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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10 views

Examples of multiply-connected compact configuration spaces

I'm a looking for examples of dynamical systems that have multiply-connected compact configuration spaces. Since I'm not a 100% sure about the correct terminology for the systems (I am sure about the ...
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4answers
113 views

How can I tell that a Lagrangian has an $SU(2)\times SU(2)$ symmetry?

this is a very basic question and it probably has a very simple answer. I was reading through some handouts when I came over something that I did not understand. One considered the simple Lagrangian ...
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0answers
18 views

the lagrangian of the axial vector field [on hold]

What is the Lagrangian of a axial vector field? How can I find it? I want to describe a model which is made by standard model with extended by an axial vector and a Dirac fermion!
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2answers
95 views

Are the partial derivatives of Lagrangian in the varied action functional derivatives?

In particle mechanics Lagrangian $L$ depends upon position, velocity (and may be explicitly on time), whereas in field theory the Lagrangian density ${\cal L}$ similarly (or analogously) depends upon ...
6
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2answers
89 views

Beyond Hamiltonian and Lagrangian mechanics

Lagrangian and Hamiltonian formulations are the bedrock of particle and field theories, produce the same equations of motion, and are related through a Legendre transform. Are there more such ...
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0answers
69 views

Geometric point of view of configuration space and Lagrangian mechanics [on hold]

Even though (I think) I understand the concept of a tangent bundle, I have trouble assimilating the idea of the configuration space being one and in relation to what that is the case. How can I ...
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51 views

Euler-Lagrange Equation in Quantum Field Theory

The quantum fields are operator valued distributions. In some sloppy books like Peskin and Schroeder the Euler-Lagrange equation are used to get the equations of motion. What does it mean to take a ...
2
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1answer
33 views

My Hamiltonian for a light ray vanishes

I have the following issue with understanding. A light ray traveling from $q(\tau_1)$ to $q(\tau_2)$ minimizes the integral $\int\limits_{\tau_1}^{\tau_2} n(q(\tau))|\dot{q}(\tau)| d\tau$, so the ...
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0answers
21 views

Change of energy of a shortening simple pendulum (Ehrenfest Pendulum) [on hold]

I've been going through Lanczos' Variational Principles of Mechanics and have been struggling with a problem that seemed pretty straightforward: A simple pendulum hangs from a fixed pulley. The ...
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0answers
46 views

Functional Differetiation of a complex functional

Suppose I have a simple functional $$F=\int{dx\;\phi^{*}(x)\phi(x)}\tag{1}.$$ Assuming $\phi(x)$ and $\phi^{*}(x)$ are independent and I take a functional differential with respect to $\phi(x)$ and $\...
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26 views

Functional variation of potential in integral form

I am trying to vary the following action, $$ S=\int_{t_0}^{t_1} \text{d}t\,(v^\mu v^\nu g_{\mu\nu} + V(t)) =\int_{t_0}^{t_1}\text{d}t\,(v^\mu v^\nu g_{\mu\nu} + \int_{t_0}^t\text{d}s T_\mu v^\mu) $$ ...
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1answer
25 views

Derivation of photon propagator from EM Lagrangian

I am following Ryder's Quantum Field Theory. In chapter 7, in order to derive the photon propagator, he first derives eq. 7.4 $$\mathcal{L}=\dfrac{1}{2}A^\mu[g_{\mu\nu}\partial^2-\partial_\mu\partial_\...
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0answers
44 views

Lagrangian of classical electromagnetism without $A_{\mu}$ field [duplicate]

Is there a Lagrangian reproducing Maxwell's equations without the use of the scalar and vector potential? Obviously $\mathcal{L} = -\frac14F_{\mu \nu}F^{\mu \nu}$ doesn't work since in terms of $E$ ...
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2answers
35 views

Lagrangian mechanics not relying on time or independent of time [closed]

If neither the potential energy nor kinetic energy depends on time, then Lagrangian is explicitly independent of time I find this statement a little bit odd because velocity is distance over time or ...
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1answer
52 views

Connection between “classical” Grassmann variables and Heisenberg Equation of motion

I have been reading di Francesco et al's textbook on Conformal Field theory, and am confused by a particular statement they make on pg 22. Let $\{\psi_i\}$ be a set of Grassmann variables. Starting ...
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0answers
41 views

Kinetic energy of a suspended mass from an overhead crane [closed]

I am using Lagrange's method to formulate the equation of motion for a 3D overhead crane that is shown below: For the formulation of the kinetic energy for the payload (suspended) mass. I have seen ...
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0answers
42 views

Non-canonical transformation

I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
4
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1answer
62 views

Vary action with respect to velocity

Variation of the action $S$ corresponding to a Lagrangian e.g. $L(x(t),\dot{x}(t))$ gives the Euler-Lagrange equations: $$ \frac{\delta S}{\delta x(t)} = 0 \\ \int du \left ( \frac{\delta L}{\delta x(...
2
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1answer
95 views

The difference between the forms of the Euler-Lagrange equations

I'm trying to learn Lagrangian mechanics and have been reading a lot of articles on it. But many of the articles write the equations in different ways, probably for different purposes. The Euler-...
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0answers
19 views

Euler-Lagrange equation no fixed endpoints

The usual way, to show the Euler-Lagrange equation is, to find the minimum of the Integral $$ I = \int_a^b L(q, \dot q, t) dt $$ and argue, that it must satisfy the following equation $$ \frac{d}{dt} \...
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0answers
47 views

From gauge invariance to charge conservation in covariant electrodynamics

I tried to solve the equations of motion using the action for the electromagnetic field interacting with a current, like $$ L = F_{\mu\nu}F^{\mu\nu} + A_{\nu}j^{\nu} $$ getting the right Maxwell's ...
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2answers
58 views

Lagrangian of an effective potential

If there is a system, described by an Lagrangian $\mathcal{L}$ of the form $$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$ where $T$ is the kinetic ...
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0answers
32 views

A question about Euler-lagrange equations

This question passes my mind so often, why do we stop at the first order of expansion of the action to get the Euler-lagrange equations and it turns out they exactly get us the Newtonian equations. ...
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1answer
22 views

Parity tranformation on Lagrangian of free fields

Free lagrangians of scalar, Dirac field and vector fields are always invariant under Parity. I am able to get this result mathematically, but I want to know if there is any obvious reason for it. ...
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3answers
86 views

The $\dot{q}$ term in the Euler-Lagrange equation

The Euler-Lagrange equation is about the functional $$ \int_{t_1}^{t_2} L(q, \dot{q}, t ) dt . $$ From a mathematical point of view, a simpler functional might be $$ \int_{t_1}^{t_2} L(q, t ) ...
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1answer
77 views

Derivation in Modern Supersymmetry by Terning

I am trying to do some calculations from Modern Supersymmetry by Terning and I am stuck on how he derived a particular term. Specifically, I am looking at 2.67 on page 27. My current work is below. $$...
5
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1answer
67 views

Electron - neutrino scattering effective Lagrangian

The electron and neutrino can interact through an intermediary Z boson, via the Lagrangian: $$ L= \frac{1}{2} \partial_\mu \phi_Z \partial^\mu \phi_Z - \frac{1}{2} m_Z ^2 \phi_Z ^2 -g_{\nu} \phi_Z \...
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0answers
29 views

Generalise Noether's theorem [closed]

I'm not sure how to generalise Noether's theorem. For this L, I think $B\cdot\dot{x}$ is conserved so I tried to relate F and K to this and try to show that that was conserved but got no where. any ...
0
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1answer
61 views

Checking the basics in QFT notation [closed]

In the very beginning of QFT we face the action (S) as a functional of the Lagrangian. I am still trying to get used to the notation used here, so I would like to check if the following makes sense: $...
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0answers
42 views

Mass term in Maxwell's Lagragian for Electromagnetism

In the scalar field Lagrangian the mass term is given by $$m^2 \phi^2.$$ But the equivalent term in Maxwell's Lagrangian for electromagnetism is $$m^2A_{\mu}A^{\mu}.$$ But I don't know why the ...
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43 views

Lagrangian of a falling rod that is free to rotate about an axis

So i came along an MIT example involving a tilted rod, length $L$ and tilted by angle $\theta$ from horizontal, whose free to fall and slid, they wrote the Lagrange equation without sweat. Now i am ...
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0answers
41 views

Reason behind $L = T - V$ (Lagrangian formalism) [duplicate]

I've been learning about the Lagrangian formulation recently, and while I'm with the process, I am still struggling somewhat with the theory behind it. As I (rather poorly) understand it, the ...
4
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1answer
97 views

Lagrangian density for Lorentz force of continuous charge distribution in external field?

It's frequently an exercise to derive the Lorentz force law for a particle with charge $q$ in an external electromagnetic field given by the following Lagrangian: $$L = -mc^2\sqrt{1-\frac{\dot{r}^2}{...
1
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1answer
76 views

Magnetic monopoles in field theory

In standard QED, we couple the electron to electromagnetism by replacing $$\partial_\mu \to \partial_\mu + i e A_\mu.$$ Upon taking the classical limit, we find that this gives electrons an electric ...
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1answer
47 views

Lagrangian in a system with a specific velocity dependent potential

I have a system of a particle moving under the generalized central potential $$ V= \frac{1}{r}(1+\dot{r}^2) \tag{1} $$ The general Euler-Lagrange equations for such type of potentials are: $$ \frac{...
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2answers
53 views

Virtual Work- Is the presentation in Cornelius Lanczos wrong?

Book: The Variational Principles of Mechanics by Cornelius Lanczos Edition: 4th Chapter: 3, The Principle of Virtual Work I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1). ...
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1answer
80 views

Why the Lagrangian $L$ is KE - PE? Why not KE + PE!

With Lagrangian, is there any way to intuitively grasp why total energy equals the difference between the kinetic and potential energy? Seems counter-intuitive - whereas Hamiltonian calculation (sum ...
3
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3answers
64 views

Why do we need $SU(2)\times U(1)$ invariant mass terms if the symmetry will be broken anyway?

In the SM we can not add fermionic mass terms like $m \overline{e}_R e_L$ to the Lagrangian since these terms are not invariant under $SU(2)\times U(1)_Y$. After introducing the Higgs in the unitary ...
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1answer
115 views

Action of a massive free point-particle in relativistic mechanics

I was reading about the formulation of mechanics in special relativity and found that the action for a massive free point-particle as $$ S = -mc\int_a^b ds $$ So, I did a few observations, ie. $$ S =...
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0answers
30 views

Lagrangian Interaction Type and Spin-Dependence

So I'm transitioning from reading particle physics books to the literature, specifically as it pertains to dark matter models. In this case I'm talking about t-channel DM-nucleon scattering. They ...
3
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1answer
51 views

Problem obtaining string equations from Polyakov action [closed]

I am trying to obtain the string equations of motion from the Polyakov action in the conformal gauge, i.e.: $$ S=T\int{d\tau d\sigma (\dot{x}^2-x^{'2})}\equiv\int{d\tau d\sigma \mathcal{L}} $$ where ...
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1answer
36 views

The components of applied forces on the masses of a Trebuchet

Background information: I was following an online mechanics document in order to learn how to derive the equations of motion for a trebuchet (shown below) using Lagrangian mechanics. At some point ...
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1answer
83 views

Number of degrees of freedom in the Standard Model Lagrangian

Consider a Lagrangian $L$ which depends on a number of fields $F_1$, $\cdots$, $F_N$ and their (spacetime) derivatives. Each of those fields $F_n$ is valued in $\mathbb{R}^{k_n}$. Is the Standard ...
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1answer
46 views

Lorentz transformation and symmetries of the Lagrangian [duplicate]

Since the Lagrangian of our quantum field theories is covariant under Lorentz transformations I'm asking myself if there is any link to some symmetries (like that we get from gauge transformations ...
3
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2answers
78 views

Gauge field and covariant derivative

To make the kinetic term in the Lagrangian for quantum field theories (for example qed) inveriant under local phase transformations we introduce the covariant derivative $D_{\mu} = \partial _{\mu} + ...
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0answers
34 views

Variation with respect to the metric and other tensors

When varying an action with respect to tensors and the metric, I'm afraid I get confused as how to one organizes the Lagrangian and then performs the variation. Take for example, the following example ...
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0answers
43 views

What to do when the fields are arranged in a matrix?

I am dealing with a Lagrangian in which the fields are arranged in an $N\times N$ matrix and i have to find the minima of the potential. Usually i would write the Lagrangian in components and then ...
0
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1answer
43 views

Correct Definition of Angular Momentum of a Charged Particle in an Electromagnetic Field? (Classical Mechanics) [duplicate]

What is the more correct definition of angular momentum $\vec{\mathbf{M}}$ in three dimensions? (I.e. classically/Lagrangian/Hamiltonian, not necessarily quantum or relativistic) $$\vec{\mathbf{M}}...
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2answers
83 views

Prove energy conservation using Noether's theorem

I wonder how you prove that energy is conserved under a time translation using Noether's theorem. I've tried myself but without success. What I've come up with so far is that I start by inducing the ...
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0answers
25 views

How to obtain the Klein Gordon equation for DBI action?

The action for DBI field is given by $$S=d^{4}x\,\sqrt{-g}\left[- V(\phi)\sqrt{1-g^{ij}\partial_{i}\phi\partial_{j}\phi}\right]$$ And the required Klein Gordon is given by $$\square \phi+\frac{\...