Lagrangian Fuzzy Dynamics of Physical and Non-Physical Systems

TL;DR

This paper introduces a fuzzy dynamics framework that generalizes Lagrangian mechanics to systems with imprecise knowledge, including non-physical systems, by incorporating set-valued functions and history dependence.

Contribution

It develops a Lagrangian-based approach to fuzzy system evolution, extending classical mechanics to include non-conservative, history-dependent, and non-physical systems.

Findings

Fuzzy dynamics are equivalent to Lagrangian mechanics in higher-dimensional space.

Lagrangian can depend on action, leading to modified equations of motion.

Principle of least action arises from causality and topology, not as an axiom.

Abstract

In this paper, we show how to study the evolution of a system, given imprecise knowledge about the state of the system and the dynamics laws. Our approach is based on Fuzzy Set Theory, and it will be shown that the \emph{Fuzzy Dynamics} of a $n$-dimensional system is equivalent to Lagrangian (or Hamiltonian) mechanics in a $n+1$-dimensional space. In some cases, however, the corresponding Lagrangian is more general than the usual one and could depend on the action. In this case, Lagrange's equations gain a non-zero right side proportional to the derivative of the Lagrangian with respect to the action. Examples of such systems are unstable systems, systems with dissipation and systems which can remember their history. Moreover, in certain situations, the Lagrangian could be a set-valued function. The corresponding equations of motion then become differential inclusions instead of differential equations. We will also show that the principal of least action is a consequence of the causality principle and the local topology of the state space and not an independent axiom of classical mechanics. We emphasize that our adaptation of Lagrangian mechanics does not use or depend on specific properties of the physical system being modeled. Therefore, this Lagrangian approach may be equally applied to \emph{non-physical} systems. An example of such an application is presented as well.