
TL;DR
This paper introduces the concept of variational openness, extending traditional variational principles to include open systems with bulk-boundary interactions, providing a unified framework for various boundary conditions.
Contribution
It formulates a new variational framework that accounts for open systems with regulated bulk-boundary exchanges, generalizing classical boundary conditions.
Findings
Defines open variational principles with boundary exchange
Derives a second-order open action quadratic form
Establishes a Rayleigh-Ritz criterion for stability thresholds
Abstract
Variational principles in mechanics, field theory and geometric analysis are usually formulated on closed admissible classes, where boundary variations are either fixed or independently cancelled through natural boundary conditions. Variational openness is formulated here as a conservative extension of this setting. Its central premise is that stationarity requires cancellation of the total first variation, not necessarily separate cancellation of bulk and boundary contributions. Separate Euler--Lagrange and boundary equations arise only when admissible variations are independently localizable. Two regimes are distinguished. In separable open systems, bulk and boundary variations remain independently testable, and stationarity yields the usual interior equation together with an open boundary balance. In regulated open systems, admissible variations form a graph subspace in which bulk…
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