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达朗贝尔原理 D'Alembert's principle

(重定向自PVW)
达朗贝尔
在这篇文章内,矢量与标量分别用粗体与斜体显示。例如,位置矢量通常用\mathbf{r}\,\!表示;而其大小则用r\,\!来表示。

达朗贝尔原理英语:d'Alembert principle)是因其发现者法国物理学家与数学家让·达朗贝尔而命名。达朗贝尔原理阐明,对于任意物理系统,所有惯性力或施加的外力,经过符合约束条件的虚位移,所作的虚功的总和等于零:

其中,\mathbf{I}_i= - m_i\mathbf{a}_i\,\!是粒子P_i\,\!感受到的惯性力,m_i\,\!\mathbf{a}_i\,\!分别是粒子P_i\,\!的质量和加速度,\mathbf{F}_i\,\!是施加于粒子P_i\,\!的外力(不包括约束力)、\delta \mathbf{r}_i\,\!是符合系统约束的虚位移。

静力学的虚功原理在动力学的版本是达朗贝尔原理。假若一个物理系统的每一个约束条件都只约束位置或时间,而不约束速度,则称此物理系统为完整系统。达朗贝尔原理比哈密顿原理的适用范围更广阔,可以用于不仅是完整系统。

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D'Alembert's principle 达朗贝尔原理

(重定向自PVW)
Jean d'Alembert (1717—1783)
Free body diagram of a wire pulling on a mass with weight W, showing the d’Alembert inertia “force” ma.
Free body diagram depicting an inertia moment and an inertia force on a rigid body in free fall with an angular velocity.
达朗贝尔

D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. It is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamilton's principle, avoiding restriction to holonomic systems. A holonomic constraint depends only on the coordinates and time. It does not depend on the velocities. If the negative terms in accelerations are recognized as inertial forces, the statement of d'Alembert's principle becomes The total virtual work of the impressed forces plus the inertial forces vanishes for reversible displacements. The principle does not apply for irreversible displacements, such as sliding friction, and more general specification of the irreversibility is required.

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更新时间:2025/6/18 8:18:10