any of several principles that find the physical trajectory of a system by minimizing or maximizing some value computed over the proposed path (for instance geometric optics can be reproduced by insisting on a minimum time principle).

learn more… | top users | synonyms

0
votes
1answer
13 views

Examples in which the light maximizes the optical path length

I posted a similar question about geodesics in the math.stackexchange. Many sources (here for instance, http://en.wikibooks.org/wiki/Waves/Fermat's_Principle) claim that the light could maximize the ...
0
votes
1answer
43 views

The einbein in the action of a relativistic massive point particles

The action of a relativistic massive point particle moving in space-time is $$S=-m\int d\tau \sqrt{g _{\nu \rho}\frac{dx^{\nu}}{d\tau}\frac{dx^{\rho}}{d\tau}}$$ [with Minkowski sign convention ...
2
votes
1answer
65 views

How to formulate variational principles (Lagrangian/Hamiltonian) for nonlinear, dissipative or initial value problems?

Although this questions is very much math related, I posted it in Physics since it is related to variational (Lagrangian/Hamiltonian) principles for dynamical systems. If I should migrate this ...
0
votes
0answers
27 views

Why are functional representations of systems important in physics or computational physics?

This was an addendum to a previous question I asked, but I figured I should make it it's own discussion. Assuming I am able derive a functional representation for any dynamical system (dissipative, ...
4
votes
1answer
68 views

Can we derive most fundamental laws from the Action Principle? [duplicate]

It is said in the book Fearful Symmetry - The Search for Beauty in Modern Physics that we can derive all basic laws in physics from a simple principle called Least Action Principle (although it may be ...
0
votes
0answers
38 views

Action principle for a faster-than-light point particle in special relativity [on hold]

When we have principle of stationary action in the Newtonian physics, we can safely choose any smooth trajectory connecting the initial and the final points because any velocity $\textbf{v}$ is ...
1
vote
2answers
73 views

Variation of a term in the Lagrangian

I don't understand why $$\frac{\delta}{\delta\phi}\left(\frac12\partial^\mu\phi\partial_\mu\phi\right)~=~\partial^\mu\partial_\mu\phi.\tag{1}$$ If we use integration by parts, there should be a minus ...
0
votes
0answers
32 views

Variational principle proof (summing over $n$)

From http://en.wikipedia.org/wiki/Variational_method_%28quantum_mechanics%29 $$= \sum_n \sum_m c_n^*c_mE_m \langle \psi_n|\psi_m \rangle$$ $$= \sum_n |c_n|^2E_n$$ I just want to better understand ...
1
vote
1answer
41 views

Variational Principle to find Energy Eigenfunctions

In Quantum Mechanics one can estimate an upper bound for the ground state energy with the following functional: $$\mathcal{F}[\psi(x)] \equiv \int_{-\infty}^\infty \psi^*(x)\hat{H}\psi(x) \,\, dx ...
6
votes
1answer
225 views

How do you determine the Lagrangian?

I have always been puzzled by how do you arrive at Lagrangians? That is, how do you know that the functional you need to get Newton's equations is $$L = T-V(x)~?$$ Do you derive the Lagrangian ...
3
votes
1answer
97 views

Using Lagrangian mechanics instead of Newtonian mechanics

When studying advanced classical mechanics, we all study about Lagrangians and the Euler-Lagrange equations and their importance. Of course, the Lagrangian is calculated based on the potential and ...
3
votes
1answer
146 views

Time-dependent Schrodinger equation from variational principle

In the paper, "Density-functional theory for time-dependent systems" Physical Review Letters 52 (12): 997 the authors mentioned that the action $$ A= \int_{t_0}^{t_1} dt \langle \Phi(t) | i ...
9
votes
1answer
489 views

Recovering all of Maxwell's equations from the variational principle

Whether you can get the first couple of Maxwell equations from a variational principle? In the second volume of the Landau theoretical physics said that it is impossible.
1
vote
0answers
34 views

Help understanding electromagnetism integral from exercise in MTW? [closed]

I was skimming through Misner, Thorne and Wheeler's book Gravitation looking for exercises to challenge myself with and came across the following exercise on page 178: Verify that the variational ...
2
votes
2answers
58 views

“Find the Lagrangian of the theory”

I've heard a few of my professors throw around the term "finding the Lagrangian of a theory". What exactly is this referring to. From what I understand it seems that you determine invariances ...
4
votes
1answer
100 views

Boundary term in Einstein-Hilbert action

Why is the boundary term in the Einstein-Hilbert action, the Gibbons-Hawking-York term, generally "missing" in General Relativity courses, IMPORTANT from the variational viewpoint, geometrical setting ...
1
vote
1answer
75 views

Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I ...
5
votes
0answers
122 views

Do “typical” QFT's lack a lagrangian description?

Sometimes as a result of learning new things you realize that you are incredibly confused about something you thought you understood very well, and that perhaps your intuition needs to be revised. ...
2
votes
1answer
129 views

Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
11
votes
1answer
2k views

Variational Derivation of Schrodinger Equation

In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't ...
4
votes
1answer
90 views

Functional derivatives as distributions

I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ...
4
votes
1answer
216 views

Equations of motion of displacement field

We have an action: $$S[\boldsymbol{u}] = \frac{1}{2} \int dt \int d^3x \left\{ \mu (\frac{\partial u_{i}}{\partial t})^{2} - \nu (u_{ii})^{2} - \rho(u_{ij})^{2}\right\} $$ Where $u_{ij} = ...
0
votes
1answer
62 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 ...
0
votes
2answers
54 views

Confusion about what the Euler-Lagrange equation says

I roughly understand the concept of the Lagrangian $L = T - V$ as well as the idea of stationary action $\delta \mathcal{S} =0$. However, I am confused what the Euler-Lagrange equation actually says. ...
2
votes
1answer
46 views

What is an “equation of motion” as used in context of geodesic equation?

I am studying general relativity and using the book Gravity by James Hartle. On page 170, he provides the following table: I don't understand what he means by "equation of motion" nor do I ...
1
vote
1answer
57 views

Geodesic equation from the proper time integral

This is something that has been bothering me for a little while. The usual procedure that I've seen is to write the proper time as the line integral $$\tau=\int_\gamma d\tau$$ along some curve ...
0
votes
0answers
38 views

Ritz Variational Method

I want to use the Ritz Variational method to find a good approximation ground state and ground state energy for the hydrogen atom. For that purpose I take two different ansatzes, do the machinery of ...
0
votes
0answers
189 views

Is the principle of least action fully equivalent to the Euler-Lagrange equations?

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 ...
0
votes
0answers
29 views

Lagrangians with higher derivatives than Klein-Gordon [duplicate]

Has anyone ever tried to work with Lagrangians involving higher derivatives? The Klein-Gordon Lagrangian only involves $(\frac{\partial}{\partial t})^2$ and $\nabla^2$ terms, what about third and ...
3
votes
1answer
59 views

Deriving the Canonical Energy Momentum Tensor

In the Mathematics for Physics of Stone and Goldbart the canonical energy momentum tensor is derived by the action principle as follows. To the action of the form $$ S=\int ...
12
votes
2answers
157 views

Is it possible to prove that planets should be approximately spherical using the calculus of variations?

Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical? Lets assume that a planet is 'born' when lots of ...
3
votes
3answers
665 views

How to obtain the field equations in Brans-Dicke theory from the action?

The action for the Brans-Dicke-Jordan theory of gravity is $$ \\S =\int d^4x\sqrt{-g} \; \left(\frac{\phi R - \omega\frac{\partial_a\phi\partial^a\phi}{\phi}}{16\pi} + \mathcal{L}_\mathrm{M}\right). ...
7
votes
2answers
666 views

Functional derivative in Lagrangian field theory

The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') ...
3
votes
1answer
46 views

Lagrangian for second-order system

Given an $n$-dimensional second-order system $$\ddot q^i-\sum_{j=1}^n A^i_j\dot q^j=0,$$ where $A$ is a constant matrix, is it possible to find a Lagrangian such that the above equation is the ...
1
vote
4answers
256 views

Why drops form spheres?

Consider a drop of water floating in an inertial frame in STP air (e.g., the ISS). Intuitively, the equilibrium shape of the drop is a sphere. How would one prove that? Is it equivalent to showing ...
3
votes
0answers
76 views

Euler-Lagrange equation (equation of motion) solution with hairy Lagrangian [closed]

I'm going through Zwiebach Chapter 6 on relativistic strings to try to solve a similar problem. I got all the way to my equation of motion \begin{eqnarray*} \delta S & = & [ p' \delta ...
1
vote
1answer
64 views

Kohn-Sham equations from variational principle

I'm trying to understand how the Kohn-Sham equations arise from the variational principle, failing. I think my problem is the inability to apply the variational principle. Or, I lack some crucial ...
1
vote
1answer
74 views

Intuition behind Hamilton's Variational Principle

Background: I am an upper level undergraduate physics student who just completed a course in classical mechanics, concluding with Lagrangian Mechanics and Hamilton's Variational Principle. My ...
1
vote
1answer
103 views

Is there an error in Susskinds' derivation of Euler-Lagrange equations?

http://imgur.com/kZO5C0V First, I believe there is a trivial error. The second equation should have another $\Delta t$ multiplying everything on the right. It is divided out later when the equation ...
1
vote
2answers
38 views

What are the end points in the action integral of field theory?

In the mechanics of particles when we apply the principle of the least action the two end points are two spatial coordinates. Therefore, if we consider the variation of the action with respect to the ...
4
votes
3answers
122 views

Geodesic Equation from variation: Is the squared lagrangian equivalent?

It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, ...
2
votes
1answer
82 views

Variation of the metric with respect to the metric

For a variation of the metric $g^{\mu\nu}$ with respect to $g^{\alpha\beta}$ you might expect the result (at least I did): \begin{equation} \frac{\delta g^{\mu\nu}}{\delta g^{\alpha\beta}}= ...
3
votes
1answer
87 views

Reference Request: Fluid dynamics/Elasticity via Lagrangians

Would there be a book that does what Landau does in Fluid Mechanics and Theory of Elasticity using Lagrangian's/Action-principles, analogous to the presentation in Landau's mechanics? I have only ...
4
votes
1answer
90 views

Total derivative in action of the field theory

Consider a classical field theory. When applying the least action I see that a term is considered total derivative. We say that $$\int \partial_\mu (\frac {\partial L}{\partial(\partial_\mu ...
32
votes
6answers
3k views

Why does calculus of variations work?

How does it make sense to vary the position and the velocity independently? Edit: Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
3
votes
1answer
98 views

Can action be unbounded from below?

While solving the problem in this question, I found cases where the numerical optimization failed, suspecting unboundedness of the function being minimized. The function approximates the action of the ...
4
votes
2answers
200 views

When is the principle of stationary action not the principle of least action?

I've only had a very brief introduction to Lagrangian mechanics. In a physics course I took last year, we briefly covered the principle of stationary action --- we looked at it, derived some equations ...
3
votes
3answers
526 views

Type of stationary point in Hamilton's principle

In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense - we all ...
9
votes
3answers
418 views

Confusion regarding the principle of least action in Landau's “The Classical Theory of Fields”

Edit: The previous title didn't really ask the same thing as the question (sorry about that), so I've changed it. To clarify, I understand that the action isn't always a minimum. My questions are in ...
5
votes
1answer
74 views

Least action principle — numerical simulation strangeness

I'm trying to get some experience with the least action principle, and for this I chose a simple 1-dimensional problem of a particle moving in some field. The least action principle would then look ...