Questions tagged [variational-calculus]
Variational calculus is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find extrema of functionals: mappings from a set of functions to the real numbers. The archetype application in physics is Lagrangian mechanics, seeking extrema of action functionals.
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Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
In Lagrangian mechanics we have the Euler-Lagrange equations, which are defined as
$$\frac{d}{dt}\Bigg(\frac{\partial L}{\partial \dot{q}_j}\Bigg) - \frac{\partial L}{\partial q_j} = 0,\quad j = 1, \...
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How does one justify Schwartz's answer to his problem 3.7 in "Quantum Field Theory and the Standard Model"?
In problem 3.7 of his textbook "Quantum Field Theory and the Standard Model", Schwartz gives a simplified Lagrangian density
$$
\mathcal{L}=- \frac{1}{2} h \Box h + \epsilon^a h^2 \Box h -\...
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Where exactly does the integral definition of the gradient come from? [migrated]
In the book "Essential mathematical methods for physicists" from Weber and Arfken, they define the integral form of the gradient,divergence and curl, althougth they give sections before an ...
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How to evaluate the action of a fractional differential momentum operator?
I need to understand how a fractional operator works if before being applied on a test function it acts on another (well known)function:
$$(-i\hbar\frac{\partial}{\partial x}\cdot\delta(x))^\epsilon\...
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Classical Mechanics proof Lagrangian constraint forces
I've got a simple mathematical question. I was studying the Lagrangian approach of classical mechanics and in this part I had the intention of proving that the differential of the Lagrangian is equal ...
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Confusion in derivation of Euler-Lagrange equations
Here's a screenshot of derivation of Euler-Lagrange from feynman lecture https://www.feynmanlectures.caltech.edu/II_19.html
My doubt is in the last paragraph. I get that $\eta = 0$ at both ends, but ...
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Derivation of the Equation for Constraint Forces
How do we derive the relationship $$F_\text{constraint}=\displaystyle\sum_i \lambda_i\nabla g_i$$ where $g$ is the constraint function, from the following relationship? $$\frac d {d\epsilon}S|_{\...
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Help deriving Maxwell's equations from the Lagrangian [duplicate]
Starting with the Lagrangian density $$\mathcal{L} = -\frac{1}{2}(\partial_\mu \mathcal{A}_\nu)(\partial^\mu \mathcal{A}^\nu)+\frac{1}{2}(\partial_\mu \mathcal{A}^\mu)^2,$$ I don't understand how to ...
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If the solution of a field vanishes on-shell does it mean anything particular?
Let us consider an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. The solution of the field $c$ is given by the expression $f(a,b)$. On taking into account the relations ...
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Feynman's Derivation for Principle of Least Action
In the Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. II, Chap. 19, Feynman demonstrates how the principle of stationary action for one particle implies Newton's second law (or ...
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Variation of action for Jackiw-Teitelboim (JT) gravity in order to get equations of motion
Im having a bit of trouble understanding the variation of action for JT gravity (for example in this article https://arxiv.org/abs/1606.01857 the equation 3.8).
I dont really get how do they get from ...
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How did Landau & Lifshitz (Mechanics) get Equation 2.5?
I understood everything in Landau & Lifshitz's mechanics book until Equation 2.4,but I'm not sure what he means when he says "effecting the variation" and gets Equation 2.5.
Can anyone ...
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Lagrange momentum for position change
After the tremendous help from @hft on my previous question, after thinking, new question popped up.
I want to compare how things behave when we do: $\frac{\partial S}{\partial t_2}$ and $\frac{\...
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For any unitary $U = e^{iA}$, question about the matrix element $A_{pq}$ of matrix $A$ and its functional derivative [closed]
Any unitary operation $U \in U(N)$ can be represented as $U = e^{iA}$, where $A$ is an arbitrary $N \times N$ Hermitian matrix. In the case of a time-independent Hamiltonian $H$, we have $A = -itH$ (...
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About the form of the cost functional for quantum optimal control theory
In quantum optimal control theory, the cost functional is often defined as (e.g, see Eq.(9) in here, as well as many other solid references such as this):
$$J = \langle \psi(T) \rvert O \lvert \psi(T) ...
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Euler-Lagrange equation for a geodesic on an elliptic paraboloid
I'm trying to set up a functional that outputs the length of a curve on the paraboloid surface. I then want to write the Euler-Lagrange equation for the geodesic.
The surface is given in the ...
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Why partial derivative in Lagrange is partial?
When trying to arrive to Euler-Lagrange equation, Susskind does terrific job but I have one problem. Let's consider motion in only $x$ direction with respect to time.
$$x(t) = \hat x(t) + \epsilon f(t)...
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Variation of 5-dimensional Lagrangian-density for 5D Einstein-Gauss-Bonnet gravity
How can I derive equation (2a) and (2b) by variation of the fundamental fundamental five-dimensional Lagrangian density (1) with respect to the connection and the vielbein as mentioned in arXiv:2109....
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Functional derivative of Ricci tensor respect to metric [closed]
I am working on following functional derivative
$$
\dfrac{\delta R_{\mu\nu}}{\delta g_{\alpha\beta}}=C^{\alpha\beta}_{\mu\nu}
$$
Intuitively, it should be nonzero but I can not work it out.
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What is considered an "acceptable" field for the variation of the action?
What fields am I allowed to use as variables of variation for extremizing the action?
For example, using the free EM action:
\begin{equation}
S_{EM}=-\frac{1}{4\mu_{o}}\int d^4x F_{\mu\nu}F^{\mu\nu}
\...
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Variation of Ricci tensor vs Riemann tensor in Gauss-Bonnet density
I don't know which steps to follow when perform an infinitesimal variation to the Gauss-Bonnet density with respect to $g^{\mu\nu}$. The first path is to perform variations with respect the Ricci ...
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On generalised potential in Electrodynamics
I'm studying Lagrangian Mechanics from Goldstein's Classical Mechanics. My question concerns Section 1.5 which talks about velocity-dependent potentials.
I am actually unsure about how Equation 1-64' ...
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Why is the Euler-Lagrange equation a necessary condition? (without counter example)
I'm currently studying Arfken's Mathematical Methods for Physicists, and I'm specifically focusing on the calculus of variations. While studying, I came across the derivation of the Euler-Lagrange ...
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QFT: Lagrangian of a field [duplicate]
I just started going through this http://www.damtp.cam.ac.uk/user/tong/qft.html of QFT and I got stuck at the very beginning.
The Lagrangian for the field is derived by:
I think that the second step ...
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A step in the derivation of the Euler-Lagrange equations using Hamilton's Principle
I am going through the derivation of the Euler-Lagrange equations from Hamilton's principle following Landau and Lifshitz Volume 1. We start by writing the variation in the action as,
$$\delta S = \...
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What does it mean when the EOM of a field is trivially satisfied if other EOMs are satisfied?
If a Lagrangian has the fields $a$, $b$ and $c$ whose equations of motion (EOM) are denoted by $E_a=0, E_b=0$ and $E_c=0$ respectively, then if
\begin{align}
E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c\tag{1}
...
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How to show that covariant derivative of null vectors under variation of the null vectors themselves is preserved?
Having two null vectors with $$n^{a} l_{a}=-1, \\ g_{ab}=-(l_{a}n_{b}+n_{a}l_{b}),\\ n^{a}\nabla_{a}n^{b}=0$$ gives $$\nabla_{a}n_{b}=\kappa n_{a}n_{b}\\\nabla_{a}n^{a}=0\\\nabla_{a}l_{b}=-\kappa n_{a}...
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Varying an action with respect to the metric in mathematica
I want to Varying an action with respect to the metric to obtain the Einstein equation $G_{\mu\nu}=k^2 T_{\mu\nu}$. Is there any way to do that in Mathematica? Can mathematica do that?
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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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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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Dimensions of functional integration [duplicate]
I'm confused by something super simple. When taking functional variation (e.g. of the action) in the context of field theory, I often see
$$ {\delta \phi(x) \over \delta \phi(y)} = \delta(x-y) \ .$$
...
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Question about integral containing derivative of Dirac delta distribution
The result of the integral of the dirac delta δ(x-a) times a function f should be f(a) right? Then why isn't the integral just the final result directly without doing all of this, where did the ...
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What is the definition of a Brachistochrone curve in a non-Euclidean space?
I have a problem where I have to study "the geometric properties of the Brachistochrone curve in non-Euclidean spaces". But I am confused about the definition of the Brachistochrone Problem/...
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$SU(N)$ Theory equations of motion
I want to derive the equations of motion from the following Lagrangian:
\begin{align}
L = -(Tr(\partial_\mu \phi \partial^\mu \phi)-mTr(\phi^2)
\end{align}
Where $\phi = \phi^a T_a$ and $T_a$ are 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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Equation of motion from a Klein-Gordon action in a curved spacetime
For my physics research course I am supposed to get the equation of motion from a given action $S$:
$$
S = \frac{-1}{2} \int d^4x \sqrt{-g} (g^{\mu\nu} \frac{\partial \phi}{\partial x^\mu} \frac{\...
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How to derive the gauge transformation of a Lagrangian with auxiliary fields?
Suppose Lagrangian $L_1(y_1,y_2)$ is a functional of fields $y_1$ and $y_2$, and Lagrangian $L_2(y_1,y_2,z_1,z_2)$ is a functional of the fields $y_1,y_2$ and the auxiliary fields $z_1$ and $z_2$. If ...
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Simple Lagrangian with free bound and constraint
Let $\alpha,\beta$ non-zero real numbers, $f$ a function of time. I define $L_1=\alpha f + \beta$ and $L_2=p(t) L_1$.
I want to minimize $\int_0^T L_2$ under the constraint $\int_0^T L_1=v$, with $T$ ...
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Finding the equation of motion for vector potential $A_{\mu}$ in topologically massive electrodynamics
Essentially I want to vary the action
$$
S_M = \int d^3x \sqrt{-g} \left[- \frac{1}{4} F^{\mu \nu} F_{\mu \nu} - \frac{\alpha}{2} \epsilon^{\mu \nu \rho} A_\mu F_{\nu \rho} \right]
$$
with respect to $...
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If two different gauge transformations of an action commute, does it imply anything?
If two different gauge transformations of a Lagrangian commute with each other, does it imply anything?
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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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Variation vs. derivative wrt a symmetric and traceless tensor
Consider a Lagrangian, $L$, which is a function of, as well as other fields $\psi_i$, a traceless and symmetric tensor denoted by $f^{uv}$, so that $L=L(f^{uv})$, the associated action is $\int L(f^{...
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Relationship of variation of the metric and its inverse
I'm reading Tong's notes on GR http://www.damtp.cam.ac.uk/user/tong/gr.html and i cannot understrand how he derived the equation that relates the varation of the metric with its inverse in page 141 ...
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Calculations with co- and contravariant formalism in QFT
i have another question regarding calculations with the co- and contravariant formalism in QFT. It is not that i don't understand all of this, but most of the time i'm missing some "middle" ...
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0
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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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0
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Semi-classical limit of Feynman path integral
I am reading Blau's note on The Path Integral Approach to Quantum Mechanics. I am troubled for the derivations of semi-classical limit of Feynman path integral, which is located on Page.50 of Blau's ...
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1
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How is the source term chosen when using path integrals?
Suppose I would like to compute (time ordered) vacuum expectation values for a quantum field theory by using the path integral approach. Using the Lagrangian for the theory, we define a generating ...
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