家 » 拉格朗日力学

拉格朗日力学

1788

Joseph-Louis Lagrange

A reformulation of classical 力学 based on the principle of stationary action. It uses a scalar quantity called the Lagrangian, defined as kinetic energy minus potential energy ([乳胶]L = T – V[/latex]). The equations of motion are derived from the Euler-Lagrange equation, [latex]\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) – \frac{\partial L}{\partial q_i} = 0[/latex], using generalized coordinates, which simplifies analysis of complex systems with constraints.

拉格朗日力学由约瑟夫-路易斯·拉格朗日创立，它为牛顿力学提供了一个强大而优雅的替代方案。它不再关注力和加速度（它们是矢量），而是关注能量（它们是标量）。这种视角的转变通常可以显著简化问题，尤其是那些涉及约束的问题。

The central concept is the principle of stationary action. It posits that the path taken by a physical system between two points in time is the one for which the ‘action’ is stationary (a minimum, maximum, or saddle point). The action is defined as the time integral of the Lagrangian function, [latex]S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt[/latex]. The Lagrangian, [latex]L[/latex], is defined as the kinetic energy [latex]T[/latex] minus the potential energy [latex]V[/latex] of the system.

By applying the calculus of variations to find the path that makes the action stationary, one derives the Euler-Lagrange equations. A key advantage of this approach is the use of generalized coordinates ([latex]q_i[/latex]). These are any set of parameters that uniquely define the configuration of the system. For example, for a double pendulum, the two angles are natural generalized coordinates. This freedom to choose the most convenient coordinate system is a major strength. Furthermore, forces of constraint (like the tension in a pendulum rod) do not appear in the Lagrangian formulation, as they do no work, meaning they can be ignored, greatly simplifying the equations of motion for constrained systems.

This formalism is not only a powerful tool in classical mechanics but also serves as the foundation for more advanced theories, including quantum mechanics (through Feynman’s path integral formulation) and quantum field theory.

控制系统, 增材制造设计（DfAM）, 优化设计, 运动学, 机械, 机器人技术

UNESCO Nomenclature: 2211

– Physics

类型

抽象系统

中断

基础

使用方法

广泛使用

前体

牛顿力学
Principle of virtual work (d’Alembert’s principle)
变分法（由欧拉和拉格朗日开发）
Maupertuis’s principle of least action

应用

机器人技术（逆运动学）
控制理论
量子场论（作为基础框架）
分子动力学模拟
带约束的复杂机械系统分析

专利：

潜在的创新想法

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Related to: lagrangian, analytical mechanics, principle of least action, generalized coordinates, euler-lagrange equation, calculus of variations, kinetic energy, potential energy.