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Is it possible for an action (the integral of a Lagrangian) to have more than one stationary value? Why or why not?
drvrm said:If there are more than one real path through which the physical system can move in phase space (q,p) then it can have multiple stationary values - but i have yet to see an example of the same
The concept of "two stationary values" for an action refers to a principle in physics known as the principle of least action. This principle states that for a physical system, the action (a measure of the system's energy) is minimized at certain points called stationary points. These points can either be a minimum or a maximum, and when there are two stationary points, one is a minimum and the other is a maximum.
There can be two stationary points for an action because the principle of least action takes into account all possible paths that a physical system can take. This means that there can be multiple paths that result in the same value for the action at a stationary point. Therefore, there can be two stationary points with the same action value, one being a minimum and the other being a maximum.
Having two stationary points for a physical system means that the system has multiple possible paths that it can take while still conserving the same amount of energy. This allows for more flexibility in the behavior of the system and can lead to interesting phenomena, such as interference patterns in wave systems.
The principle of least action is used in physics to determine the path that a physical system will take between two points in space and time. By finding the stationary points for the action, we can determine the most probable path that the system will take. This principle is used in various fields of physics, including classical mechanics, electromagnetism, and quantum mechanics.
Yes, there can be more than two stationary points for an action. In some cases, a physical system may have multiple paths that result in the same action value, leading to more than two stationary points. Additionally, in certain systems, there may be multiple variables that contribute to the action, resulting in multiple stationary points in a multidimensional space.