# Principle of Least Action - not always valid?

• Master J
In summary, the Principle of Least Action, as stated in Landau & Lifshitz Mechanics, is only valid for a small segment of a system's path in phase space. When deriving Lagrange's Equation, it is important to ensure that the principle is valid for the entire path. This may require examining other quantities to determine if the extremal of the action is a minimum, saddle point, or maximum. The equation is generally correct, but the name "least action" may be misleading. Specific examples and conditions for deriving differential equations from a variational principle can be found in additional sources.
Master J
It is stated in Landau & Lifgarbagez Mechanics that the Principle of Least Action is not always valid for the entire path of a system in phase space, but only for a sufficiently small segment of the path.

Can anyone expand on this?

How can we be sure that when we derive Lagrange's Equation that the principle is valid?

I guess the statement has slightly confused me - what are the consequences of this?

Basically, the equation gives a geodesic in some space. On a sphere, the great circles are geodesic. But between two points, you can go either direction on the geodesic, the short way or the long way. But between any two nearby points on the geodesic, it is always the short way. So some other quantity needs to be examined to figure out whether the extremal of the action we get is a minimum, saddle point, or maximum. The equation is almost always right, just the name "least action" isn't. Some specific examples are given by:
http://www.people.fas.harvard.edu/~djmorin/chap6.pdf
http://www.eftaylor.com/pub/Gray&TaylorAJP.pdf

The conditions under which differential equations can be derived from a variational principle are discussed by http://www.dic.univ.trieste.it/perspage/tonti/
http://arxiv.org/abs/1008.3177

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## 1. What is the Principle of Least Action?

The Principle of Least Action is a fundamental concept in physics that states that a system will always follow the path of least action, meaning that it will take the path that minimizes the difference between its potential and kinetic energy. This principle is often used to derive the equations of motion for physical systems.

## 2. Is the Principle of Least Action always valid?

No, the Principle of Least Action is not always valid. It is a classical principle and only applies to systems that follow classical mechanics, meaning that they are not subject to quantum effects. In certain cases, such as with very small particles, the principle may not accurately predict the behavior of the system.

## 3. What are some limitations of the Principle of Least Action?

One limitation of the Principle of Least Action is that it only applies to conservative systems, meaning that there is no external force acting on the system. It also does not take into account any constraints or boundary conditions that may affect the system's motion. Additionally, the principle assumes that the system is in a state of equilibrium, which may not always be the case.

## 4. Can the Principle of Least Action be used for all physical systems?

No, the Principle of Least Action is not applicable to all physical systems. It only applies to systems that can be described by classical mechanics and does not account for other important phenomena such as quantum mechanics or relativity. It is also limited to systems with a finite number of degrees of freedom.

## 5. Are there any alternative principles to the Principle of Least Action?

Yes, there are alternative principles to the Principle of Least Action that have been proposed in physics. Some examples include the Principle of Least Time, which states that a system will take the path that minimizes the time it takes to reach its destination, and the Hamilton's Principle, which is a more general form of the Principle of Least Action that applies to non-conservative systems.

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