generalized coordinates

简明释义

广义座标

英英释义

例句

1.The concept of generalized coordinates allows for the inclusion of constraints in mechanical systems.

概念上的广义坐标允许在机械系统中包含约束条件。

2.In robotics, generalized coordinates can help in defining the position and orientation of robotic arms.

在机器人技术中，广义坐标有助于定义机器人手臂的位置和朝向。

3.In classical mechanics, the motion of a system can be described using generalized coordinates, which simplify the equations of motion.

在经典力学中，系统的运动可以使用广义坐标来描述，这简化了运动方程。

4.The Lagrangian formulation often utilizes generalized coordinates to derive the equations of motion for complex systems.

拉格朗日公式通常利用广义坐标来推导复杂系统的运动方程。

5.When analyzing a pendulum, we can use generalized coordinates to represent its angle instead of Cartesian coordinates.

在分析摆时，我们可以使用广义坐标来表示其角度，而不是使用笛卡尔坐标。

作文

In the field of physics and engineering, the concept of generalized coordinates plays a crucial role in the analysis and description of mechanical systems. Generalized coordinates are parameters that can be used to uniquely define the configuration of a system relative to some reference configuration. Unlike traditional coordinates, which may only describe positions in space, generalized coordinates can represent a wide variety of configurations, including angles, distances, and even more abstract quantities.To understand generalized coordinates, it is essential to consider their application in classical mechanics, particularly in Lagrangian mechanics. In this framework, instead of using Cartesian coordinates (x, y, z), we can use generalized coordinates such as angles of rotation or lengths of springs. This flexibility allows for a more straightforward formulation of the equations of motion for complex systems.For example, consider a simple pendulum. The position of the pendulum bob can be described by its angle from the vertical. In this case, we can define a single generalized coordinate θ (theta) to represent the angle, rather than using both x and y coordinates. This simplifies the analysis significantly. The dynamics of the pendulum can then be derived using this single generalized coordinate, leading to a clearer understanding of its motion.Moreover, generalized coordinates are not limited to just one dimension. In systems with multiple degrees of freedom, such as a double pendulum or a robotic arm, several generalized coordinates may be needed to describe the entire configuration. Each coordinate corresponds to a degree of freedom in the system, allowing us to capture the complexities of motion in a more manageable way.The choice of generalized coordinates is often guided by the symmetries and constraints of the system. For instance, in a system with spherical symmetry, spherical coordinates (radius, polar angle, azimuthal angle) might be chosen as generalized coordinates. This choice not only simplifies the equations of motion but also makes it easier to apply conservation laws, such as conservation of angular momentum.In addition to their importance in mechanics, generalized coordinates have applications in various fields, including robotics, computer graphics, and even economics. In robotics, for example, the configuration of a robot arm can be described using generalized coordinates that represent joint angles and link lengths. This allows for efficient control algorithms that can manipulate the arm with precision.Furthermore, the concept of generalized coordinates extends beyond physical systems. In optimization problems, we often use generalized coordinates to represent variables that need to be optimized. By transforming the problem into a higher-dimensional space defined by generalized coordinates, we can apply various mathematical techniques to find optimal solutions.In conclusion, generalized coordinates are a powerful tool in the study of mechanical systems and beyond. They provide a flexible and efficient way to describe complex configurations, leading to deeper insights into the dynamics of systems. Whether in classical mechanics, robotics, or optimization, the use of generalized coordinates simplifies our understanding and enhances our ability to solve intricate problems.

在物理和工程领域，广义坐标的概念在机械系统的分析和描述中发挥着至关重要的作用。广义坐标是用于独特地定义系统相对于某个参考配置的配置的参数。与传统坐标仅能描述空间中的位置不同，广义坐标可以表示各种配置，包括角度、距离甚至更抽象的量。要理解广义坐标，必须考虑它们在经典力学中的应用，特别是在拉格朗日力学中。在这个框架中，我们可以使用广义坐标，例如旋转角度或弹簧的长度，而不是使用笛卡尔坐标（x, y, z）。这种灵活性使得对复杂系统的运动方程的公式化更加简单。例如，考虑一个简单的摆。摆锤的位置可以通过其相对于垂直方向的角度来描述。在这种情况下，我们可以定义一个单一的广义坐标θ（theta）来表示这个角度，而不是使用x和y坐标。这大大简化了分析。然后，可以使用这个单一的广义坐标推导出摆的动力学，从而更清楚地理解其运动。此外，广义坐标不仅限于一个维度。在具有多个自由度的系统中，例如双摆或机器人手臂，可能需要多个广义坐标来描述整个配置。每个坐标对应于系统中的一个自由度，使我们能够以更可管理的方式捕捉运动的复杂性。广义坐标的选择通常受系统的对称性和约束的指导。例如，在具有球对称性的系统中，可能会选择球坐标（半径、极角、方位角）作为广义坐标。这种选择不仅简化了运动方程，还使得应用守恒定律（如角动量守恒）变得更加容易。除了在力学中的重要性，广义坐标在机器人技术、计算机图形学甚至经济学等各个领域都有应用。例如，在机器人技术中，机器手臂的配置可以使用表示关节角度和连杆长度的广义坐标来描述。这使得能够有效地控制算法，以精确操纵手臂。此外，广义坐标的概念超越了物理系统。在优化问题中，我们经常使用广义坐标来表示需要优化的变量。通过将问题转换为由广义坐标定义的高维空间，我们可以应用各种数学技术来找到最优解。总之，广义坐标是研究机械系统及其他领域的强大工具。它们提供了一种灵活而高效的方式来描述复杂配置，从而深入洞察系统的动态。不论是在经典力学、机器人技术还是优化中，使用广义坐标都简化了我们的理解，并增强了解决复杂问题的能力。