For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.

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Path Integral Quantization in General Relativity

In Ref. 1 I have seen that the action must contain only the first derivative of the metric as required by the path integral approach. I don't understand why. I mean why the path integral approach of ...
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0answers
29 views

General approach to Mechanics? [duplicate]

So, I know that this question may be tough to answer, but I am asking this question in all seriousness, and I don't consider myself a newbie... Lately, I am trying to find a way to "generalize" my ...
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1answer
60 views

Usage of total time derivative within first Euler-Lagrange equation

As one key argument in Introductoryt QM Class, we've been taught to use a Lagrangian and Hamiltonian generalized description of a dynamic systems, which follows the Euler-Lagrange or Hamilton equation ...
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3answers
92 views

Gauge invariance in classical electrodynamics

I think that I don't fully understand concept of gauge invariance. Suppose we have a Lagrangian for classical ED which is: $$\mathcal{L} = -\frac{1}{4} (F_{\mu \nu})^2 - j^{\mu}A_{\mu}.$$ First part ...
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1answer
82 views

Why can you make $V$ stationary with respect to a parameter of the field in Derrick's theorem?

I'm going over Coleman's derivation of Derrick's theorem for real scalar fields in the chapter Classical lumps and their quantum descendants from Aspects of Symmetry (page 194). Theorem: Let ...
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0answers
41 views

Good books for understanding Lagrangian formulation of classical fields?

I want to understand Lagrangian formulation for classical fields and apply it to understand constrained dynamics. Currently I am referring to "A modern approach: Classical Mechanics" by ECG Sudarshan, ...
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37 views

Gravity as a gauge theory - Cartan-Killing form?

First, let me state the form of Lagrangian for YM and GR \begin{align} L_{YM} = \alpha \textrm{tr}(F^2), \qquad L_{GR} = \beta R \end{align} I heard, YM is a gauge theory but GR isn't a really gauge ...
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1answer
73 views

A question regarding $f(R)$ Lagrangians

Consider the class of Lagrangian known as $f(R)$ Lagrangians where the Lagrangian is some function $f(R)$, \begin{equation} S=\int\sqrt{g}d^4x\ f(R) \end{equation} assuming there are no (or ...
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1answer
34 views

Calculating motion of equation in tensor form

for the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2$$ how can I get this $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$ ...
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1answer
77 views

Non-holonomic constraints in Dirac-Bergmann theory

The Dirac-Bergmann algorithm effectively isolates the physical degrees of freedom of a system, by changing from Poisson brackets $\{\cdot,\cdot\}_\mathrm{PB}$ to Dirac brackets ...
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0answers
46 views

QFT: prove Dirac lagrangian is invariant under C, P, T separately

As it is stated in Peskin, $\mathcal{L}=\bar\Psi(i\gamma_{\mu}\partial^{\mu}-m)\Psi$ is invariant under C,P and T transformation separately. I have some problems to see how the partial derivative is ...
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1answer
48 views

Principle of Stationary Action and Euler-Lagrange Equation

Principle of Stationary Action: Given a mechanical system, there exists an action $S$ such that it is extremitized, or $\delta S=0$, for the actual motion of the system. $$S = ...
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44 views

Confused about something in Landau classical mechanics

On page 7 in Landau and Lifshitz Mechanics He writes: We have $L'=L(v'^2)=L(v^2+2ve+e^2)$ now the confusing part comes (for me): He writes: Expanding this expression in powers of e and neglecting ...
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0answers
55 views

Lagrangian Mechanics + Account for friction of block

$\newcommand{\dd}{\mathrm{d}}$ I am trying to work out the Lagrangian mechanics for a pendulum problem in order to animate it. I'm working on one of the examples in the Wikipedia page on Lagrangian ...
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0answers
68 views

Are there any conditions under which the Christoffel symbols can be treated as a damping term in a harmonic oscillator?

(Mathjax did not seem to be working as I composed this question. Hopefully it will kick into action once I post.) Note I am a novice at tensor notation. I am working with the following Lagrangian ...
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0answers
19 views

Max & inflection point in the principle of least action [duplicate]

Short question: What is the physics interpretation of max & inflection points in the principle of least action? Long question: If $$L(q_1,q_2;t)=K-V$$ then let $$S = \int^{t_1}_{t_2} ...
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0answers
29 views

kinetic energy in analytical mechanics

I have a question about the derivation of kinetic energy in analytical mechanics. What is the physical diffrence bettwen the 3 components (T0 T1 and T2), and when should we use those components ...
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1answer
99 views

Polyakov From Nambu-Goto Directly, for Strings?

The following derivation, for a classical relativistic point particle, of the 'Polyakov' form of the action from the 'Nambu-Goto' form of the action, without any tricks - no equations of motion or ...
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2answers
133 views

Does the conservation of the Wronskian follow from Noether's principle?

Noether's principle is the paradigm that symmetries of Hamiltonian and Lagrangian systems correspond to conservation laws of various kinds. Consider a one-dimensional harmonic oscillator $$\tag{*} ...
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1answer
90 views

Use partial or covariant derivatives when deriving equations of a field theory?

I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu ...
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2answers
120 views

Finding geodesics: Lagrangian vs Hamiltonian

I have a question referring to how to compute geodesics of a given spacetime (say, Kerr). I know that the direct way is via the geodesic equation ...
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3answers
74 views

How simultaneous information of coordinates and velocities sufficient to completely determine the subsequent motion of a mechanical system?

I somehow could not find the answers to the question in Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system? to ...
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4answers
597 views

Why is the Lagrangian approach preferred over the Hamiltonian approach in QFT? [duplicate]

Going from non-relativistic quantum mechanics(QM) to QFT there is a marked change in the approach used. QM almost exclusively uses Hamiltonains. Lagrangian based methods like the path-integrals are ...
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0answers
44 views

Friction in Lagrangian Method [duplicate]

A uniform, flexible chain of length $l$, mass $m$, hangs off a frictionless table-top of height greater than $l$. The length of the part of rope hanging off is $x$. Gravity accelerates the part of the ...
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0answers
34 views

Finding the constraint equation

I am trying to solve a problem on Constraint equations for a triple pendulum model, but was not able to derive a constraint equation for the last mass. I solved constraint equations for Masses 1 ...
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1answer
27 views

Hamilton's equations of motion

One of hamilton's equations is $(\frac{\partial H}{\partial q} )_t = -(\frac{\partial p}{\partial t}) _q$. But isn't it $\frac{\partial L}{\partial q} = \frac{dp}{dt}$? If H = L(i.e. V = 0), what ...
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64 views

Are there (interesting) Poincare-invariant QFTs with non-invariant Lagrangian densities?

In all QFTs I know, the Lagrangian density is completely invariant under the Poincare group, $$ \mathcal L \to \mathcal L. $$ On the other hand, the action would be invariant even if the Lagrangian ...
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1answer
38 views

How to scale variables in a classical Hamiltonian?

So I looked at some research articles where one has a classical Hamiltonian $H(p,q,t) = p^{2}/2 + V(q,t)$. If one introduces the scaling transformation $$t \mapsto t/\sqrt{s}, \quad H \mapsto Hs, ...
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1answer
67 views

How did he find the “lambda” value in this question? [closed]

There is a pdf i found when searching about Lagrangian Multpliers, but i was not able to understand how he derived lambda from two differential equations. If anyone can walk me through it, i would be ...
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1answer
63 views

How do I derive geodesic equation using variational principle? [duplicate]

I am trying to derive the geodesic equation using variational principle. My Lagrangian is $$ L = \sqrt{g_{jk}(x(t)) \frac{dx^j}{dt} \frac{dx^k}{dt}}$$ Using the Euler-Lagrange equation, I have got ...
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1answer
31 views

Relation between homotopy theory and symmetry transformation of the Lagrangian

What is the relation between the symmetry transformations of the Lagrangian and homotopy theory? If yes, how? Not sure if this is a math or physics questions. References would be very helpful.
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1answer
91 views

Why can the bra and ket be varied independently?

Given a functional which depends on a function (ket), and its complex conjugate (bra), e.g. $$F[\varphi] = \langle \varphi|\hat{F}|\varphi\rangle = \int \varphi^{*}(\mathbf{r}) \hat{F} ...
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50 views

What is the path taken by a “cable car”?

A well known result in variational calculus & Lagrangian Mechanics is the solution to the "brachistochrone" problem, where it is found the path connecting two points, A & B such that the time ...
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0answers
64 views

Eigenfrequencies of a truss

I want to calculate the eigenfrequencies of a 3D truss using the finite element method. The beams should be modelled as Timoshenko beams and I want to use arbitrary shape functions. I know how ...
4
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1answer
83 views

Superficial degree of divergence on Weinberg

Reading volume 1 of Weinberg's QFT book, chapter 12, page 505 he says that if you consider a diagram with degree of divergence $D\geq{}0$, its contribution can written as a polynomial of order $D$ in ...
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22 views

How can intuitively guess what conserved quantities has the system that I am studying?

I'm taking a course in Classical Electrodynamics and in one problem my teacher introduced us to a triplet of fields ($\phi^a$) invariant under internal rotations, i.e. transformations like: ...
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1answer
70 views

How to calculate the effective action in general?

Considering the scalar field, we have the effective action $$\tag 1 \Gamma[\phi_{cl}]=\int ...
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1answer
40 views

Complex Coordinate change

I have a simple question where I must change the coordinates of a system however I am unsure whether I am correct. I am changing from Cartisian to complex coordinates. Let's say I only have $x$ and ...
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2answers
96 views

Anomalous Slavnov-Taylor identity

I will be happy if someone could clarify the mystery here. Consider the following derivation of the anomalous Slavnov-Identity. It's based on lecture notes by Adel Bilal. Suppose we have an action ...
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2answers
183 views

Quantum Anomalies and Quantum Symmetries

In Quantum Field Theories (QFT) there is a well known phenomenon of anomalies, where a classical symmetry is broken in the quantum theory due to a so called anomaly. This symmetry breaking can be ...
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0answers
37 views

Help in writing down Feynman rule? [duplicate]

I have a term in my Lagrangian that looks like: $A^\mu B^{*\nu} \partial_\mu B_\nu - A^\nu B^{* \mu} \partial_\mu B_\nu$ where A is the photon field, and B is a charged, massive spin-1 boson. I am ...
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1answer
84 views

A physical system is described by the following Lagrangian: $ L = \frac{m}{2} (\dot{\rho}² + \rho ² \dot{\phi} ² + \dot{z} ²) + a \rho² \dot{\phi}$ [closed]

Where $a$ is a constant and $(\rho,\phi,z)$ are cylindrical coordinates. I found the following Hamiltonian $ H =\frac{m}{2}(\dot{\rho}² + \dot{z}² + \rho²\dot{\phi}²)$. The problem asked me to find ...
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0answers
32 views

Rotating disk moving on a circular path

A uniform disk with radius $R$ and mass $M$ is in earth's gravitational field (i.e $\vec{g}$). A point $A$ on the perimeter of the disk is attached to a circular path with radius $L$, and the disk is ...
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1answer
91 views

Relationship between locality, causality, and free theories

This text on QFT defines a free theory as that in which dynamics of the field for each degree of freedom evolves independently from all the other. In principle we have an infinite degrees of freedom, ...
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1answer
12 views

Center of Instantaneous velocity in two degree of freedom problem

I have a problem like this I have two DoF where the force F is acting on the car and another force could be acting on the coordinate $q_1$. The force acting on $q_2$, the coordinate of the car will ...
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2answers
431 views

Can one write down a Hamiltonian in the absence of a Lagrangian?

How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to ...
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1answer
75 views

Fayet-Iliopoulos terms

It is mentioned in first page of this paper by Seiberg and Komargodski that the Lagrangian in superspace of a $U(1)$ gauge SUSY theory with FI terms is not gauge invariant. However, the FI terms in ...
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1answer
109 views

Differential holonomic constraints

Differential holonomic constraint is an integrable homogeneous first order differential equation: $$\sum_{i}\mathcal{E}_{i}(q)\frac{dq_{i}}{d\tau}=0;$$ in which $\sum_{i}\mathcal{E}_{i}(q)dq_{i}$ is ...
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1answer
57 views

Existence of lagrangians at strong coupling

It is well known that some QFT do not admit a lagrangian formulation (like the $(2,0)$ SCFT in $d=6$). Up to my understanding, all the examples that I know of non lagrangian theories are always ...
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1answer
41 views

Lagrangian mechanics - small oscillations around equilibrium diagonalization

In my analytical mechanics class, we have been taught that normal modes of small oscillations around equilibrium are given by the solution of $$ p(\omega) = \det(K-\omega^2M) = 0 $$ Where $K_{ij} = ...