Principle of least action

• Ragnar
In summary, the principle of least action states that the motion of a system is minimised by following the path of least effort. This is why nature always tries to move in the most efficient way possible.f

Ragnar

Could someone explain why the principle of least action is true?

Before learning this principle, you should have a good foundation with Newton's 2nd law.

Can you explain why the second law is true?

Before learning this principle, you should have a good foundation with Newton's 2nd law.

Can you explain why the second law is true?

I suppose. Yes I can.

I suppose. Yes I can.

If you tell me, then I will give you an equally satisfying answer to your original question.

I found this thread through a search looking for an answer to the same question as the OP. I find the responses a bit mysterious and wondered if anyone had a fairly straightforward physical explanation as to exactly what action is and why it is minimized.

what action is and why it is minimized

As I promised, action is the quantity which when minimized reproduces the same motions as would Newton's 2nd law. Similarly, the lagrangian is the thing which when substituted into lagrange's equations reproduces Newton's law.

It has been said that the principle of least action is the only law of physics.
I don't think anyone knows why nature should move in such a way as to minimise the action. If anyone knows, please tell us.

We know nature hates potential energy and all motion is just potential energy
converting to kinetic or heat energy. But the transfer between the energies is alway subject to least action.

The question is not so much - is the LAP true - but why it works so well.

We know nature hates potential energy and all motion is just potential energy
converting to kinetic or heat energy. But the transfer between the energies is alway subject to least action.

i can't understand what u mean when u say nature hates p.e??
p.e is just a definition and why should it be converted to heat or kinetic energy

as far as fermat's principle of least time goes it's a beautiful and intutive observation like Newton's third law and many other laws of physics...

As I promised, action is the quantity which when minimized reproduces the same motions as would Newton's 2nd law. Similarly, the lagrangian is the thing which when substituted into lagrange's equations reproduces Newton's law.

I am not doubting the equivalance of Langrangian and Newtonian mechanics. Indeed, most classical dynamic books provide a fairly staightforward dirivation of this equivalance. However, why not just give this dirivation in reverse so that you begin with Newtons Law's and end with the Lagrange equations? Why do classical dynamic books go to all the trouble of outlining in detail the calculus of variation and then applying it to this mysterious quantity action in order to derive the Lagrange equations? Clearly a point is being made: nature is once again minimizing a quanity. This time that quantity is the time integral of action. Which begs the question, all equivalence aside: what does action physically represent and, in its own right, why does it make sense that nature would minimize it?

i can't understand what u mean when u say nature hates p.e??
Mentz is referring to the fact that any system will always seek to minimize its potential energy. This is why caculating the stability of a system only requires calculating the 2nd derivative of its potential energy, ie: its concavity at a particular point. If the system can shed additinal potential energy the it will do so until it reaches a local or absolute minimum.

as far as fermat's principle of least time goes it's a beautiful and intutive observation .
This post is on Hamilton's Principle, not on Fermat's Principle. Both require the calculus of variation to derive a useful result. In the context of this post it is Langrange's equations, whereas Fermat's Principle leads to Snell's Law. However, I'm not so sure I find Fermat's Principle that intuitive either.

Thank you, Harriger, for clarifying my remark about potential energy.

It might help to know that action is the work done by summing infinitessimal forces that guide a body on a certain path. The path of least action minimises the work. So it is really the path of least effort. Why nature does this is not clear.

I'm looking at a derivation now that starts with F=ma and ends with Lagrange's equations, so at least one book does it this way.

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