发表机构
University of Virginia（弗吉尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从哈密顿表示推导出二维Lotka-Volterra系统的正则拉格朗日量，揭示其非二次动能结构产生位置相关阻尼，并通过Noether定理验证哈密顿量为守恒量，连接了捕食者-猎物动力学与力学。
AI 中文摘要
哈密顿力学和拉格朗日力学是分析物理系统的强大框架。先前的工作已将这些形式体系扩展到生态系统，例如捕食者-猎物Lotka-Volterra（LV）系统。在本文中，我们直接从其哈密顿表示推导出二维LV模型的正则拉格朗日量。我们发现二维LV系统允许具有一个自由度和非二次动能结构的标准正则拉格朗日表述。该表述允许在势阱中运动的粒子的力学解释，其中非标准动能结构产生一个位置相关的阻尼项，该阻尼项反而可以充当“加速”（revving）。该推导提供了捕食者-猎物动力学与力学动力学正则表述之间的直接联系。作为构建的验证，我们将Noether程序应用于显式时间无关的推导拉格朗日量，并揭示LV系统的众所周知哈密顿量是相应的守恒量。我们还发现了与正则动量选择及其与原始种群变量识别相关的微妙冗余。
英文摘要
Hamiltonian and Lagrangian mechanics are powerful frameworks for analyzing physical systems. Previous work has extended these formalisms to ecological systems, such as the predator-prey Lotka-Volterra (LV) system. In this Article, we derive a canonical Lagrangian for the two-dimensional LV model directly from its Hamiltonian representation. We find that the two-dimensional LV system admits a standard canonical Lagrangian formulation with one degree of freedom and a non-quadratic kinetic structure. This formulation admits a mechanical interpretation in terms of a particle moving in a potential well, where the non-standard kinetic structure produces a position-dependent damping term that can instead act as "revving." The derivation provides a direct connection between predator-prey dynamics and a canonical formulation of mechanical dynamics. As a verification of the construction, we apply Noether's procedure to the explicitly time-independent derived Lagrangian and reveal that the well-known Hamiltonian of the LV system is the corresponding conserved quantity. We also uncover a subtle redundancy associated with the choice of canonical momentum and its identification with the original population variables.
Comments21 pages, 4 figures; for corresponding computational work, see the GitHub repository available at https://github.com/Woofmagic/stat_mech_extensions/tree/main/lv_lagrangian
