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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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15 views

Real Lagrangian with complex variable

I have a general question concerning real valued Lagrangians that take complex arguments. I have seen in many works of physicists and lecture books where extremal problems are discussed in Lagrangians ...
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18 views

In what sense does variational calculus help us graph the path of a moving object represented by the stationary action formula? [on hold]

This is off the beaten path if you allow me to build a short history of the question. Let us assume we have the function in hand we know that function will produce a stationary action so I assume if ...
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0answers
29 views

Integrating Out Auxiliary Field of point-particle Polyakov Action

The Polyakov action of a point-particle is $$S[X,e]=\frac{1}{2}\int d\tau\left(\frac{\dot{X}^{2}}{e}-m^{2}e\right)$$ with the $(−,+,+,+)$ Minkowski sign convention. How to perform the path-integral ...
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14 views

Gravity train solution through symmetries

I've been having trouble feeling okay with all of the solutions I've found to the Brachistochrone problem inside earth thus far. To me the way to do it is to show: The time that it takes a path $\...
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0answers
53 views

Einstein Notation in QFT and GR [on hold]

My question is about Einstein notation. Given a field theory, when we set up the Lagrangian $L = T-V$, the kinetic part $T$ contains a 2nd order derivative. Thus, let us just say we have the kinetic ...
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14 views

$USp$ global symmetry in $d=3$, ${\cal N}=4$ supersymmetric QFT

Define a 3-dimensional QFT with $N=4$ supersymmetry (4 supercharges), and the field content is $g$ $N=4$ hyper-multiplets (that are in a representation $R$ of some group $G$). Each hyper-multiplet is ...
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2answers
108 views

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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2answers
830 views

Is the Lagrangian in the Standard Model exact or approximate?

We say that the Standard Model has $SU(3)_{C} \otimes SU(2)_{L} \otimes U(1)_{Y}$ symmetries. However, the $SU(2)_{L}$ symmetry of the doublet ($u, d$) is not exact because $u$ and $d$ have different ...
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32 views

Derivation of Born-Infeld (BI) for D-brane

I am trying to reproduce derivation of the Born-Infeld (BI) action for D-branes as it is presented in R.J. Szabo, BUSSTEPP Lectures on String Theory, arXiv:hep-th/0207142. In particular, I don't ...
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30 views

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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1answer
55 views

Does a Lagrangian being independent to a spacial coordinate imply spatial translation symmetry?

I’m trying to delve into lagrangian mechanics, mainly because i want to have a deeper understand of things lik Noether’s theorem. What i don’t fully understand is what the lagrangian of a system ...
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0answers
54 views

Can the Euler-Lagrange equation be used to derive the stationary action formula? [duplicate]

From what I understand I can use the Euler-Lagrange equation to find the function ( Let us call L. ) where L can be the function as stated in the action formula. But how difficult is it to actually ...
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27 views

Lagrangian of Charged Particle Evaluated On-Shell

I am trying to calculate the Lagrangian of a charged particle in background gauge field evaluaed on-shell. Let $A^{\mu}(x)$ be a gauge field. The action of a charged particle in this background gauge ...
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1answer
76 views

Canonical quantization of time-dependent lagrangians

I have a lagrangian $$ L(x^{a}, \dot{x}^{a}, t), $$ which is non-degenerate, quadratic in the fields, and contains an explicit dependence on the evolution parameter $t$. If $L$ was time-independent,...
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47 views

Can it be shown that the principle of least action applies to problems with no analytical solutions?

After thinking about the earlier version of my question, I realize that what I’m really asking about is whether it can be shown that the principle of least action applies to systems without an ...
2
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1answer
78 views

Hydrogen and the neutron

I have a question about Hydrogen and the neutron. So a neutral hydrogen atom has structure ${}^1H\equiv[p^+ e^-]$, a bound proton and electron. From a quantum theoretical viewpoint, one would write ...
2
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1answer
57 views

Euler-Lagrange equation in General Relativity

In Relativity the Lagrangian of a free particle is \begin{align} \mathcal L=\sqrt{g_{ab}\frac{dx^a}{d\tau}\frac{dx^b}{d\tau}}\end{align} Since $\mathcal L$ is parameterization invariant we can always ...
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2answers
396 views

What is the relation between a symmetry and the invariance of the Lagrangian?

While proving that homogeneity of space implies conservation of momentum, we use the fact that homogeneity of space means that the Lagrangian of the system remains invariant under translation. Why is ...
2
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2answers
79 views

$q\mathbf{A}\cdot\mathbf{v}$ term in potential energy

In the famous Goldstein mechanics book, there is an example about a single (non-relativistic) particle of mass m and charge q moving in an E&M field. It says the force on the charge can be ...
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1answer
52 views

$L^2$ and $L_z$ of a Quantum System

Let's assume I have a 3D Box problem (which is considered as a particle a box).I'm considering that I have a fermion which is in the 3rd excited state in the box. What can infer about the $L_z$ ...
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1answer
68 views

Help with Chrstoffel symbols for geometric mechanics problem?

I am working through the book Geometric Control of Mechanical Systems by Bullo and Lewis https://www.amazon.com/gp/product/0387221956/ and I am stuck on a problem, E4-18. The problem was evidently at ...
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1answer
20 views

Disparity in the equations of motion for a particle in a central field [duplicate]

Hello! I am considering the motion of particle in a central field and appear to derive inconsistent equations of motion when the Euler Lagrange equations are applied directly to (1) the Lagrangian $L(...
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1answer
54 views

Theory invariance after substitution of theory's field equations back into theory's action functional?

Suppose I have a theory $A$ concerning the evolution of a set of fields $T_1, \dots, T_n$. Let the action functional for this theory be $S[T_1, \dots, T_n]$. Suppose in the action, in addition to ...
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0answers
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How to find all Lagrangians that produce given laws of motion? [duplicate]

I've learned from Landau's book that you can produce Newton's laws given the principle of least action and a certain function for the Lagrangian. My question is: How can we find (and prove) all ...
2
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1answer
81 views

Potential energy and conservation law

I'm preparing for my masters entrance exam on pure mathematics (thought some problems are devoted to classical/lagrangian mechanics). I would be grateful to clarify some basics regarding the ...
3
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1answer
59 views

What is meant by the coupling term $g_{\mu\nu}T^{\mu\nu}$ in Supergravity?

In the "Cambridge Lectures on Supersymmetry and Extra Dimensions" of F.Quevedo it is written on page 59 ($T^{\mu\nu}$ stands for the energy-momentum tensor): The metric $g_{\mu\nu}$ as gauge field ...
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1answer
47 views

From non-relativistic to relativistic action

There is a derivation of relativistic action that treats space and time symetricaly which is just playing arround with the square of kinetic energy in the non-relativistic action and plugging in speed ...
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0answers
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New-minimal vs old-minimal supergravity

New-minimal set of supergravity auxiliary fields includes a two-form field, whereas the old-minimal auxiliary set includes a vector and a complex scalar. Is anyone aware of how to transform between ...
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0answers
145 views

Particle in electromagnetic field Lagrangian

Given the two definitions of $\vec E$ and $\vec B$ by scalar potential $\phi$ and vector potential $\vec A$: $$\vec B=\vec \nabla \times \vec A$$ $$\vec E=-\vec \nabla \phi -\frac 1 c\frac {\partial \...
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0answers
30 views

Noether's theorem for fields and infinitesimal transformations [duplicate]

I'm starting to learn QFT by myself using many references, mostly (QFT for the Gifted Amateur, and Tong's lectures) and both present a proof of Noether's theorem using infinitesimal tranformations, ...
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0answers
28 views

Acceleration from scalar-matter coupling in classical field theory

I have came across a text where an interaction term in a classical Lagrangian is presented that couples a matter density $\rho$ and a scalar field $\phi$ as \begin{equation} \mathcal{L}_{\text{int}} =...
2
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1answer
56 views

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
94 views

Legendre Transformation [closed]

Can someone explain how the Legendre transformation for multivariable functions work? I have been trying to find Legendre transformation for a function like $$F(x, y) = x^{2}+y^{2}+xy.$$ But I don't ...
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2answers
67 views

Degrees of freedom of a constrained vector

I have to handle with this lagrangian of a real vector $\chi^\mu$ $$ \mathcal{L} = -\frac{1}{4}F_{\mu\nu}^2 + B^\mu \square \chi_\mu + C\, \partial_\mu \chi^\mu + \mathcal{L}_{int} $$ where $B^\mu$ ...
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0answers
39 views

Matrix Euler’s rigid-body equation

Define the action $$S[g]=\displaystyle\frac{1}{2}\int^1_0 Tr(I(g^{-1}\dot g)~g^{-1}\dot g)~dt.$$ $I:SO(N)\to SO(N)$ denotes the endomorphism $\omega \to I(\omega)$ with $I(\omega)_{ij}=\omega_{ij}/...
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1answer
32 views

Why is the formula for stationary action expressed as kinetic minus potential energy instead of potential minus kinetic energy?

I am sure this is a duplicate but I could not spot it exactly. And I am sure folks have covered this topic online here in great detail. I am referring to the Lagrangian here in the "Action" formula ...
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46 views

Fields in Lagrangian

This Lagrangian was mentioned in Schwartz book of quantum field theory: Consider the Lagrangian $$L = -\frac{1}{4}(F_{μν})^2− |φ|^2 − ieA_μ(φ^*∂_μφ − φ∂_μφ^*) \tag{3.67} $$ where $φ$ represents ...
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27 views

Lagrangian density of the electromagetic field [duplicate]

Why the Lagrangian density of the electromagetic field like $$\mathcal{L}(A_\mu,\partial_\nu A_\mu)=c_1F_{\mu\nu}F^{\mu\nu}+c_2A_\mu j^\mu$$ defines the only possible structure consistent with gauge ...
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1answer
32 views

Why the Dirac Lagrangian is globally invariant under U(1) symmetry and the QED Lagrangian is locally invariant under U(1)?

I don't understand why this is true The Dirac Lagrangian is globally invariant under U(1) symmetry and the QED Lagrangian is locally invariant under U(1). Electric charge is the conservation ‘law’ ...
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0answers
21 views

Determining the stress-energy tensor from the equations of motion

I have a question on finding the stress-energy tensor from the equations of motion in general relativity. Given the Einstein-Hilbert action+matter Lagrangian, it is straightforward to then determine ...
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1answer
44 views

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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1answer
76 views

Problem 6.38 from David Morin (classical mechanics)

This problem is from Introduction to Classical Mechanics by David Morin. This is my solution: The solution is weird. Is it incorrect? if yes then can someone give me any hint on how to solve the ...
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2answers
50 views

Confusion about Noether's theorem on conservation of energy

Assume a gravitational field, with area A having some gravitational forces while area B having no gravitational force. The Lagrangian of particle moving along this field is obviously time invariant, ...
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2answers
93 views

How does one formally express the statement that the electromagnetic Lagrangian is Lorentz invariant?

The electromagnetic Lagrangian (density) is given by $$\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu\nu}$$ It is said that the Maxwell's equations (and hence this Lagrangian) are $\text{SO}(1,3)$ ...
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0answers
36 views

Why does transformation matrix commute with $\gamma$ matrices?

In Paul Langacker's The Standard Model and Beyond, equation 3.80 says the following $$ \mathcal{L}' = \overline{\psi} \mathrm{i} \partial ^{\mu} \gamma _{\mu} e^{- \mathrm{i} \beta ^i L^i} e^{\mathrm{...
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3answers
134 views

Is the Euler-Lagrange equation a special case of the principle of least action?

Is the Euler-Lagrange equation a special case of the principle of least action? I have some confusion after reading a few dozen stackexchange articles of the "principle of least action". I follow the ...
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0answers
49 views

How to prove the change in Lagrangian is related to divergence of Noether current?

In Paul Langacker's The Standard Model and Beyond, equation 3.50 states that, for a Lagrangian $\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_1$, where $\left[ T^i , \mathcal{L}_0 \right] = 0$ and $\left[ ...
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1answer
44 views

Fermion-Fermion scattering in Yukawa theory

The interaction term in the Lagrangian for Yukawa theory is given by $$ \mathcal{L}_\text{int} = -g\phi\bar{\Psi}\Psi, $$ where $g$ is the coupling constant, $\phi$ some scalar field and $\Psi$ a ...
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1answer
46 views

Probability interpretation of electron charge

Consider the QED. When we write the covariant derivative of the theory and couple vector field of gauge boson and the spinor of fermion we think of electric charge of electron $e$ as a coupling ...