Questions tagged [variational-principle]
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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Slave boson mean field theory of spin liquid: minimization of energy
The following are summarized from the book:
Quantum Field Theory of Many-Body Systems by Xiao-Gang Wen
The Heisenberg model on a lattice is
$$
H = \sum_{\langle ij \rangle} J_{ij}
\mathbf{S}_i \cdot \...
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Energy-Momentum tensor of Polyakov action vanishes
In the lecture notes of David Tong on String Theory he defines the energy momentum tensor of the polyakov action as
\begin{align*}
T_{\alpha\beta}=-\frac{2}{T}\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta ...
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Deriving the Lagrangian for an Arbitrary Equation [closed]
Suppose one knows that for an arbitrary tensor $U_{\mu\nu}$, the equation $\partial_{\nu}U_{\mu\nu}=0$ is satisfied. Suppose that one seeks the Lagrangian associated with this equation.
One try could ...
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van der Waals interaction using variational method [closed]
2 hydrogen atoms separated by the distance R, find the total energy of the system using variational method
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Does $δS = 0$ mean that "the small changes in the actions equal to zero"?
Please correct me if I'm wrong.
What I understood from the Principle of Stationary Action is that for an object moving from point A to point B, at every point of the path with the least action, the ...
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Hamilton's Principle extended to deal with non-conservative forces [closed]
I have seen both authors stating that there is an extension of Hamilton's Principle to systems with non-conservative forces, and others stating that there is no variational principle available for ...
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Deriving Hamilton's Principle from Lagrange's Equations
I'm trying to derive Hamilton's Principle from Lagrange's Equations, as I've heard they're logically equivalent statements, and am stuck on a final step. For simplicity, assume we're dealing with a ...
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Why is the time taken for light propagation between two points in anisotropic media independent of $y$?
Background
Light propagating in an anisotropic medium does not (in general) take a straight-line path between two points. The propagation time between those points, then, is dependent on the total ...
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Numerically approximate the ground state wave function of the finite potential well problem [closed]
I am trying to numerically approximate the ground state wave function of an embedded square well, but when I plot the wave function, I get an unreasonable result.
Problem Setup
Here is the set up of ...
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Auxiliary field with a special property
Consider an action $S(y, z_1,z_2)$, a functional of the field variables $y, z_1$ and $z_2$, where $z_1$ and $z_2$ are auxiliary fields. The equations of motion of the fields are denoted by $E_y, E_{...
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Can action never be a maximum extremal for classical systems?
"For classical (non-quantum) systems, the action is an extremum that can never be a maximum; that leaves us with a minimum or a saddle point, and both are possible."
The above statement is ...
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Is optical length maximized for any ray reflected off a concave mirror? [duplicate]
Statement: "In optics, you can take the example of a concave mirror: the optical path chosen by the light to join two fixed points A and B is a maximum."
The statement gives the impression ...
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How to find the 'best' path(s) between two points on a surface? [migrated]
I was watching a TV programme about hiking trails; they were talking about this one, and mentioned that the guy who directed the construction work was a mathematician, 'hence' he could calculate the '...
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Change in number of gauge symmetries after adding auxiliary fields to the action
As per part (c) of Ex. (3.17) in Ref. 1, the number of gauge symmetries of an action does not change after adding auxiliary fields to it. But we know that a Stueckelberg field is an auxiliary field, ...
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How can I derive the equations of motion with the least action principle from the action of $p$-Form Electrodynamics? [closed]
I know this is the correct formula for the action for a arbitrary $p$. I know how to obtain the equations of motion for $p=1$, but I struggle to find a way to do this with an arbitrary $p$. I also ...
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Are equations of motion invariant under gauge transformations? [duplicate]
We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?
If yes, can you show a mathematical proof (instead ...
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Boundary conditions for Lagrangian formulation of General Relavitiy
I am reading section 4.1.3. of Poisson's book "A relativist's toolkit" and I am a bit perplexed by condition (4.13), namely that the variational principle for General Relativity has to be ...
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Simple Lagrangian with free bound and constraint
Let $\alpha,\beta$ non-zero real numbers, $f$ a function of time. I define $L_1=\alpha f + \beta$ and $L_2=p(t) L_1$.
I want to minimize $\int_0^T L_2$ under the constraint $\int_0^T L_1=v$, with $T$ ...
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Conceptual problem with incorporating constraints to a particular variational principle problem
Consider the following problem:
A vector field $\boldsymbol{F}(x)$ is defined over a finite region $V$. A functional of the form
\begin{equation}
U = \int_V u(\boldsymbol{F})\ d^3x
\end{equation}
is ...
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How to prove that a drop of water in the weightlessness of space is round in shape?
How to prove that a drop of water in the weightlessness of space is round in shape theoretically? More specifically, how to prove that a drop of water in the weightlessness of space is round in ...
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Variation of action for null geodesics in GR [duplicate]
I have a question about applying the variational principle to obtain the geodesic equation for null geodesics. Specifically, I am unsure about the justification for the choice of the Lagrangian. I ...
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Special cases of action integral $\delta S=0$ that do not satisfy the Euler-Lagrange equation
One way of deriving the Euler-Lagrange equations is to require that the action integral is stationary under a virtual displacement $\delta S=0$. One then usually arrives at the equation
$$
\delta S=-\...
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Is the principle of stationary action a deterministic or probabilistic principle? [duplicate]
I was reading Why the Principle of Least Action? and the top voted answer says
You can go further mathematically by learning the path integral formulation of nonrelativistic quantum mechanics and ...
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Why can't there be terms in the Lagrangian that are differentiated twice? [duplicate]
We often say the Lagrangian is a function of some coordinates and only their first derivatives,
$$
\mathcal{L}(q,\dot{q}).
$$
Even in quantum field theory, the fields are only differentiated once,
$$
\...
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Directly solving Hamilton's principle as an initial-value problem? [duplicate]
Can Hamilton's principle, i.e. the principle of stationary action, be posed as an initial-value problem instead of the usual boundary-value problem and still produce the correct equations of motion?
...
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Fermat's principle and a non-physical conclusion
Fermat's Principle is the statement that a ray will follow a minimum-time path between a point, A, to a point, B.
So, if I have a block of material of high refractive index, so that it slows the light ...
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How to distinguish a trivial gauge transformation from a non-trivial one?
Two days ago I posted a post that discusses a very generic gauge transformation. I repeat it here. Suppose we have an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. We ...
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Question about Trivial Gauge Transformation
Suppose we have an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. We denote the variation of $S$ wrt to a given field, say $a$, i.e. $\frac{\delta S}{\delta a}$, by $E_a$....
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Why does additional term to electromagnetic Lagrangian leave Maxwell's equations unchanged?
The addition of
$$\mathcal{L}' = \epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma} \propto \vec{E}\cdot\vec{B}$$
to the electromagnetic Lagrangian density leaves Maxwell's equations unchanged (shown ...
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Tensionless string in Nambu-Goto action
I am studying string theory from the book "String theory and M-theory", written by Becker, Becker and Schwartz. My question is:
We are taught that one of the advantages of introducing a ...
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Nambu-Goto action and the World-Sheet Area
I am studying string theory from the book "String theory and M-theory", written by Becker, Becker and Schwartz. My question is:
We are told that the Nambu-Goto action is simply the one that ...
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Integration by parts on generic tensors
I try to rephrase here a my question (https://math.stackexchange.com/q/4661784/), explaining more specifically the case.
Given a lagrangian $L=L(\theta_{\mu\nu},\phi)$ , the conserved Noether current ...
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Why do we put factors of zero in a Lagrangian that is to be extremized?
According to the Wikipedia page on Lagrange multipliers under the section - Example 3: Entropy, it is written that:
$$f(p_1,p_2,\ldots,p_n) = -\sum_{j=1}^n p_j\log_2 p_j$$
For this to be a ...
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Variational principle and the medium equation for photon paths in general relativity
I am reading a 2015 paper by Rogers on Frequency-dependent effects of gravitational lensing within plasma. He gives the relation between the components of the photon four-momentum and the refractive ...
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Variation of Action and Border Terms
I need to compute the following very general (piece of) variation:
\begin{equation}
\int d^4x \delta (\sqrt{-g} R ) f
\tag{1}
\end{equation}
where $R$ is Ricci scalar and $f$ a generic scalar ...
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Does there exist a square root of Euler-Lagrange equations of a field? (Factorization)
Does there exist a square root of Euler-Lagrange equations $\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0$ in the sense that $(x+...
user356874
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Feynman physics on least time of light
What does light checks all paths mean by Feynman? Especially the statement is labeled by yellow. Why there is only one path that leads radiowaves to D’? And how wave check all paths, that is, why it ...
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Simplicity constraints from $SO(4)$ Plebanski action
The $SO(4)$ Plebanski action yields a first order formulation of Euclidean General Relativity as a constrained (topological) BF-theory. It depends on a $so(4)$ connection 1-form $\omega^{IJ} = \omega_{...
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Geodesics: Energy functional vs length functional
The Wikipedia article about geodesics talks about the equivalence of obtaining the geodesic by either minimizing the length functional $L$, or by minimizing the energy functional $L^2/2$, cf. the Phys....
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From EH action to Newtonian Mechanics action? [closed]
How does one start from the Einstein Hilbert action go to the action of a (point particles + some field) in special relativity and then Newtonian mechanics for a local neighborhood around a point for ...
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Different relativistic actions [duplicate]
I am slightly confused about different action integrals in relativity. When you work through some introductions to general relativity, you usually get in contact with the relativistic action integral
\...
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Is calling it "The Principal of Extremal/Stationary Action" pedantry? [duplicate]
I understand that the equations appear to permit paths of maximal action, but is there any real physical case where this actually occurs? Would it not be more sensible to refer to this as the ...
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Gauge Symmetry of the Lagrangian
My teacher told the following statement to me during office hours. Is it correct and if so, how could one go about proving it?
Given a material system subject to holonomic and smooth constraints ...
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Why does a complete four-divergence term in a Lagrangian density not affect the equation of motion in special relativity? [duplicate]
The classical theory of fields by Landau and Lifshitz, page 68 says:
As for the quantity $\epsilon^{iklm}F_{ik}F_{lm}$ (§ 25), as pointed out in the footnote on p. 63, it is a complete four- ...
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Using the principle of inertia to motivate the principle of least action?
Can we motivate the principle of least action with the principle of inertia that causes a mass particle to resist changes in its momentum? After all, the principle of inertia is the starting point and ...
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Is action an extensive quantity - always?
Action, the integral over time of the kinetic minus the potential energy seems to be an extensive quantity. (There is nothing serious coming up in Google on this issue. Neither on Google Scholar.)
In ...
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Do solutions of the Schrödinger equation for multiple particles automatically obey spin-statistics?
Consider the Hamilitonian for a general two-electron system subject to an external potential $V_\mathrm{ext}$ and an interaction potential $V_\mathrm{ee}$. In this case
$$H\psi(x, y) = -\frac{1}{2} \...
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How to derive the free rigid body equations from Euler-Lagrange? [closed]
I'm trying to retrieve the equations of motion for a free rigid body:
$$
I(t)\dot{\omega}(t)+\omega(t)^T \times (I(t)w(t)) = 0
$$
where $$ I(t)=R(t)I_{0}R(t)^T $$
I know that Euler-Lagrange equations ...
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Thermodynamic interpretation of the cosmological constant?
So I'm going through this paper of emergent gravity where T. Padmanabhan provides a thermodynamic interpretation of Gravity. What I'm failing to understand is what exactly is the thermodynamic ...
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Understanding this Lagrangian calculation
I was trying to understand this section of a Wikipedia article:
$$0 = \delta \int \sqrt{2T} d\tau =
\int \frac{\delta T}{\sqrt{2T}} d\tau =
\frac{1}{c} \delta \int T d\tau$$
For the life of me, ...
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