Why Is Action Defined as Energy Minus Potential Energy in Physics?

[Andrea2]

Hi! I'm new of this forum and I'm searching a way to understand why the action is E-U, but in this moment i don't know how to do...ther's someone who can help me? Thank you

 

[hunt_mat]

It turns out that if you set the Lagrangian to be T-V then you recover Newton's laws of motion.

 

[Cleonis]

Hi! I'm new of this forum and I'm searching a way to understand why the action is E-U, but in this moment i don't know how to do...ther's someone who can help me? Thank you

The question is of course: why does T-V recover Newton's laws of motion? As it turns out: that question is not hard to answer.

An abbreviated discussion is in post https://www.physicsforums.com/showpost.php?p=2975435&postcount=10" of the recent thread called 'Lagrangian'.

A more detailed version (more diagrams) is available on my website: http://www.cleonis.nl/physics/phys256/least_action.php" .

 

[Andrea2]

ok, thank you very much!

 

[PJC01]

Hello and welcome to the forum! The principle of least action is a fundamental principle in physics that states that nature tends to take the path of least resistance or least action. This means that out of all the possible paths that a system could take, it will take the path that minimizes the action, which is a mathematical quantity that represents the total energy of the system.

To understand why the action is equal to the energy minus the potential energy (E-U), we need to look at the mathematical formulation of the principle. The action (S) is defined as the integral of the Lagrangian (L) over time (t):

S = ∫ L dt

The Lagrangian is a function that describes the kinetic and potential energies of a system. It is defined as:

L = T - U

Where T is the kinetic energy and U is the potential energy. The kinetic energy is the energy that a system has due to its motion, while the potential energy is the energy that a system has due to its position in a force field.

When we plug in the definition of the Lagrangian into the equation for action, we get:

S = ∫ (T-U) dt = ∫ T dt - ∫ U dt

The first term in this equation represents the change in kinetic energy over time, and the second term represents the change in potential energy over time. The principle of least action states that the system will take the path that minimizes the total change in energy over time, which is equivalent to minimizing the action. This is why the action is equal to the energy minus the potential energy (E-U).

I hope this helps you understand the principle of least action better. If you have any further questions, please feel free to ask.