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### In Fermat's Principle of Least Time, how do we know that light is able to reach the end point? [duplicate]

From my understanding of Fermat's Principle, you decide a start point and an end point for a light ray to travel between, and the light 'chooses' whichever path takes the least time (or technically ...
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### Doubt regarding Fermat's principle [duplicate]

Which two points are we talking about in Fermat's principle? Are those points decided by light or decided by us? Can we take any two points?
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### How can we predict how a system evolves using the stationary action principle even though we need to specify the final state? [duplicate]

The stationary action principle states that a system evolves between a fixed initial and fixed final configuration in such a way that the action is stationary. But isn't the final configuration what ...
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### In the Principle of Least Action, how does a particle know where it will be in the future?

In his book on Classical Mechanics, Prof. Feynman asserts that it just does. But if this is really what happens (& if the Principle of Least Action is more fundamental than Newton's Laws), then ...
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### Question about the apparent loophole in principle of least action

In Lagrangian formalism, given two points $(x_1,t_1)$ and $(x_2,t_2)$, we ask the question which paths $x(t)$ make the action $S=\displaystyle \int_{t_1}^{t_2}L\ \mathrm dt$ stationary and satisfy the ...
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Mathematically speaking, is there any essential difference between initial value problems and boundary value problems? The specification of the values of a function $f$ and the "velocities" $\frac{\... 3answers 730 views ### Can the Euler-Lagrange equations be derived from an infinitesimal Principle of Least Action? The Euler-Lagrange equations can be derived from the Principle of Least Action using integration by parts and the fact that the variation is zero at the end points. This has a mystical air about it, ... 3answers 712 views ### Is the path of stationary action unique? What are the physical implications of$L_{\dot{x}}=L_x$Below, for any function$Q$the notation$Q_x$means$\frac{\partial Q}{\partial x}$, and$Q_{xx}$means$\frac{\partial^2 Q}{\partial x^2}$. In physics, the trajectory of a particle is given by the ... 1answer 609 views ### “Principle of least action” and “Principle of conservation of energy”: Which one is fundamental and which one is derived? [closed] Suppose I throw a ball upwards. First it will rise under gravity and then fall under gravity. During the rising part the kinetic energy gradually decreases and the potential energy increases until ... 1answer 597 views ### Lagrangian mechanics and initial conditions vs boundary conditions It bothers me that many basic books on the classical mechanics don't discuss the following difference between "Newton's laws" and the "Principle of stationary action". Newton's laws can predict the ... 1answer 472 views ### Hamilton-Jacobi theory and initial value problem? Having read through some recent posts regarding the Lagrangian formulation being interpreted into an initial value problem rather than the familiar boundary condition problem we are familiar with, I ... 1answer 314 views ### Maximum aging and path of rock When a rock falls from a ledge, why does it head to the surface and not up to where time runs faster? If a rock, free from forces, follows a worldline of maximum aging, why would that rock approach ... 2answers 223 views ### Why can we consider the endpoint fixed in the derivation of the Euler-Lagrange equation in mechanics? In mechanics, we obtain the equations of motion (Euler-Lagrange equations) via Hamilton's principle by considering stationary points of the action $$S = \int_{t_i}^{t_f} L ~ dt$$ where we have$L=T-...
I am citing from Landau and Lifschitz, this statement that will seem to you well-known, trivial, etc: "Between these positions, (i.e. $q_1$ and $q_2$) the system moves then in such a way that the ...