Is the Action Always a Minimum in the Principle of Least Action?

In summary, the principle of least action requires that the action S be an extremum, usually a minimum, when applying it. This terminology was influenced by Fermat's principle of least time in optics. While both minimum and maximum can work in this context, stationary is a more appropriate term to use. This concept has also inspired the development of wave mechanics. The full explanation and history of this terminology can be found in Gray and Taylor's paper "When action is not least."
  • #1
BookWei
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Hello, When we applying the principle of least action, we require ##\delta S=0##, which corresponding to the action S being an extremum. I am just wondering why do we say that the action is a minimum instead of a maximum for a physical path? Can I use the path integral to explain this problem?
Thanks for all responses.
 
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  • #3
I think that long ago, when Fermat's principle of least time came about, that people so liked the idea that they asked themselves why couldn't such a thing work for mechanics. The principle of least action was then made intentionally analogous to Fermat's principle. These optical considerations would then inspire others to develop wave mechanics.
 
  • #4
If S is minimum , -S is maximum and anyway the both work. Stationary is more appropriate word to say.
 

FAQ: Is the Action Always a Minimum in the Principle of Least Action?

1. What is the Principle of Least Action?

The Principle of Least Action is a fundamental principle in physics that states that a physical system will always choose the path of least action, or the path that requires the least amount of energy or effort to reach a certain outcome.

2. How does the Principle of Least Action relate to Newton's laws of motion?

The Principle of Least Action is closely related to Newton's laws of motion, specifically the law of inertia. It states that an object in motion will continue moving in a straight line unless acted upon by an external force. The path of least action is the path that follows this law of inertia.

3. What is the significance of the Principle of Least Action in physics?

The Principle of Least Action is significant because it allows us to predict the behavior and motion of physical systems by minimizing the amount of energy or effort required. It has been used to explain a wide range of phenomena, from the motion of planets to the behavior of subatomic particles.

4. Is the Principle of Least Action always applicable?

No, the Principle of Least Action is only applicable in certain situations. It is most commonly used in classical mechanics, but it also has applications in quantum mechanics and other areas of physics. In some cases, it may not accurately predict the behavior of a system, and other principles and laws may need to be considered.

5. How is the Principle of Least Action used in practical applications?

The Principle of Least Action is used in a variety of practical applications, including in engineering, robotics, and optimal control systems. It is also used in theoretical physics to develop mathematical models and equations that accurately describe the behavior of physical systems.

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