40
Here's what I perceive to be a mathematically and logically precise presentation of the theorem, let me know if this helps.
Mathematical Preliminaries
First let me introduce some precise notation so that we don't encounter any issues with "infinitesimals" etc. Given a field $\phi$, let $\hat\phi(\alpha, x)$ denote a smooth one-parameter family of fields ...
30
Assume for simplicity that the speed of light $c=1$. The existence of the gauge $4$-potential $A^{\mu}=(\phi, \vec{A})$ alone implies that the source-free Maxwell equations
$$\vec{\nabla} \cdot \vec{B} ~=~ 0 \qquad ``\text{no magnetic monopole"}$$
$$ \vec{\nabla} \times \vec{E} + \frac{\partial \vec{B}}{\partial t} ~=~ \vec{0}\qquad ``\text{Faraday's law"}$$...
30
In general relativity, it's not entirely clear what "least time" means, since you have to ask "whose time are you talking about"? Are you talking about the time as measured by the emitter? The receiver? Someone else "far away" from both of them? All three of these quantities might be different.
Rather, it's more productive to say that light rays travel ...
20
As that lovely article linked by dfan says the virial theorem comes from varying the action $S[x]$ by $x\rightarrow(1+\epsilon)x$
$$\frac{1}{T}\delta S = \frac{1}{T}\epsilon\int_{0}^{T} dt\{m\dot{x}^2 -x\frac{\partial V}{\partial x}\}$$
This is a variation of the action and therefore must vanish up to some boundary terms if $x$ is a solution of the ...
20
I) Initial value problems and boundary value problems are two different classes of questions that we can ask about Nature.
Example: To be concrete:
an initial value problem could be to ask about the classical trajectory of a particle if the initial position $q_i$ and the initial velocity $v_i$ are given,
while a boundary value problem could be to ask ...
20
More generally, Lagrange equations$^1$ read
$$ \frac{d}{dt}\frac{\partial (T-U)}{\partial \dot{q}^j}-\frac{\partial (T-U)}{\partial q^j}~=~Q_j-\frac{\partial{\cal F}}{\partial\dot{q}^j}+\sum_{\ell=1}^m\lambda^{\ell} a_{\ell j}, \qquad j~\in \{1,\ldots, n\}, \tag{L}$$
where
$q^1,\ldots ,q^n,$ are $n$ generalized position coordinates;
$T$ is the kinetic ...
18
It does follow from calculus. Here's the standard way this is treated (I'm not going to be explicit about mathematical details such as smoothness assumptions here).
Definition of $\delta q$.
Given a parametrized path $q:t\mapsto q(t)$, we consider a deformation of the path which we call $\hat q:(t, \epsilon)\mapsto \hat q(t,\epsilon)$ satisfying $\hat q(t,...
18
I) Let there be given a local action functional
$$\tag{1} S[\phi]~=~\int_V \mathrm{d}^nx ~{\cal L}, $$
with the Lagrangian density
$$\tag{2} {\cal L}(\phi(x),\partial\phi(x),x). $$
[We leave it to the reader to extend to higher-derivative theories. See also e.g. Ref. 1.]
II) We want to study an infinitesimal variation$^1$
$$\tag{3} \delta x^{\mu}~=~\...
17
A way to understand this, is to imagine that light actually follows all paths. However, most paths experience destructive interference with other paths. The only paths that do not experience destructive interference are those in the neighbourhood of paths with stationary (e.g., minimal) action (time).
I strongly recommend reading Feynman's QED: The Strange ...
15
I) Not all equations of motion (eom) are variational. A famous example is the self-dual five-form in type IIB superstring theory. In classical point mechanics, frictional forces typically lead to non-variational problems.
II) Consider for instance $n$ variable $q^i$ and $n$ eoms,
$$\tag{1} E_i~\approx~ 0, \qquad i~\in~\{1, \ldots, n\}. $$
A simplified ...
15
Contrary to your claim near the end of your question, I claim that the time-derivative of the field is being treated as an "independent" argument of the Lagrangian. I'll try to convince you of this by showing you how this independence leads to everything working out the way you think it should. Some of the key points are at the end, so please read all the ...
15
Before developing the theory, I decided to first make an experiment in order to understand, what we are dealing with. A cylinder with a diameter of 11.5 cm is mounted on the motor shaft (I used an old popcorn machine). I attached a 12.5 cm length of clothesline with a screw, so that exactly 11.5 cm leaves the cylinder. When the rope hangs freely, it forms a ...
14
We bend light all the time - using lenses.
Light bends when going from one material to another, due to conservation of momentum.
Snell's law describes how light bends.
Light is also bent when traveling past massive objects - look into "gravitational lensing" if you are interested.
Light can be effectively bent into a parabolic path using materials that ...
14
Quantum systems are essentially defined by their symmetries. For example, in QFT's you expect all terms not forbidden by the symmetries of the problem to appear in the Lagrangian, with irrelevant operators suppressed by large scales, etc.
So I think your first step in this approach would be to write down the most general 2D QFT respecting the 2D Diff and ...
13
I) The closest cosmetic resemblance between the Nambu-Goto action and the Polyakov action is achieved if we write them as
$$\tag{1} S_{NG}~=~ -\frac{T_0}{c} \int d^2{\rm vol} ~\det(M)^{\frac{1}{2}} , $$
and
$$\tag{2} S_{P}~=~ -\frac{T_0}{c}\int d^2{\rm vol}~ \frac{{\rm tr}(M)}{2} , $$
respectively. Here $h_{ab}$ is an auxiliary world-sheet (WS) metric ...
13
Perhaps a simple example is in order. Consider a harmonic oscillator
$$\tag{1} S~=~\int_{t_i}^{t_f} \! dt~L, \qquad
L~=~\frac{m}{2}\dot{x}^2 - \frac{k}{2}x^2, $$
with characteristic frequency
$$\tag{2} \frac{2\pi}{T}~=~\omega~=~\sqrt{\frac{k}{m}}, $$
and Dirichlet boundary conditions
$$\tag{3} x(t_i)~=~x_i \quad\text{and}\quad x(t_f)~=~x_f. $$
For ...
13
I) In Palatini $f(R)$ gravity, the Lagrangian density is$^1$
$$ {\cal L}(g,\Gamma)~=~ \frac{1}{2\kappa}\sqrt{-g} f(R) + {\cal L}_{\rm m}; \tag{1}$$
with matter Lagrangian density ${\cal L}_{\rm m}$;
with scalar curvature
$$R~:=~ g^{\mu\nu} R_{\mu\nu}(\Gamma);\tag{2}$$
with Ricci curvature $R_{\mu\nu}(\Gamma)$; and where
$$\Gamma^{\lambda}_{\mu\nu}~=~\...
13
The problem with your approach is that your proposed action
$$S = \int |\mathbf{v}| \, dt$$
is not invariant at all. While Landau's action is invariant under Lorentz transformations, and in fact completely coordinate independent, yours is not invariant under even Galilean transformations, which add a constant to $\mathbf{v}$. The space-only analogue of a ...
12
The Lagrangian density for a Dirac field is
$$
\mathcal{L} = i\bar\psi\gamma^\mu\partial_\mu\psi -m \bar\psi\psi
$$
The Euler-Lagrange equation reads
$$
\frac{\partial\mathcal{L}}{\partial\psi} - \frac{\partial}{\partial x^\mu}\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\right] = 0
$$
We treat $\psi$ and $\bar\psi$ as independent dynamical ...
12
Comment to the question (v2): P&S is using the notation of a 'same-spacetime' functional derivative. To illustrate this notation, let us for simplicity stay within first variations, and leave it to the reader to generalize to higher-order variations.
I) First of all, functional/variational derivatives should not be confused with partial derivatives. In ...
12
Indeed the problem with boundary conditions, generally speaking, is not well-posed.
There are boundary conditions admitting no curves or admitting many curves, satisfying both these conditions and Euler-Lagrange equations.
Examples.
(1) Think of a particle constrained to stay on a smooth sphere where it can freely move. If you assign the North and the ...
12
The Wikipedia quote appears to be lifted from this Solid State Physics text by CTI Reviews, and then plastered all over the web. The text does not give any citation of Lanczos, however. Here is the only passage in Lanczos's 300 pages long Variational Principles of Mechanics that contains the word "self-adjoint":
"Schroedinger, on the other hand, ...
12
The Lagrangian and Hamiltonian approaches are frameworks, and not theories. It is certainly true that a wide variety of systems are susceptible to such an approach.
However, there are many theories which do not possess Lagrangians. For example, it is believed that a certain set of six-dimensional superconformal field theories may be able to describe all ...
12
It is the least-time path.
Fermat's principle was itself built on two earlier observations: on the one hand the Greeks had noticed that in reflections, light traveled along the least-distance path; on the other hand Snell had discovered and Descartes had popularized that in refraction light obeyed a "law of sines" which said that the ratio of the speed of ...
12
The Euler Lagrange equation is a differential equation resulting from the search for the extremum of a functional: this extremum is given by the first variation only.
This is similar to the condition for finding a point where a function $f$ is extremum: the condition $df/dx=0$ is on the first derivative only.
In both cases, one does not seek to ...
11
Yes, there is a link, both are examples of extremum principles. And yes it is possible to derive the principle of least action from the law of maximum entropy. The derivation is lengthy and I will only sketch the main steps needed:
We start from the law of maximum entropy $dS/dt \geq 0$. As we know this law is only valid for isolated systems [i]. For ...
11
The term vanished because we can translate this term to one making a statement about the fields at the boundary and assume that the fields themselves vanish in spatial and temporal infinity.
By Stokes' Theorem, we can translate volume integrals into surface integrals. More specifically Gauss' Theorem states that the integral of a divergence of a field over ...
11
Fermat's principle is a bit more complicated than what you state: it says that in travelling from $A$ to $B$, light will go along the paths that will minimize the time taken to get there - and these may or may not be straight lines. (See e.g. Wikipedia.)
That said, the gravitational lensing of light does not operate quite like that. Since it is in vacuum, ...
11
I) The equation of motion for a scalar massless relativistic point particle on a Lorentzian manifold $(M,g)$ is
$$ \tag{A} \dot{x}^2~:=~g_{\mu\nu}(x)~ \dot{x}^{\mu}\dot{x}^{\nu}~\approx ~0, $$
where dot denotes differentiation wrt. the world-line parameter $\tau$ (which is not proper time). [Here the $\approx$ symbol means equality modulo eom.] Thus a ...
11
The more common names for what you are talking about are the abbreviated action
$$S_0[q] := \int p \mathrm{d}q$$
versus the action
$$ S[q] := \int_{t_1}^{t_2}L(q,\dot q,t)\mathrm{d}t$$
Both are used in different formulations of classical mechanics, and deliver a different "flavor" of solutions. On both one can do variations calculus and obtains the ...
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