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).

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85 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 ...
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0answers
224 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 ...
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0answers
44 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 ...
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1answer
189 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 \mathcal{L}(\varphi,\...
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1answer
204 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 ...
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1answer
56 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 ...
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4answers
366 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 ...
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0answers
255 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 \...
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1answer
206 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 ...
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1answer
137 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 ...
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1answer
179 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 ...
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2answers
84 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 ...
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1answer
439 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 ...
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4answers
620 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 $\mathbb{R} \to M$, $...
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1answer
542 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}}= \delta^\...
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1answer
233 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 \phi)}\...
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1answer
119 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 ...
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2answers
796 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 ...
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1answer
190 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 ...
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1answer
166 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 ...
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2answers
595 views

Variational Theorem proof

I have been trying to prove variational theorem in quantum mechanics for a couple of days but I can't understand the logic behind certain steps. Here is what I have so far: \begin{equation} E=\frac{\...
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0answers
58 views

Action principles and covariant equations [duplicate]

Can every physically sound differential equation, that is covariant, deterministic etc. be derived by extremising a suitable action using a suitable lagrangian, that may be arbitary. Is this a ...
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1answer
267 views

One step in deriving the Einstein-Hilbert action

In this amazing first principles derivation of the Einstein-Hilbert action there is one small manipulation needed to show $$c g_{ab,cd}\left(\eta^{ac}\eta^{bd} - \eta^{ab}\eta^{cd}\right)$$ is ...
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1answer
733 views

Why is it so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection?

There are two ways to do the variation of Einstein-Hilbert action. First one is Einstein formalism which takes only metric independent. After variation of action, we get the Einstein field equation. ...
3
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1answer
163 views

Is it possible to have the principle of least action and multiple solutions?

This is possibly a silly question but when we derive the equations of motion of a particle using the principle of least action. We must assume that there is a single minimum (for a fixed choice of ...
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1answer
42 views

Relevance of this equation

The book 'The variational princples of mechanics' by Lanczos was recommended to me and I am reading it now. So far it is an enjoyable journey, but I don't see the significance of this equation: $t = \...
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2answers
765 views

Finding 3-Sphere Christoffel connection coefficients using variational calculus, Sean Carrol problem

I have A 3-Sphere with coordinates $x^{\mu} = (\psi,\theta,\phi)$ and the following metric: \begin{equation} ds^2 = d\psi^2 + \text{sin}^2\psi(d\theta^2 + \text{sin}^2\theta d\phi^2) \end{equation} ...
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1answer
189 views

Hamilton's equations from the action with boundary conditions involving position and momentum

Generally, when you are given the action $$ S=\int_{t_1}^{t_2}\mathrm dt (p\dot q - \mathcal H )$$ the boundary conditions are $q(t_1)=q_1$ and $q(t_2)=q_2$. This is useful because to calculate $\...
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1answer
437 views

Variational form of Euler's incompressible fluid equations?

I am trying to derive Euler's incompressible fluid equations in terms of a variational stationary principle. Given Euler's flow equations: $$\frac{\partial v}{\partial t} = -\nabla p$$ $$\nabla\cdot ...
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1answer
261 views

Variational principle for a point particle (massive or massless) in curved space

We know that for a point particle, the action is $$ S[x,e] ~=~ \frac{1}{2}\int_{\lambda_A}^{\lambda_B} d\lambda\left[e^{-1}(\lambda)~g_{\mu\nu}(x(\lambda))~\dot{x}^\mu(\lambda)~\dot{x}^\nu(\lambda) -...
4
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1answer
353 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 ...
3
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1answer
137 views

First variation of the action in relativistic notation - Landau & Lifshitz “Classical theory of fields”

In Landau & Lifshitz's book, Classical theory of fields, the action for a free particle is defined as: $$\tag{8.1} S= \int ^b _a {-mc \ \text d s}=0,$$ where $$\text d s=c\,\text d t\sqrt{1-\...
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1answer
190 views

Use of the term first order dependency

In a question I am doing it says: Show explicitly that the function $$y(t)=\frac{-gt^2}{2}+\epsilon t(t-1)$$ yields an action that has no first order dependency on $\epsilon$. Also my textbook ...
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1answer
129 views

The principle of stationary action?

In proving that the action $$S\equiv \int^{t_2}_{t_1}L(x, x',t)dt$$ has a has a stationary point $x_0$ that satisfies the following: $$\frac{d}{dt}(\frac{\partial L}{\partial x'_0})=\frac{\partial L}{\...
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1answer
191 views

Is the zero acceleration path also the shortest path between two points?

In flat, free, Euclidean space, the shortest path and the zero acceleration path are the same path, which is a straight line. However, in general relativity, is the zero acceleration path also the ...
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1answer
653 views

How to calculate the refracted light path when refraction index continuously increasing?

Suppose an incident light from vacuum ($n_1=1.0$) into some media ($n_2=n_1+\mu\; x^2$) as in the figure below. How to calculate the refracted light path curve in closed form? Update: Try to set ...
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3answers
272 views

Virtual Work: How is the applied force related to the coordinates chosen?

I have a question after reading a section from Goldstein's Classical Mechanics. The question deals with equation 1.43 in the text (given below): $$ \tag{1.43} \sum\limits_{i} {\bf F}_i^{(a)}\cdot \...
2
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1answer
742 views

Proof that total derivative is the only function that can be added to Lagrangian without changing the eom

So I was reading this: Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time and while the answers for the first question are good, nobody gave much ...
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1answer
830 views

Why are D'Alembert's Principle and the Principle of Least Action Related?

Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common. $$\sum _i ( \...
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94 views

General relativity from helicity 2 massless field theory by using Deser's arguments

Recently I have discovered the method of constructing of GR from massless field with helicity 2 theory. It is considered here, in an article "Self-Interaction and Gauge Invariance" written by Deser S. ...
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0answers
150 views

Variations of S-matrix functional and Feynman diagrams in Weinberg QFT

Weinberg on p. 287 of his QFT vol. 1 introduces the extended interaction operator: $$ \tag 1 \hat{V}(t) \to \hat{V}(t) + \sum_{a}\int d^{3}\mathbf x \hat{o}_{a}(\mathbf x ,t)\varepsilon_{a}(x). $$ ...
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136 views

BTZ Black Hole Central Charge and Conformal Weight

I have been trying to reproduce a calculation (equation 4.12) in this paper http://arxiv.org/pdf/1107.2678v1.pdf by Carlip reviewing the derivation of the effective central charge of the BTZ Black ...
3
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0answers
91 views

Is the “Force” of Gravity Simply Hamilton's Principle on a Curved Spacetime?

It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it ...
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1answer
118 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 ...
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2answers
556 views

Action and Action integral: Different kinds of variational principles

What are the difference between: the action $\int_{t_{1}}^{t_{2}}(L+H) dt$ that we use in the principle of least action, and the action integral $\int_{t_{1}}^{t_{2}}L dt$ that we use in Hamilton'...
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4answers
1k views

Can you explain Fermat's Principle to me?

The textbook(F.A.Jenkins and H.E.White Fundamentals of Optics) states that the Fermat's principle is that the path taken by a light ray in going from one point to another through any set of media ...
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1answer
280 views

Proca Lagrangian manipulation

How can I show that the Lagrangian density $$\mathcal{L} = -\frac{1}{2}\partial_\alpha \varphi_\beta \partial^\alpha \varphi^\beta + \frac{1}{2} \partial_\alpha \varphi^\alpha \partial_\beta \varphi^\...
5
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2answers
643 views

Functional Derivative in the Linear Sigma Model

In the linear sigma model, the Lagrangian is given by $$ \mathcal{L} = \frac{1}{2}\sum_{i=1}^{N} \left(\partial_\mu\phi^i\right)\left(\partial^\mu\phi^i\right) +\frac{1}{2}\mu^2\sum_{i=1}^{N}\left(\...
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3answers
1k 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 ...
11
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5answers
715 views

What is the best path for a given initial and final state?

I am trying to calculate an efficient acceleration curve given starting and final positions and velocities. I'm assuming no friction, and that the acceleration can be applied in any direction at any ...