generalized coordinates

简明释义

广义坐标

英英释义

例句

1.In classical mechanics, we often use generalized coordinates (广义坐标) to describe the configuration of a system.

在经典力学中，我们常常使用广义坐标来描述系统的配置。

2.When analyzing a pendulum, we can use the angle as a generalized coordinate (广义坐标) instead of Cartesian coordinates.

在分析摆时，我们可以使用角度作为广义坐标而不是笛卡尔坐标。

3.The choice of generalized coordinates (广义坐标) can greatly affect the complexity of the problem.

选择广义坐标可能会大大影响问题的复杂性。

4.The equations of motion can be simplified by choosing appropriate generalized coordinates (广义坐标).

通过选择合适的广义坐标，可以简化运动方程。

5.In robotics, the position and orientation of a robot arm can be described using generalized coordinates (广义坐标).

在机器人技术中，机器臂的位置和方向可以用广义坐标来描述。

作文

In the realm of classical mechanics, the concept of generalized coordinates plays a pivotal role in describing the motion of systems. These coordinates extend beyond the traditional Cartesian coordinates, allowing for a more flexible representation of complex systems. By utilizing generalized coordinates, we can simplify the analysis of mechanical systems, particularly those with constraints or multiple degrees of freedom.To understand generalized coordinates, consider a simple pendulum. In a Cartesian framework, we might describe the pendulum's position using x and y coordinates. However, this approach can become cumbersome when dealing with more complex systems, such as a double pendulum or a robotic arm. Here, generalized coordinates come into play, enabling us to use a single angle to describe the position of the pendulum. This reduction in variables not only simplifies calculations but also enhances our understanding of the system's dynamics.The utility of generalized coordinates is further exemplified in Lagrangian mechanics, where the equations of motion are derived from a function called the Lagrangian. The Lagrangian is defined as the difference between kinetic and potential energy, and it is expressed in terms of generalized coordinates and their time derivatives. This formulation allows physicists and engineers to apply powerful mathematical techniques to solve complex problems in mechanics.Moreover, generalized coordinates provide a systematic way to incorporate constraints into our models. For instance, if we have a system with a constraint that limits the motion of a particle, we can introduce a generalized coordinate that captures this restriction. This approach not only streamlines the modeling process but also ensures that we account for all relevant factors influencing the system's behavior.In addition to their application in mechanics, generalized coordinates are also prevalent in fields such as robotics, where they are used to represent the configuration of robotic arms and other mechanisms. By employing generalized coordinates, engineers can design control systems that effectively manage the movement and interaction of robotic components within their environment.Furthermore, the concept of generalized coordinates extends to other areas of physics, including thermodynamics and quantum mechanics. In thermodynamics, for example, generalized coordinates can represent different state variables, such as temperature and pressure, which are essential for understanding the behavior of systems at equilibrium. In quantum mechanics, the notion of generalized coordinates is crucial for formulating wave functions and understanding particle behavior in various potentials.In conclusion, the concept of generalized coordinates is fundamental to the study of mechanics and other branches of physics. By providing a flexible and efficient means of representing complex systems, generalized coordinates facilitate the analysis and solution of problems that would otherwise be intractable. As we continue to explore the intricacies of the physical world, the importance of generalized coordinates will undoubtedly remain a cornerstone of scientific inquiry and technological advancement.

在经典力学领域，广义坐标的概念在描述系统运动中起着关键作用。这些坐标超越了传统的笛卡尔坐标，允许对复杂系统进行更灵活的表示。通过利用广义坐标，我们可以简化机械系统的分析，特别是那些具有约束或多个自由度的系统。为了理解广义坐标，可以考虑一个简单的摆。在笛卡尔框架中，我们可能会使用x和y坐标来描述摆的位置。然而，当处理更复杂的系统时，例如双摆或机器人手臂，这种方法可能变得繁琐。在这里，广义坐标发挥了作用，使我们能够使用一个角度来描述摆的位置。这种变量的减少不仅简化了计算，还增强了我们对系统动态的理解。广义坐标的实用性在拉格朗日力学中得到了进一步体现，在这种情况下，运动方程是从称为拉格朗日函数的函数中推导出的。拉格朗日函数被定义为动能和势能之差，并以广义坐标及其时间导数的形式表达。这种表述使物理学家和工程师能够应用强大的数学技术来解决复杂的力学问题。此外，广义坐标提供了一种系统的方法，将约束纳入我们的模型。例如，如果我们有一个限制粒子运动的约束系统，我们可以引入一个广义坐标来捕捉这一限制。这种方法不仅简化了建模过程，还确保我们考虑到影响系统行为的所有相关因素。除了在力学中的应用外，广义坐标在机器人技术等领域也很常见，它们用于表示机器人手臂和其他机制的配置。通过采用广义坐标，工程师可以设计有效管理机器人组件在其环境中移动和交互的控制系统。此外，广义坐标的概念扩展到物理学的其他领域，包括热力学和量子力学。在热力学中，例如，广义坐标可以表示不同的状态变量，如温度和压力，这对于理解平衡状态下系统的行为至关重要。在量子力学中，广义坐标的概念对于制定波函数和理解粒子在各种势能中的行为至关重要。总之，广义坐标的概念是力学和其他物理学分支研究的基础。通过提供一种灵活和高效的方式来表示复杂系统，广义坐标促进了对本来难以处理的问题的分析和解决。随着我们继续探索物理世界的复杂性，广义坐标的重要性无疑将仍然是科学探究和技术进步的基石。