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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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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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
$$
H = \sum_{\langle ij \rangle} J_{ij}
\mathbf{S}_i \cdot \...
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Deriving the Lagrangian for an Arbitrary Equation [closed]
Suppose one knows that for an arbitrary tensor $U_{\mu\nu}$, the equation $\partial_{\nu}U_{\mu\nu}=0$ is satisfied. Suppose that one seeks the Lagrangian associated with this equation.
One try could ...
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Clarifications regarding the Maupertuis/Jacobi principle
I'm slightly confused regarding the Maupertuis' Principle. I have read the Wikipedia page but the confusion is even in that derivation. So, say we have a Lagrangian described by $\textbf{q}=(q^1,...q^...
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Energy in dynamical variational principle
In quantum mechanics we use variational principle in order to find approximate expression for the ground state. Lets assume our probe wavefunction $|\Psi\rangle$ can be expanded in orthonormal basis
$...
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van der Waals interaction using variational method [closed]
2 hydrogen atoms separated by the distance R, find the total energy of the system using variational method
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Does $δS = 0$ mean that "the small changes in the actions equal to zero"?
Please correct me if I'm wrong.
What I understood from the Principle of Stationary Action is that for an object moving from point A to point B, at every point of the path with the least action, the ...
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Derivation of Lagrange's equations for non-conservative systems using Hamilton's Principle [duplicate]
Consider $\vec{F}$, as the total force applied on the system, $U$ the potential energy of the existent field, and $\vec{Q}$ a non-conservative force.
We have that:
$$F_k=-\frac{\partial U}{\partial ...
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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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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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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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Question about the apparent loophole in principle of least action: boundary condition vs initial condition
In Lagrangian formalism, given two points $(x_1,t_1)$ and $(x_2,t_2)$, we ask the question which paths $x(t)$ make the action $S=\displaystyle \int_{t_1}^{t_2}L\ \mathrm dt$ stationary and satisfy the ...
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Example in motivation for Lagrangian formalism
I started reading Quantum Field Theory for the Gifted Amateur by Lancaster & Blundell, and I have a conceptual question regarding their motivation of the Lagrangian formalism. They start by ...
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Is there some connection between the Virial theorem and a least action principle?
Both involve some 'averaging' over energies (kinetic and potential) and make some prediction about their mean values. As far as the least action principles, one could think of them as saying that the ...
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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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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 ...
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Why are there only derivatives to the first order in the Lagrangian?
Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded?
Is there a good reason for ...
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Confusion regarding the principle of least action in Landau & Lifshitz "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 ...
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Action max, min, or saddle?
It is well known that $\delta S = 0$ lays the foundation for variational mechanics. But I am confused as to whether or not this S is a minimum, a maximum, or a saddle point. Some books address this ...
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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 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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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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Equation of motion of an auxiliary field
I'm a newbie in the field of QFT and SUSY, so I'm warning you: this might be a stupid question.
I'm working with auxiliary fields to describe supersymmetric models and I understand that upon ...
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Point of Lagrange multipliers
I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
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How do non-conservative forces affect Lagrange equations?
If we have a system and we know all the degrees of freedom, we can find the Lagrangian of the dynamical system. What happens if we apply some non-conservative forces in the system? I mean how to deal ...
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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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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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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 ...
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Why can we consider the endpoint fixed in the derivation of the Euler-Lagrange equation in mechanics?
In mechanics, we obtain the equations of motion (Euler-Lagrange equations) via Hamilton's principle by considering stationary points of the action
$$ S = \int_{t_i}^{t_f} L ~ dt $$
where we have $L=T-...
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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 Euler-...
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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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The einbein in the action of a relativistic massive point particles [closed]
The action of a relativistic massive point particle moving in space-time is
\begin{equation}
S=-m\int d\tau \sqrt{g _{\nu \rho}\frac{dx^{\nu}}{d\tau}\frac{dx^{\rho}}{d\tau}}\tag{1}
\label{eq1}
\end{...
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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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Virtual Work - Is the presentation in Cornelius Lanczos wrong?
Book: The Variational Principles of Mechanics by Cornelius Lanczos
Edition: 4th
Chapter: 3, The Principle of Virtual Work
I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1).
...
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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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Does light always take the shortest path?
Does light always take the shortest path?
And is it possible to change the probability of a photon travelling to a point by only disturbing the paths that are far away from the shortest path?
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