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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votes
1answer
71 views
Derivation of Dirac equation using the Lagrangian density for Dirac field
How can I find Dirac equation using the Lagrangian density for Dirac field?
1
vote
1answer
80 views
Retrieving Maxwell's equations from the minimum action principle
I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps.
Starting with the action:
$$S = \int dt \int ...
1
vote
1answer
53 views
Why vary the action with respect to the inverse metric?
Whenever I have read texts which employ actions that contain metric tensors, such as the Nambu-Goto, Polyakov or Einstein-Hilbert action, the equations of motion are derived by varying with respect to ...
1
vote
1answer
47 views
Total energy is extremal for the static solutions of equation of motions
In physics total energy is extremal for the static solutions of equation of motions.
Can anyone explain this sentence to me?
0
votes
1answer
95 views
proper variation of action term
I have a term I want to vary by a field, $\phi$.
$$
`S' = \frac{-1}{2}\,\sqrt{-g}\,g^{\mu\,\nu}\,\delta\left[h(\phi)\,\partial_{\mu}\phi\,\partial_{\nu}\phi \right].
$$
Is it correct to get this?
...
5
votes
2answers
265 views
English translation of Helmholtz' paper: “On the Physical Significance of the Principle of Least Action”
I am asking about an English translation of a Helmholtz paper:
Ueber die physikalische Bedeutung des Princips der kleinsten Wirkung. Journal für die reine und angewandte Mathematik (Crelle's ...
4
votes
1answer
86 views
What is the action for an electromagnetic field if including magnetic charge
Recently, I try to write an action of an electromagnetic field with magnetic charge and quantize it. But it seems not as easy as it seems to be. Does anyone know anything or think of anything like ...
1
vote
1answer
56 views
White Dwarf radius
So I've been reading this about white dwarves, and various other sites about white dwarves. In all of them, they say that we can find the radius of a white dwarf by minimizing its total energy. I know ...
2
votes
1answer
157 views
How light know which path is smallest?
We know from fermat's principle that light follows the smallest path. But how light know that which path is smallest?
1
vote
2answers
120 views
How the boundary term in the variation of the action vanishes
Can someone explain a little more that why the last term in equation (1.5) vanishes?
Reference:
David Tong, Quantum Field Theory: University of Cambridge Part III Mathematical Tripos, Lecture ...
2
votes
3answers
220 views
Noether's current expression in Peskin and Schroeder
In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence.
But if we ...
0
votes
0answers
67 views
Minimum energy magnetic field has zero current?
Following section 9.2 of Bellan's "Fundamentals of Plasma Physics" suppose we have some domain (I assume simply connected) $D$ with a magnetic field $\mathbf{B}$ inside, but no electrostatic field. ...
3
votes
3answers
124 views
Virtual differentials approach to Euler-Lagrange equation - necessary?
I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for the Euler-Lagrange equation. The whole notion of, and ...
0
votes
0answers
32 views
Suggestions for a physics oriented book on Variational Calculus [duplicate]
I would like to buy a good book on Variational Calculus. Most of the books that I find seem to be rather formal in a mathematical sense, which is not necessarily bad, but makes the studying a bit ...
1
vote
3answers
91 views
Formalities of the variational integral
Usually when the variational principle is introduced one starts by defining a Lagrangian density
$${\mathscr L}(x,\phi(x),\partial_{\mu}\phi(x))$$
and an action
$$S[\phi]=\int_R d(x) {\mathscr L}$$
...
6
votes
4answers
244 views
Least-action classical electrodynamics without potentials
Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which ...
3
votes
2answers
88 views
How is the physical Lagrangian related to the constrained minimization Lagrangian?
If we're minimizing an energy $V(q)$ subject to constraints $C(q) = 0$, the Lagrangian is
$$L = V(q) + \lambda C(q).$$
I have fairly solid intuition for this Lagrangian, namely that the energy ...
2
votes
0answers
171 views
Prove that the first order perturbation theory overestimates fundamental state [closed]
This was a question on my exam and I don't know how to solve it.
Use the variational principle to prove that the first order perturbation theory always overestimates the energy of the fundamental ...
2
votes
1answer
199 views
Electrodynamics and the Lagrangian density
Could anyone tell me what equations can I obtain from the Lagrangian density
$${\cal L}(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i,\,\,A_{i,j})~=~\frac{1}{2}|\dot A+\nabla\phi|^2-\frac{1}{2}|\nabla \times ...
5
votes
3answers
552 views
Entropy and the principle of least action
Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ?
5
votes
1answer
145 views
Finding interplanetary flight trajectory using calculus of variations?
Consider two orbits $x(t),\space y(t)$ representing the origin and destination for some spaceflight of interest. These could be, for example, cycloids describing LEO and another orbit circling, say, ...
1
vote
0answers
77 views
Are there any good reading materials for variational approach in many-body theory? [closed]
I need something like a summary of existing results, including the treatment of BCS Hamiltonian and Hubbard model. Auerbach's book is a good one but I still hope to get more comprehensive review. My ...
1
vote
1answer
153 views
What's the motivation behind the action principle? [closed]
What's the motivation behind the action principle?
Why does the action principle lead to Newtonian law?
If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and ...
3
votes
3answers
117 views
What is the meaning of the word “Principle” in Physics?
What is the meaning of the word principle in Physics?
For example in the "action principle". Is it an action law, an action equation, or an unproved assumption? (I have an idea what an action is).
...
0
votes
0answers
67 views
What are the details of this variational calculus solution?
This answer includes the problem:
Suppose you have a 2-d bullet going very fast through a 2-d gas. The gas molecules reflects specularly off the bullet, making glancing collisions. What shape of ...
1
vote
1answer
153 views
Questions regarding solving the Brachistochrone problem using Lagrangian
brachistochrone problem: Suppose that there is a rollercoaster. There is point 1 ($0,0$) and point 2 ($x_2, y_2)$. Point 1 is at the higher place when compared to the point 2, so the rollercoaster ...
2
votes
3answers
193 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, ...
1
vote
2answers
226 views
Can cos(x) or sin(x) be the function of stationary action?
Is there a way to express $\cos(x(t))$ (or $\sin(x(t))$) as the solution to the Euler-Lagrange equation, in other words is there a sense in which this function is the path of stationary action?
3
votes
3answers
200 views
Is the Lagrangian “math” or “science”?
I've seen in class that we can get from Lagrangian to derive equations of motion (I know its used elsewhere in physics, but I haven't seen it yet). It's not clear to me whether the Lagrangian itself ...
14
votes
4answers
457 views
Is the principle of least action a boundary value or initial condition problem?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:
In analytic (Lagrangian) mechanics, the derivation of the ...
5
votes
2answers
129 views
What shape of track minimizes the time a ball takes between start and stop points of equal height?
I was at my son's high school "open house" and the physics teacher did a demo with two curtain rail tracks and two ball bearings. One track was straight and on a slight slope. The beginning and end ...
3
votes
1answer
253 views
Gauss law in classical U(1) gauge theory
I can see that $a_{0}$ is not an independent field and Gauss law is a constraint on the theory arising from field equations. But, I don't get the geometrical picture.
Let $A$ be the space of all ...
4
votes
3answers
259 views
Is it safe to ignore derivatives of velocity w.r.t. position and vice versa?
In a certain textbook a function is given as:
$$f=f(x(t))$$
And then this is differentiated w.r.t. $t$ to get:
$$f_t=\dot{x}f_x$$
(Where the notation $f_u=df/du$, $f_{uu}=d^2f/du^2$, etc.)
This ...
2
votes
2answers
206 views
What is the significance of action?
What is the physical interpretation of
$$ \int_{t_1}^{t_2} (T -V) dt $$
where, $T$ is Kinetic Energy and $V$ is potential energy.
How does it give trajectory?
4
votes
4answers
1k views
How to bend light?
As we all know that light travels in rectilinear motion. But can we bend light in parabolic path? If not practically then is it possible in paper? Has anyone succeeded in doing that practically ?
4
votes
1answer
120 views
Variational wavefunctions and “spread” of potential in quantum mechanics
A particle in a box has an energy that decreases with the size of the box. In the general case, it is often said that a variational solution for a "narrow and deep" potential is higher in energy than ...
3
votes
1answer
202 views
Brachistochrone problem for 3 points
I wonder how I can solve the Brachistochrone problem for 3 points?
The matter starts from point A that is the highest point and it must pass from B and must finish with point C. (No any friction in ...
5
votes
4answers
727 views
Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
2
votes
2answers
197 views
How do I show that there exists variational/action principle for a given classical system?
We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
2
votes
1answer
148 views
Differentiation of the action functional
In the QFT book by Itzykson and Zuber, the variation of the action functional $I=\int_{t_1}^{t_2}dtL$ is written as:
$$\delta I=\int_{t_1}^{t_2}dt\frac{\delta I}{\delta q(t)}\delta q(t)$$
How is ...
8
votes
2answers
76 views
More general invariance of the action functional
I will formulate my question in the classical case, where things are simplest.
Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the ...
4
votes
1answer
451 views
Lagrangian for Relativistic Dust derivation questions
In most GR textbooks, one derives the stress energy tensor for relativistic dust:
$$
T_{\mu\nu} = \rho v_\mu v_\nu
$$
And then one puts this on the right hand side of the Einstein's equations. I ...
3
votes
2answers
274 views
What are some interesting calculus of variation problems? [closed]
That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks (catenary, brachistochrone, Fermat, etc..)
12
votes
6answers
2k views
Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
2
votes
1answer
320 views
Snell's Law of Refraction
I was told that "Snell's law of refraction implies that a light ray in an isotropic medium travels from point a to point b in stationary time." Why is this true?
Thanks.
2
votes
1answer
206 views
What variables does the action $S$ depend on?
Action is defined as,
$$S ~=~ \int L(q, q', t) dt,$$
but my question is what variables does $S$ depend on?
Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$?
In ...
4
votes
1answer
294 views
Fluid Mechanics from a variational principle
It is posible to define a good variational principle to describe Fluid Mechanics? if so, wath is the correct tratement of the issue. I guess something like:
$I=\int d^4x (\frac{1}{2}\rho v^2-P-\rho g ...
6
votes
2answers
603 views
Hydrostatic friction: why do water droplets stay at rest on an inclined glass surface?
Tjis is a non-expert question on a (seemingly simple) text-book topic.
The question is about "hydrostatic friction", defined as follows.
Consider a drop of water resting on a flat surface. If the ...
2
votes
2answers
204 views
Must the action for a relativistic charge be a Lorentz scalar?
Page 580, Chapter 12 in Jackson's 3rd edition text carries the statement:
From the first postulate of special relativity the action integral must be a Lorentz scalar because the equations of ...
1
vote
1answer
479 views
2nd order variation of Hilbert-Einstein action + Gibbons-Hawking-York boundary term
While the first order metric variation of Hilbert-Einstein action plus Gibbons-Hawking-York boundary term is well-known and takes the form:
$\delta S_{HE}+\delta S_{GHY}=-\frac{1}{16\pi G}\int d^3x ...
