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Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm

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Lagrangian formulation to optimize interactions in virtual environments.

Andrea Afify¹, Alessandro Vicini², Andrea Bellacicca³

¹Università degli Studi di Milano, Dipartimento di Fisica, Via Celoria 16, 20133 Milano, Italy.

Physical Review. E

|February 20, 2026

Summary

This study introduces a mathematical model using Lagrangian formalism to optimize agent numbers and computational costs in virtual environments. The Euler-Lagrange equations provide a universal method for scaling agent interactions with environmental complexity.

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Area of Science:

Computational Science
Mathematical Modeling
Virtual Environments

Background:

User-agent interactions in virtual environments present complex dynamics.
Optimizing agent allocation and computational cost is crucial for efficient task completion.

Purpose of the Study:

To develop a mathematical framework for optimizing user-agent interactions in virtual environments.
To derive scaling laws for agent interactions that minimize computational cost.

Main Methods:

Utilized Lagrangian formalism and derived Euler-Lagrange equations of motion.
Defined an action functional over interaction trajectories and enforced its minimization.
Developed a universal procedure to compute scaling laws for arbitrary interaction rules.

Main Results:

The Euler-Lagrange equations naturally emerge as the optimal agent allocation strategy.
A universal procedure for computing scaling laws ensures systems scale with environmental complexity.
Methodology illustrated with two specific examples.

Conclusions:

The proposed formalism intrinsically couples dynamics and cost for optimized agent allocation.
Results provide a foundation for multi-agent dynamics research in virtual environments.
Offers a universal approach to managing agent populations and computational expense.