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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?
I suppose. Yes I can.
what action is and why it is minimized
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.
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.
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.i can't understand what u mean when u say nature hates p.e??
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.as far as fermat's principle of least time goes it's a beautiful and intutive observation .