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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 ...

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Derivation of EL equations for real scalar field

I am looking to derive Eq. (1.11) from these notes on QFT from Tong’s notes: http://www.damtp.cam.ac.uk/user/tong/qft/one.pdf But Im the second equation, is there not suppose to be a factor of 1/2 ...
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Feynman Lecture Principle of Least Action: Glossed over Taylor expansion?

His initial one dimensional derivation of Newton's Second Law using the Principle of Least Action, I believe is fairly concise and easy to read. However, I did get hung up on his use of the Taylor ...
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Periodic motion(s) on a torus

I recently came across the Lyusternik-Fet theorem concerning closed geodesics on a compact manifold. For simplicity of description, take the 2-torus, and imagine it represents the configuration space ...
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Definition of integral functional [duplicate]

I'm reading the section of Marion and Thornton devoted to basics on the Calculus of Variations, and came across this definition for the functional: $$J = \int f(y(x), y'(x);x) dx$$ implying that $f$ ...
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Is it possible for the Action $S$ to *not* have a stationary point?

So the path of an object in configuration space is given by Hamilton's principle, which states that the path which the particle travels on is the one on which the action is stationary: $$\delta S = \...
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Equation for surface of a rotating liquid [duplicate]

The surface of rotating coffee in a mug after it has been stirred takes on a characteristic shape, depending on angular velocity, viscosity of the liquid, etc and I've been interested in knowing what ...
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Proof of treating $\psi^*$ as a field independent from $\psi$?

Given $L=\partial_\mu\psi^*\partial^\mu\psi -m^2\psi^* \psi$. The field equation is obtained by applying Euler–Lagrange equation on $L$ by treating $\psi^*$ independent of $\psi$. I am trying to ...
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Equation of motion for a massless scalar in 2 dimensions

I'm working through Polchinski's String Theory (Volume 1, Chapter 2, page 34), with $D$ scalar fields: The action is given by $$ S = \frac{1}{2 \pi \alpha'}\int d^{2}z \partial X^{\mu} \overline{\...
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Varying the Einstein-Hilbert action without reference to a chart

In most treatments of General Relativity, when the the Einstein-Hilbert action over some manifold $\mathcal{M}$ (plus Gibbons-Hawking-York term if $\mathcal{M}$ has a boundary), given by $$S=\frac{1}{...
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How to deal with explicit time dependence of the Lagrangian?

Clearly, if the Lagrangian in explicitly time dependent, the Euler-Lagrange equations being satisfied does not extremise the action. I am unclear as to how to deal with systems with an explicitly time-...
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Why can we choose affine parameterization?

In general relativity when deriving the geodesic equation $$\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0\tag{1}$$ from the action $$S = \int d\tau \sqrt{|g_{\mu\nu} \dot{x}...
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Equations of motion for a Weyl spinor in the context of SUSY

I'm learning supergravity from the textbook of Antoine Van Proeyen (this is from page 114). Suppose I'm given a Lagrangian $$ \mathcal{L} = - \partial^{\mu} \bar{Z} \partial_{\mu} Z - \bar{\chi} \...
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Polyakov Lagrangian and Lagrange multipliers

I'm reading Polchinski's Introduction to String Theory (volume I) and something got me quite puzzled in the beginning (At the top of page 19 to be precise). This part is about the open string and the ...
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Solve equations of motion to find $\Phi(t)$ for a pendulum with an infinite period

Consider a pendulum of mass $m$ and length $l$ that can rotate in the vertical plane subject to the gravitational field $g$. Write down the Lagrangian and solve the resulting equation of ...
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Variation/Differential in Critical Radius of Homogeneous Nucleation

Getting the critical radius during nucleation from G(r) is straightforward - but in our lecture notes, a notation is used that I cannot quite wrap my head around: The molar Gibbs Free Energy $G(r)$ ...
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Variation with respect to a traceless symmetric tensor

Suppose we have an action variation like $$\delta S[G]=\int \mathfrak{H}^{\mu\nu}\delta G_{\mu\nu} \,\, d^Nx,$$ where $\mathfrak{H}^{\mu\nu}$ is a tensor density. If the variation with respect to $...
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Maxwell equations of motion from $S = \frac{-1}{2} \int F \wedge \ast F$

I'm trying to understand the following equation, used in the derivation of the equations of motion. Let $S = \frac{-1}{2} \int F \wedge \ast F$ and $F = dA$. Let $\delta$ denote variation. Then $$...
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Calculus of Variations. Finding the extremals of a perturbed Lagrangian [closed]

Im trying to solve the following problem: Approximate with an error of $O(\epsilon ^3)$ the extremals of the Lagrangian $$L(y,y',x) = y^2 + (y')^2 - 2y \sin(x) + \epsilon y^3$$ with $y(0)=1$ ...
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Schwinger's variation of the action of point particle with *both* time and position as independent variables

In Chapter 8, pages 86-87, equations (8.5)-(8.11) of Julian Schwinger et al., Classical Electrodynamics, the equations of motion for the following action principle of a point particle in an external ...
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Trouble understanding why fields are unnaffected by translations

The following sentence appears in my classical field theory notes Fields are Lorentz tensors and spinors, and as such unaffected by translations: $ \dfrac{\delta \phi}{\delta a^{\alpha}}=0.$ Where ...
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Is action for free particle really minimal?

On my mechanics classes I have a problem: show, that the action for free non-relativistic particle $$S=\int\limits_{t_i}^{t_f}\frac{m\dot{x}^2}{2}dt\tag{1}$$ is really the least (but not maximal). ...
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When and why is $\frac{d}{dt}\delta q^{i}=\delta \frac{dq^{i}}{dt}$ true? [duplicate]

Apparently my question is different from Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}δq=δ\frac{dq}{dt}$. I hadn't noticed because the answer given in the comments to this question was ...
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Functional Poincaré's lemma and the inverse Lagrangian problem

I have only encountered the inverse Lagrangian problem in mathematics books that treat Lagrangian field theory using jet bundles and homological algebra, and while I am studying this approach, I still ...
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Relativistic EM Lagrangian and the derivation of equations of motion

As mentioned in my other post, I am attempting to learn from Gross'"Relativistic quantum mechanics and field theory", and I have a question concerning the manipulation of the antisymmetric 4x4 tensors ...
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Explicit form for $\frac{\delta u^\alpha}{\delta g^{\mu\nu}}$?

The title says it all: Is there a closed form expression for the following variational derivative, $$ \frac{\delta u^\alpha}{\delta g^{\mu\nu}}, $$ where $u^\alpha$ is the four-velocity of a massive ...
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For the Lagrangian stationary action formula does the eta function for a specific path vary the distance from the true path? [closed]

This question can apply to any variation calculus problem although it has come up in my case for the stationary action principle so I will stick to the application I am using it for. The action is ...
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1answer
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Schrodinger matter-field equation? [closed]

Given the Lagrangian Density: $$L=i \hbar \dot{\psi} \psi^* -\dfrac{\hbar^2}{2m} \nabla\psi^* \nabla\psi-V\psi\psi^*$$ and the Lagrange equation: $$\dfrac{\partial L}{\partial \psi}-\nabla\cdot \dfrac{...
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What's the variation of a product of two metrics? [closed]

I was trying variate an action in General Relativity, and I come to the next calculus: $\delta(g^{\alpha\beta}g^{\mu\nu})$ And I did: $\delta(g^{\alpha\beta}g^{\mu\nu})=g^{\alpha\beta}\delta g^{\mu\...
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Schrödinger's variational method

In Schrödinger's Quantisation as an Eigenvalue Problem he solves the Hydrogen atom through a precursor of Schrödinger's Equation, derived from the Hamilton-Jacobi equation through a variational method ...
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Gravity train and fastest path

As a fun exercise to kill some time, I have been thinking in the gravity train. For a point mass falling through a tunnel dug from pole to pole in an spherically symmetric planet of radius $R$, the ...
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Equations of motion from Polyakov action, before choosing the conformal gauge

My question is the following: It is usual in the standard textbooks to firstly choose a gauge (usually the conformal gauge) and then extract the equations of motion from the Polyakov action by ...
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In what sense is the stress-energy tensor the derivative with respect to the metric?

In Di Francesco et al (the big yellow book), section 2.5.2, it is suggested that the (symmetrized) stress energy tensor can be interpreted as the functional derivative of the action with respect to he ...
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Functional derivative

I am not able to derive Eq. 21 of this paper F. Zahariev, S. S. Leang, and Mark S. Gordon, "Functional derivatives of meta-generalized gradient approximation (meta-GGA) type exchange-correlation ...
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Derivation of Lorentz law in Curved Spacetime?

In the presence of external forces or rather presence of fields to which a particle "couples" in curved spacetime. Ex: $$S[\gamma;A]=\int d\lambda m\sqrt{g_{\gamma(\lambda)}(v_{\gamma,\gamma(\lambda)}...
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Palatini action: variation of spin connection: show that torsion vanishes

Consider the tetrad-Palatini action: $$S[e,\omega] = \int e \wedge e \wedge F[\omega]^\star,$$ where $\star$ denotes the Hodge dual, i.e. $F_{IJ}^\star = \frac{1}{2} \varepsilon_{IJKL} F^{KL}$. The ...
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Why is the shape of a hanging chain not a “V”?

From Wikipedia: To answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy. ...
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How to take into account the symmetry of the metric tensor when doing the Functional derivative in GR? [duplicate]

I have a Straightforward question. When the functional derivative of the Ricci scalar to get the GR field equations. As the derivative is done using the metric which is symmetric do I have to ...
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General relativity - scalar gravitational field, variation principle

I have a basic question about the variation principal when applied to a scalar gravitational field in general relativity. Consider the action $$S_M = \int d^4 x\sqrt{|g|}g^{uv}\partial_u \phi\...
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Does the integral in the action formula regarding the principle of stationary action represent an area or a length?

I am referring to the Feynman Lectures. The second volume has the "Principle of Least Action" as one of his lectures. (See after the 2nd paragraph below figure 19-6.) Although he does not explicitly ...
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1answer
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Why isn't it important, after which coordinates the Variation of the action integral is done?

I often read,that if the lagrangian $L=p\dot{q}-H$ of a pair of coordinates in phase space $(q,p)$ and $P\dot{Q}- K $, for some new pair of coordinates $(Q,P)$ only differ by a total time derivative $...
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Variational calculus and KKR method for band structure calculation

I’ve been studying the KKR method from the original Kohn and Kostoker’s paper (https://journals.aps.org/pr/abstract/10.1103/PhysRev.94.1111). On the text, they use variational calculus for dealing ...
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1answer
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Notation question in calculus of variations — QFT

these two integrals below are equal, but I am not understanding where the $x'$ variable comes from. \begin{align} I_0&=e^{ i\int d^4x \left\{ \frac{1}{2}\left[ \left( \partial\varphi(x) \right)^...
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Equation of motion from $D=3$ Lorentz Chern-Simons action

In three dimensions, the well known Lorentz Chern-Simons action is $$ S_{\text{CS}}=\int\text{d}^3x\varepsilon^{\mu\nu\rho}\bigg(\omega_{\mu}{}^{ab}R_{\nu\rho ab}+\frac{2}{3}\omega_{\mu a}{}^{b}\...
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Reduction of the mechanical equilibrium condition given by Gibbs

In the "The scientific papers of J.Willard Gibbs: Volume One Thermodynamics" p.276-281, the following mechanical equilibrium condition is given for a heterogeneous liquid system with gravity ...
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Derivative of the electromagnetic tensor invariant $F_{\mu\nu}F^{\mu\nu}$

The electromagnetic field tensor is $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$. I am trying to calculate the quantity $$ \frac{\partial(F_{\alpha\beta}F^{\alpha\beta})}{\partial(\partial_{\...
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How to find the Lagrangian of this system?

I am trying to find the Lagrangian $L$ of a system I am studying. The equations of motion is: $$\left\{ \begin{array}{c l} r \ddot{\phi} + 2\dot{r} \dot{\phi}+k(r) \cdot r \dot{r} \dot{\phi} = ...
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Lagrangian gauge invariance $L'=L+\frac{df(q,t)}{dt}$

So, I have to prove directly (e.g. by substitution) that if a path satisfies the Euler-Lagrange equations for the Lagrangian $L$ it does so for $$L'=L+\frac{df(q,t)}{dt}.$$ Let me tell you what I have ...
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Is there a formal proof that the equilibrium figure of fluid under its own gravity is a sphere? [duplicate]

Every textbook that presents the hydrostatic equilibrium condition of stars starts by assuming a spherical distribution of matter. This all makes sense intuitively, yet I haven't been able to find a ...
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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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Energy-momentum tensor from Matter Field Action

I'm currently in the process of familiarising myself with some basic concepts of general relativity and have stumbled upon a problem that is probanly quite simple. I'm referring to the book by Hobson, ...