# Action Definition: Integral of Lagrangian Over Time

• o_neg
In summary, the conversation discusses the concept of action defined as the integral of the Lagrangian over time and its relation to the (kinetic - potential) energy. The integral is seen as a way of measuring how much energy is being used up and the path a particle takes minimizes this. A more detailed explanation can be found in the Feynman lectures on physics vol 2.
o_neg
Hi,

why is the action defined as the integral of the Lagrangian over time?
i don't see the meaning of integral on the (kinetic - potential) energy

thanks,
ori.

The integral has no meaning by itself, but setting its variation equal to zero leads to the equations of motion.

You can see the integral as a way of measuring how much of something is used. So in this case, the integral measures how much (K-U) energy is being used up, and the path that a particle takes minimizes this (this isn't technically correct... it takes a 'stationary' path, but looking at it this way could be beneficial to the student visually).

thanks for the replay.

I found a detailed explanation in the Feynman lectures on physics vol 2

## 1. What is the "Action Definition: Integral of Lagrangian Over Time"?

The action definition is a mathematical concept used in classical mechanics to describe the motion of a system over time. It is defined as the integral over time of the Lagrangian, which is a function that describes the energy of a system.

## 2. How is the action definition used in physics?

The action definition is used to derive the equations of motion for a system in classical mechanics. By finding the path that minimizes the action, we can determine the most likely trajectory of a system and predict its future behavior.

## 3. What is the difference between the action definition and the principle of least action?

The action definition is a mathematical concept, while the principle of least action is a physical principle that states that the actual path a system takes is the one that minimizes the action. In other words, the principle of least action is based on the action definition.

## 4. Can the action definition be applied to all physical systems?

Yes, the action definition can be applied to any system in classical mechanics, including particles, rigid bodies, and fields. It is a fundamental concept that is used to describe the behavior of all physical systems in classical mechanics.

## 5. Are there any real-world applications of the action definition?

Yes, the action definition has many practical applications in physics, engineering, and other fields. It is used to study the motion of celestial bodies, design efficient trajectories for spacecraft, and analyze the behavior of complex mechanical systems, among other things.

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