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
195 views

What are the equations of motion for the scalar field in the tetrad formalism?

The action of a massless scalar field in curved spacetime is given by: \begin{equation} S(\phi)=\int d^{4}x \sqrt{-g}\left(g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right) \end{equation} Now the action can ...
0
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1answer
25 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.
1
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1answer
134 views

Understanding Noether's theorem rigorously

I've known about Noether's theorem for some time and reading some things about it recently I've realised I haven't completely understood it. In that case I've been trying to understand a more rigorous ...
0
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3answers
186 views

Problem in Euler-Lagrange imply Newton

I'm self-studying Mechanics and I have a little problem: We can see that in Landau's book or in Wikipedia that when we inject the lagrangian in Euler Lagrange equation the term $\frac{\partial ...
30
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3answers
2k views

Why treat complex scalar field and its complex conjugate as two different fields?

I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar ...
4
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1answer
66 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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2answers
355 views

Kepler's laws to determine radius of circular orbit

"In nonrelativistic limit of general relativity there is a correction to the Newtonian gravitational potential energy $−h/r^3$ with $h = αL^2/(mc)^2$, where $c$ is the speed of light, $α = GMm$ ...
-1
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0answers
50 views

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

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
15 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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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 ...
42
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6answers
4k views

Why does calculus of variations work?

How does it make sense to vary the position and the velocity independently? Edit: Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
4
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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
51 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 ...
16
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2answers
398 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 ...
0
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1answer
161 views

Degrees of freedom in double Atwood machine?

Why the degree of freedom in double Atwood machine (one block on one side and a pulley with one block in its each side on other side) is 2 and not 1? According to the formula $s=3*n-m$; where ...
4
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1answer
381 views

Does time invariance conclude conservation of energy? [closed]

I find it hard to understand that time-translation invariance necessarily implies conservation of energy. As I understand it, Noether's theorem says that there is an energy conservation because the ...
4
votes
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 ...
1
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1answer
52 views

How to calculate the effective action in general?

Considering the scalar field, we have the effective action $$\tag 1 \Gamma[\phi_{cl}]=\int ...
15
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10answers
8k views

Book about classical mechanics

I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...
0
votes
1answer
37 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 ...
8
votes
2answers
158 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 ...
3
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2answers
290 views

Hamiltonian from a Lagrangian with constraints?

Let's say I have the Lagrangian: $$L=T-V.$$ Along with the constraint that $$f\equiv f(\vec q,t)=0.$$ We can then write: $$L'=T-V+\lambda f. $$ What is my Hamiltonian now? Is it $$H'=\dot q_i p_i ...
0
votes
1answer
277 views

Euclidean classical action

This is the Euclidean classical action $S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))$. It would be nice if somebody could explain the structure of the potential. I don't ...
0
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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 ...
-1
votes
1answer
79 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 ...
9
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2answers
225 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 ...
1
vote
1answer
76 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, ...
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 ...
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0answers
25 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 ...
3
votes
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 ...
0
votes
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 ...
8
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1answer
348 views

Pressure and density using a general Lagrangian

Given a lagrangian of a form: \begin{equation}\mathcal{L}=f(\phi,\partial_{\mu}\phi\partial^{\mu}\phi)\end{equation} where $f$ is a function, I need to derive pressure and density in a FLRW universe ...
3
votes
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 ...
3
votes
1answer
408 views

Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
13
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4answers
9k views

Derivation of Maxwell's equations from field tensor lagrangian

I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density $L = ...
0
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1answer
146 views

Acoustical wave equation from Hamilton's principle

It is common to show the features and power of the Hamilton's principle by deriving the equation of vibrating string, membrane etc. using this principle. But I have never seen that used for deriving ...
1
vote
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' ...
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 ...
3
votes
1answer
310 views

How do I treat the Lagrangian in the case of a rigid body?

Here's Exercise 1.11 from Goldstein's Classical Mechanics 3rd edition (the first one after having derived the Lagrangian basically): Exercise 1.11: Consider a uniform thin disk that rolls without ...
3
votes
1answer
167 views

Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I ...
5
votes
1answer
350 views

Determinant for a coupled fluctuation Lagrangian

Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the ...
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vote
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 ...
3
votes
1answer
184 views

Parity violating Dirac particle

We normally write down the Dirac Lagrangian as \begin{equation} {\cal L} _D = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi \end{equation} but are the Lagrangian's, \begin{equation} ...
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 $t_{initial}, \ t_{final}$. ...
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2answers
37 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 $$\begin{align} \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2. \end{align}$$ ...
0
votes
1answer
34 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 ...
2
votes
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 ...