem.

Contribution

It introduces a canonical decomposition of conservative systems based on minimal extensions, providing insights into the coupling and decoupling of subsystems.

Findings

Only a small part of the total oscillatory motion is coupled to the subsystem.

A canonical decomposition of the conservative system is constructed.

The minimal conservative extension uniquely describes the entire system.

Abstract

The subject under study is an open subsystem of a larger linear and conservative system and the way in which it is coupled to the rest of system. Examples are a model of crystalline solid as a lattice of coupled oscillators with a finite piece constituting the subsystem, and an open system such as the Helmholtz resonator as a subsystem of a larger conservative oscillatory system. Taking the view of an observer accessing only the open subsystem we ask, in particular, what information about the entire system can be reconstructed having such limited access. Based on the unique minimal conservative extension of an open subsystem, we construct a canonical decomposition of the conservative system describing, in particular, its parts coupled to and completely decoupled from the open subsystem. The coupled one together with the open system constitute the unique minimal conservative extension. Combining this with an analysis of the spectral multiplicity, we show, for the lattice model in particular, that only a very small part of all possible oscillatory motion of the entire crystal, described canonically by the minimal extension, is coupled to the finite subsystem.