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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Can ground states computed through the variational method not be eigenvectors of a Hamiltonian?
Suppose you have to apply the variational principle to compute the ground state of a system whose Hamiltionian is $H_s$ and your ansatz is a linear combination of ground states of a simpler system ...
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Variation of the square of the Weyl tensor
After reading a bit about Conformal gravity, I came across the Lagrangian of the form: $$L=C_{abcd}C^{abcd} \sqrt{-g}$$
Where C is the Weyl tensor, I am interested in finding the field equations that ...
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Spacetime dimension by variational principle
I'm asking it out of curiosity: if hypothetically speaking spacetime had the freedom to "choose" its dimension, and it can be described by an action, is it possible to do an analytic ...
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In principle, "how much" of the path integral is required to match the electron $g$-factor experiment?
As Dirac was the first to realize (Dirac 1933, page 69), the reason the quantum path integral converges to the classical action principle as $h\rightarrow 0$ is that
The only important part in the ...
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Geodesic: maximal aging versus extremal aging
From Exploring Black Holes, by Taylor and Wheeler, page 1-7:
Purists insist that we say not maximum reading but rather extremal reading: either maximum or minimum. This book contains only examples of ...
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What is the variational energy of two spinless bosons with given interaction potential?
There was a question on my exam quantum mechanics that I wasn't able to solve and I am curious how it is done, I cannot find any reference in the section of pertubation theory that describes systems ...
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Is there a deeper reason behind the Principle of Stationary Action? [duplicate]
I do not know if this is an answerable question but...
Is there a deeper proof/ reason behind the Principle of Stationary Action? As the only proof I have seen is showing that, using the Euler ...
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Deriving Euler-Lagrange equation [duplicate]
I have derive the Euler-Lagrange equation which is equation (2) for a condition in which generalised velocity is independent on the generalised coordinate but when generalised velocity is dependent on ...
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Hamilton's equations and Euler-Lagrange equation [closed]
This paper is about deriving hamilton's equations from Euler-Lagrange equation, what i don't understand is equation 19. In equation 18 the process involved is if we substitute lagrangian $L$ for ...
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In the context of field-theoretic constrained dynamics, do we have the freedom to choose the Lagrange multipliers to be time-independent?
Let us work in a box $(t,\overrightarrow{x}) \in [0,1] \times [0,1]^3$. For any function on this box, we impose some Dirichlet boundary condition on the temporal direction and periodic boundary ...
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Euler-Lagrange equation in curved spacetime
The action of a field $\phi^\mu$ in flat $n$-dimensional spacetime is
$$ S = \int \text{d}^n x \mathscr{L}(\phi^\mu(x),\partial_\alpha \phi^\mu(x)) $$
From an infinitesimal variation of field ...
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How to find the second quantized form of Hamiltonian for particle in a box?
A single particle Hamiltonian can be written in the second quantized form as follows:
$$F_{1}= \sum f_i(r_i,p_i)$$
$$F_1=\sum(l|f_i|l')a_{l}^{\dagger}a_l$$
When we use this for the hamiltonian of a ...
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Relativistic Euler-Lagrange equation
I am confused from the equation 6, why we get Euler-Lagrange equation from equation 8 but not from equation 6?
Why we need to use $\zeta$ as invariant parameter in equation 8 even we already have ...
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Energy minimization of complex periodic scalar field with modulus constraint
I'm looking for an elegant method to (in general numerically) minimize an energy functional $E(\psi)$ for a complex field $\psi(x,y,z)$ which I know will be periodic in $z$ direction (due to self ...
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Geodesics Equation from Lagrangian [duplicate]
In the book Introduction to General Relativity Blackholes and Cosmology by Yvonne Choquet-Bruhat, she defines the length of a causal curve as $$\ell\equiv \int_a^b \left( -g_{\alpha \beta} \frac{d \...
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Deriving Euler-Lagrange equation for vector field in curved spacetime
I'm trying to derive covariant Euler-Lagrange equations for a vector field. The variation of the action should be
\begin{gather*}
\delta S
=
\int
\text{d}^n{x} \sqrt{|g|}
\left(
\delta\phi^\mu \frac{\...
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Why is the action guranteed to have unique extrema in classical mechanics? [duplicate]
Reading any classical mechanics book which introduces the Lagrangian formalism of mechanics, a one particle system is introduced to show that we obtain the euler-lagrange equations from Newton's ...
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Why is the action guranteed to have one unique extrema in classical mechanics? [duplicate]
Reading any classical mechanics book which introduces the Lagrangian formalism of mechanics, a one particle system is introduced to show that we obtain the euler-lagrange equations from Newton's ...
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When does a variation of the metric correspond to a variation of the gravitational field only?
When I look at changes in the metric tensor (for instance, when I am deriving Einstein's field equations via Hamilton's principle), how do I know they describe actual changes in the gravitational ...
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Does nature perform optimal linear quadratic control?
Given any linear quadratic control problem
$$\min \sum_t c_x(x_t, t) + c_u(u_t, t)$$
where $u_t$ is a "control variable" (think of it as an adjustable velocity) and $x_t$ is a "state ...
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Infinitesimal geodesic motion directly from the metric?
How can I see---directly from the Schwarzschild metric---that initially stationary (w.r.t. Schwarzschild coordinates) inertial test clocks will begin to fall toward e.g. the Earth (i.e. far outside ...
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Path Integral of Photon
I am having issues recalling how to perform integration by parts for the path integral of the photon, namely the term, $$Z[J] = \int\mathcal{D}[A_\mu]\exp(i\int\mathcal{L}\:dx)$$ where $\mathcal{L} = -...
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Does a constant in the action always have unobservable consequences in classical mechanics?
Background
So in classical mechanics, my understanding is that for the action by using a the principle of least action one can get the equations of motion. Adding a constant to the action does not ...
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Questions regarding the derivation of Euler-Lagrange Equation from Taylor's Classical Mechanics
I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)\tag{...
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Hilbert's criticism on the first version of Einstein's field equations
Crossposted at HSM SE
I have read once that Hilbert had some reservations regarding the first form of the field equations
$$ R_{\mu\nu}= k T_{\mu\nu}$$
because it was not possible to retrieve them ...
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How fundamental Physics is constructed? [duplicate]
I was wondering, how do theoretical physicist arrive to such fundamental things like Lagrangians or actions.
For example, the QED, action is given by:
$$
\mathcal S_{QED} = \int_{\mathcal M} {\mathrm ...
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Steps taken to differentiate action in wave equation [closed]
I'm currently reading Blundell and Lancaster's "Quantum Field Theory for the Gifted Amateur."
In chapter 1, example 1.4, they talk about how the action and Lagrangian density ideas are super ...
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Path variations in deriving Euler-Lagrange equation
This is a basic question about Euler-Lagrange equations that I could not find being addressed anywhere else (some similar questions: Q1 and Q2).
In deriving the Euler-Lagrange equations from the ...
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Domain of definition of a Lagrangian in classical field theory
In classical field theory one has the action:
$$S[\phi] = \int_{t_{0}}^{t_{1}}\int_{\Omega}\mathcal{L}(t,x,\phi(t,x),\dot{\phi}(t,x),\nabla\phi(t,x))dxdt$$
and we want to obtain the Euler-Lagrange ...
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Can Lagrangians model all possible dynamics? [duplicate]
We use Lagrangians and variational calculus for almost all of physics, from Newtonian mechanics to QFT.
Is there any theorem in mathematics that guarantees that all possible dynamics of objects (say ...
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Are principle of virtual work and principle of minimum potential energy same? And how is it related to Calculus of variation?
I am studying Finite element method and Classical Mechanics. I have come across three important terms
Principle of virtual work (found in Classical Mechanics)
Principle of minimum potential energy (...
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Is the value of the action important?
I know that the action, in Classical Mechanics, is a functional of the path of a physical system, such that
"the path actually followed by a physical system is that for which the action is ...
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Solving 3D Kepler Problem substitution goes wrong
I'm trying to arrive at the effective potential equation in Kepler Problem using Routh reduction method. We can procede in two ways, either using polar coordinates in the plane where the orbit happens ...
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How would someone discover the Einstein-Hilbert Action?
Usually in textbooks or on online resources, when you are learning General Relativity, propose the following:
$$
\mathcal S[g_{\mu\nu}] =\frac{1}{2 \kappa}\int_{\mathcal M} {\mathrm d^4 x \; R\sqrt{-g}...
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Derivation of gravitational dynamics using Lagrangian?
The standard textbook approach in Newtonian gravitational dynamics is to derive the particle dynamics using the particle Lagrangian:
$$L = T-V = \frac 1 2 m \dot x_u \dot x_u -m\phi(x_u)\tag{1}$$
With ...
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Why do we discount higher-order variations when applying variational methods in analytical mechanics? [duplicate]
In No-Nonsense Classical Mechanics, the calculus of variations is introduced with what I'm sure is a standard example. We try to find the minima of function $f(x) = x^2$ by evaluating it at $x + \...
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Variation of Pontryagin density $*R^{abcd}R_{abcd}$ with inverse metric $g^{ab}$
I'am computing the variation of
$$\int *R^{abcd}R_{abcd} \sqrt{-g}\,d^4x$$
with $g^{ab}$ and find it is difficult. Is this a known result?
$$*R^{abcd}=\frac{1}{2\sqrt{-g}}\epsilon^{abij}R_{ij}{}^{cd}$$...
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How is proper time extremized?
I just completed an exercise that asked me to prove that, in special relativity, free particles move with uniform velocity on geodesics that are straight lines. After doing this problem, I was ...
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Boundary condition from extremization
I have an AdS Schwarzschild blackhole spacetime where the metric is given by,
$$ds^2 = \frac{1}{z^2} \left( -f(z) dt^2 + \frac{dz^2}{f(z)} + dx^2 \right) \tag{1}\label{1}$$
There is a plane embedded ...
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Scalar field equation of motion in FRW metric
Consider a scalar field $\phi$ with the following Lagrangian density:
$$\mathscr{L}=-\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi-V(\phi),$$
and consider a FRW metric, whose line element is ...
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Fierz-Pauli action as an effective action from Einstein-Hilbert acition?
The Fierz-Pauli action
$$
S=\frac{1}{16 \pi G} \int d^{4} x\left[-\frac{1}{4} (\partial_{\rho} h_{\mu \nu})( \partial^{\rho} h^{\mu \nu}) + \frac{1}{2} (\partial_{\rho} h_{\mu \nu}) (\partial^{\nu} h^{...
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How does nature know Hamilton's principle? [duplicate]
I have gone through some of the questions asked here re Hamilton's principle, but could not readily find an answer to the following:
Hamilton's principle states that paths particles follow extremizes ...
user228424
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What is the intuitive implication behind $L' = L + \frac{df}{dt}$ not affecting equations of motion?
I am referring to another post on the same question as this post: Lagrangian $L' = L + \frac{df}{dt}$ gives the same equations of motion
In the first paragraph, the poster says:
"It is well ...
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Is there more than one GR action (if we include boundary terms)?
The action of GR is proportional to the Einstein-Hilbert action
$$S_1=\int \sqrt{-g}R dx^4.$$ Now, $R$, contains terms of the form $\partial^2 g$. Using integration by parts, one can write this ...
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Tight-Binding method and orthogonality of Bloch functions
Tight binding summary
When computing the electronic bands of a crystal in the tight-Binding approximation, the standard way to do it is to construct Bloch solution as
$$
\Psi_n(\textbf{r}, \textbf{k}) ...
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Variational derivative of a Lagrangian
I would like to know how exactly to calculate the variational derivative of:
$$L = \oint dx \: \frac{m}{a} (\dot{u}(x,t))^2 - ca(u'(x,t)^2\tag{1}$$
with respect to $u$, ($\delta L / \delta u$).
The ...
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Is there a Lagrangian $L$ (equivalently an action functional $S$) which yields the Navier-Stokes equation?
The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws.
However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\...
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Magnitude of the variations $\delta q_i$ in the principle of stationary action
To determine the equation of motion using the principle of stationary action, one has to consider the variation of the action due to variations $\delta q_i$ in all the generalized coordinates $q_i$. ...
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Proof of principle of stationary action when the Lagrangian is not $L=T-V$
The principle of stationary action claims that the action $S$ takes a stationary value in a real system, where:
$$S = \int_{t_1}^{t_2} L dt\tag{1}$$
and $L$ is the Lagrangian of the system. It can be ...
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Why does the trajectory of a relativistic particle "minimises its Minkowski distance"?
The action of a relativistic free particle is
$$\mathcal{S}=\int^{t_{1}}_{t_{0}} L dt\tag{1},$$
for
$$L=-\frac{mc^{2}}{\gamma}.\tag{2}$$
I understand that a particle will follow the trajectory of ...
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