Could someone explain why the principle of least action is true?
If you tell me, then I will give you an equally satisfying answer to your original question.I suppose. Yes I can.
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.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.
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?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 .