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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1answer
29 views

How do I derive geodesic equation using variational principle?

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 ...
2
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3answers
44 views

Harmonic oscillator with squared damping term

Does a solution exist for a harmonic oscillator with a squared damping term? $$m\ddot{u}+c\dot{u}^2+ku=0$$
0
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1answer
26 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.
4
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1answer
73 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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0answers
50 views

Show that function $F$ is a constant of motion for a Lagrangian [closed]

Function $$F=(\dot{z}R-z\dot{R})\sin\varphi-zR\dot{\varphi}\cos\varphi$$ should be a constant of motion for the Lagrangian $$ ...
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0answers
40 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
53 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
votes
1answer
51 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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0answers
16 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
54 views

How to calculate the effective action in general?

Considering the scalar field, we have the effective action $$\tag 1 \Gamma[\phi_{cl}]=\int ...
0
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1answer
39 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 ...
4
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2answers
90 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 ...
8
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2answers
160 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
36 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
80 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
26 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
77 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
11 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 ...
16
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2answers
400 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 ...
3
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1answer
64 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 ...
1
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1answer
104 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 ...
3
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1answer
49 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 ...
0
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1answer
38 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} = ...
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1answer
53 views

Equivalence between principle of least action and minimum potential energy

Are the principle of least action and the principle of minimum potential energy equivalent? How does one show that? Also, are Newton's laws of motion equivalent to the principle of least action? How ...
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0answers
61 views

Does this vertex equal 0?

If I have an interaction term in my Lagrangian that looks like: $\mathcal{L}_{int} = (\partial_\mu B_\nu)(A^\mu B^\nu - A^\nu B^\mu)$ where B is a massive spin-1 field. Am I correct in thinking that ...
2
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0answers
26 views

Higher order Lagrangians [duplicate]

Recently I have read some papers in which the authors considered higher order lagrangians. For example, in this paper "A path integral leading to higher-order Lagrangians" by C.Acatrinei the higher ...
2
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0answers
44 views

Principle of Most Action? [duplicate]

In Landau-Lifshitz - Vol 1. Mechanics, right after the introduction of the principle of leas action, there is the following comment: It should be mentioned that this formulation ($S = ...
2
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1answer
62 views

Conceptual problem with action considered as function of endpoints

I am having some trouble with understanding why it makes sense to consider action in classical mechanics as function of endpoints $q_{initial}, \ q_{final}$ and endtimes $t_{initial}, \ t_{final}$. ...
0
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1answer
35 views

Calculate lapse function from the metric

I have a technical question about the lapse function: Assume I have some given (Lorentzian) metric $g$. I have seen the following definition of the lapse function $\alpha^{-2}=-g(\nabla f, \nabla ...
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2answers
38 views

Where does the factor of half appear from in the Klein-Gordon Lagrangian?

The lagangian density of a scalar field or a Klein-Gordon field has the form of $$\begin{align} \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2. \end{align}$$ ...
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2answers
63 views

Variation of a Lagrange density Symmetries

So I am reading Goenner's Spezielle Relativitästheorie and I am currently in chapter §4.9.1 Variation under Inclusion of Coordinates p. 129. So basically we have: $$\delta W_\zeta=\int d^4x' ...
2
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1answer
56 views

Functional Derivative of action

Consider the action of free Klein-Gordon theory $S[\phi]=\frac{1}{2}\displaystyle\int d^4y(\partial_\mu\phi(y)\partial^\mu\phi(y)-m^2\phi^2(y))$ Integrating by parts in the first term gives me ...
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0answers
34 views

When the equations of motion are not unique (eg. when they are given by eigenvectors), which will the free particle adhere to?

For this question I think it will be easier to express the usual equation describing the motion of a "free particle,"--viz. $g_{ij}\dot{x}^i\dot{x}^j$--in matrix form as follows: ...
0
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0answers
13 views

Why is an angle undergoing forced rotation not a generalized coordinate?

In a textbook it is using the following example. A bead of mass m slides freely on a light wire of parabolic shape, which is forced to rotate with angular velocity ω about a vertical axis. The ...
3
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1answer
61 views

Are there conserved quantities in field theory which don't arise from Noether's Theorem?

In some QFT texts one writes down the number operator $N$ for free theories, such that when acting on an $n$-particle state $|n\rangle$ we have $$N|n\rangle=n|n\rangle$$ In free theories this is a ...
2
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1answer
38 views

Lagrangian of a block connected to a circular track [closed]

Could someone help me? I am having trouble with obtaining the same result in part b) for this problem: Using the Lagrange Equation with respect to $\theta$, I obtained ...
9
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2answers
226 views

Symmetry of the Polyakov action?

Let us look at the Polyakov action for a string moving in a spacetime with metric $g_{\mu \nu}(X)$:$$S_P = -{1\over{4\pi \alpha'}} \int d^2 \sigma \sqrt{-\gamma} \gamma^{ab} \partial_a X^\mu ...
3
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1answer
58 views

Application of Euler-Lagrange equations (Trivial problem, instructive one)

I have some doubt about a really trivial and simple problem in which I have to use ELE. Supposing I have a pendulum, in which the rope is a spring, so it's length may change in time. I have a mass ...
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0answers
29 views

Show that a vector field is a symmetry for a Lagrangian [closed]

Let Lagrange function be $$ L=\frac{1}{2}m(\dot{x_1}^2+\dot{x_2}^2+\dot{x_3}^2)-U((x_1^2+x_2^2,x_3)). $$ Show, that vector field $\vec{Y}(\vec{x})=(-x_2,x_1,0)$ comply $$ ...
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0answers
25 views

Majorana fermions and the continuum limit of the Ising model

In Paolo Moligini's Analyzing the two dimensional Ising model with conformal field theory lecture notes, it is shown at the end of chapter 3 that the Lagrangian of the continuum limit of the Ising ...
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0answers
43 views

How does satisfying the Euler-Lagrange equation put a Classical Path on-shell?

I am thinking of what the Euler-Lagrange equation, $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0 $$ specifically represents in satisfying the ...
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0answers
24 views

Sign of elastic potential from a spring [closed]

When writing the Lagrangian of a problem, how do I determine the sign of an elastic potential from a spring?
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0answers
18 views

Equations of motion for normal modes

I really need some help understanding how to find normal modes. So I brought the euler-lagrange equation of my probelm to this form: $X'' = -AX.$ Where $X$ is the coordinates vector. So I found the ...
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0answers
44 views

Is the Symmetry factor different in Path integral Formalism?

Is the Symmetry factor different in Path integral Formalism and the Perturbation theory (canonical) formalism? For example, the order-1 4-point cross X diagram in the $\phi^4$ theory has symmetry ...
3
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2answers
98 views

Principle of least action: $\frac{d S_{cl}}{dt_b} = \frac{\partial S_{cl}}{\partial t_b} + \frac{\partial S_{cl}}{\partial x_b}\dot{x}_b$

Question I cannot see how I can obtain the yellow highlighted section on the RHS from that of the LHS. The following equation can be found in both my lecture notes(*1) (page 9, equation 2.7) and is ...
3
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1answer
70 views

To derive the relation between work function and potential energy

I'm reading "The variational principles of mechanics- Lanczos", The author mentions a relation between Work-Function $U(q_1,q_2,\cdots,q_n,\dot q_1,\dot q_2,\cdots,\dot q_n)$ and the potential ...
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0answers
25 views

How to obtain calculus of variation of Einstein summation?

I have the Lagrange density for Maxwell field, which is $\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_{\nu}A_{\mu}$. How can I obtain ...
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2answers
65 views

Are the generalized coordinates in Lagrangian mechanics really independent?

In Goldstein's Classical Mechanics, Chapter 2.3: Derivation of Lagrange's Equations From Hamilton's Principle part of the derivation involves each of the generalized coordinates being independent. $$ ...
3
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1answer
99 views

Why do we require local gauge invariance

My thought on this are somewhat scattered so I apologise in advance. Maxwell's equations are gauge invariant. The physical Electric and Magnetic fields don't depend on whether we use $A_\mu$ or ...
3
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0answers
47 views

Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = ...