What is the Principle of Least Action?

In summary, the conversation discusses the Lagrangian of a mass in a uniform gravitational field and how to show that the action is a minimum for the function defining the true motion. One method mentioned is to add a perturbation to the equation of true motion and show that the action of the perturbed equation is greater than the original. Another method is to use calculus of variations to find the function that minimizes the action and see that it is the same as the true motion.
  • #1
HalfManHalfAmazing
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Homework Statement


The Lagrangian of a mass in a uniform gravitational field can be written as follows: [tex] L = \frac{1}{2}m\dot{y}^2 + mgy [/tex]

Consider all differentiale functions y(t) such that y(t1) = y1 and y(t2) = y2 where y1 and y2 are fixed values. Show that the action is a minimum for the function defining the true motion.


Homework Equations



I believe this simply an 'example' of prooving the principle of least action.

The Attempt at a Solution



I am wondering where to start. I derived the equation of true motion of the particle. Should I know plus in the given Lagrangian to the action integral and then then add a pertubation to the equation of true motion and then plug THAT into the action and show that the action of the perturbed equation of motion must be greater then the action of the original?

Thanks guys!
 
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  • #2
Your idea is one way of doing it.

Another way would be to find the function that minimizes the action using calculus of variations, and see that it is the same as the true motion which you derived indepedently.
 
  • #3
We have not been taught calculus of variations yet, though I may look into that. Thanks!
 

What is the Principle of Least Action?

The Principle of Least Action is a fundamental concept in physics that states that a physical system will always choose the path between two points that requires the least amount of energy. It is a key principle in the study of mechanics and is used to determine the motion and behavior of objects in the universe.

Who developed the Principle of Least Action?

The Principle of Least Action was first proposed by French mathematician and physicist Pierre Louis Maupertuis in the 18th century. It was later refined and popularized by Swiss mathematician Leonhard Euler and French physicist Joseph-Louis Lagrange in the 18th and 19th centuries, respectively.

How is the Principle of Least Action applied in physics?

The Principle of Least Action is applied in various fields of physics, including classical mechanics, quantum mechanics, and electromagnetism. It is used to determine the equations of motion for a system and to predict the behavior of objects in a given environment. It is also used in the study of thermodynamics and statistical mechanics.

What is the significance of the Principle of Least Action?

The Principle of Least Action is significant because it provides a fundamental understanding of the laws of motion and energy conservation in the universe. It is also a key principle in the study of physics and is used to solve complex problems and predict the behavior of objects in a given system.

Are there any limitations to the Principle of Least Action?

The Principle of Least Action is a powerful and widely applicable concept, but it does have some limitations. It is based on the assumption that a physical system will always choose the path of least energy, which may not always be the case in certain situations. Additionally, it does not take into account the effects of quantum mechanics, which can be significant at the atomic and subatomic level.

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