Linked Questions

12
votes
6answers
2k views

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 ...
8
votes
2answers
1k views

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 ...
7
votes
3answers
3k views

What's the difference between “boundary value problems” and “initial value problems”?

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{\...
3
votes
2answers
284 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-...
3
votes
0answers
188 views

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 ...
2
votes
3answers
742 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, ...
2
votes
2answers
117 views

How does Hamilton's Principle give us the path taken?

We defined the action as: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $q_i(t_1)$ and $q_i(t_2)$ are known and fixed. Hamilton's principle states that the path that is ...
2
votes
1answer
651 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 ...
2
votes
3answers
750 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 ...
1
vote
1answer
71 views

If the path integral formulation includes future events, why doesn't that imply retrocausality?

I know that such events would cancel out in the math, but if an extreme event were to happen in the future (say a black hole forming or something on that par), would a particle in the present react to ...
1
vote
3answers
257 views

How can an action be dependent on both its past and future?

Is it true that whenever an action takes place it is dependent on both its past and future? I mean if we already know that whatever we are doing is dependent on future as much as it is dependent on ...
1
vote
2answers
151 views

Principle of least action and greedy algorithm

Is the principle of least action sort of a greedy algorithm that all mechanical systems follow?, sometimes to minimise and sometimes to maximise the quantity we call action, at each individual step.
1
vote
1answer
494 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 ...
1
vote
0answers
31 views

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?
0
votes
1answer
679 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 ...

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