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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92 views

From Lagrangian to equations of motion [closed]

I have a given Lagrangian: $$L= e^{st}\cdot\frac12\cdot(mv_y^2-ky^2)$$ And are asked to identify the equations of motions, the constants of motions and physical system. Without the exp-time-term, ...
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80 views

What is meant by a local Lagrangian density?

What is meant by a local Lagrangian density? How will a non-local Lagrangian look like and what is the problem that we do not consider such Lagrangian densities?
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168 views

Einstein action and the second derivatives

I have naive question about Einstein action for field-free case: $$ S = -\frac{1}{16 \pi G}\int \sqrt{-g} d^{4}x g^{\mu \nu}R_{\mu \nu}. $$ It contains the second derivatives of metric. When we want ...
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2answers
185 views

Einstein equation and scalar field stress-energy tensor

Let's have interaction between gravitational and scalar real fields. For an action of gravitational field in vacuum I add term $S_{m} = \int d^{4}x\sqrt{-g}L_{m}$, where $$ L_{m} = \frac{1}{2}g^{\mu ...
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90 views

Expansion of a function

In Landau-Lifschitz, following expansion is given, We have, $$L(v'^2)~=~L(v^2+2\textbf{v}\cdot\epsilon+\epsilon ^2)$$ expanding this in powers of $\epsilon$ and neglecting powers of higher order, ...
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157 views

Dimensions in lagrangian potential

According to Mankowski flat space dimensions We can write, $$L= \int \text{dt} \text d^d{x} \left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
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296 views

Questions regarding solving the Brachistochrone problem using Lagrangian

brachistochrone problem: Suppose that there is a rollercoaster. There is point 1 ($0,0$) and point 2 ($x_2, y_2)$. Point 1 is at the higher place when compared to the point 2, so the rollercoaster ...
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212 views

Improved energy-momentum tensor

While still dealing with this issue, I've stumbled upon this answer to a question asking about the conserved quantity corresponding to a scaling transformation. It mentions that in accordance with ...
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439 views

M-theory no lagrangian?

Is there any formulated lagrangian (density) for M-theory? If not, why is there no lagrangian? If not, is this related to many vacua existing? Thnx.
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438 views

Relation between between linear momentum and translational kinetic energy

The momentum $m v$ of a particle is formally the same as the derivative its translational kinetic energy $\frac{1}{2} m v^2$ with respect to $v$. Similarly the angular momentum $I \omega$ is the ...
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50 views

Alternative symmetries for the Maxwell Lagrangian?

I'm wondering about how to show that $A_a\rightarrow A_a+\alpha\partial_0A_a$, with $\alpha$ infinitesimal, is an infinitesimal symmetry of $\mathcal L=-\frac14F_{ab}F^{ab}$. \begin{equation} ...
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62 views

How does isotropy of free space imply $L(v^2)$ for a free particle? [duplicate]

From Mechanics; Landau and Lifshitz, it's stated on page 5: Since space is isotropic, the Lagrangian must also be indpendent of the direction of $ \mathbf{v}$, and is therfore a function only of ...
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56 views

How to obtain Maxwell's Lagrangian from complex scalar fields?

I've looked in several books and they all show how to obtain electrical interactions by forcing local gauge invariance of any complex scalar field Lagrangian (like Klein-Gordon or Dirac). I manage to ...
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37 views

The exact definition of conjugate momentum density

After checking various websites, I've seen the conjugate momentum density defined as either: \begin{align*} P_r ~=~ \frac{\partial \mathcal{L}}{\partial \dot{A}_r} \end{align*} or \begin{align*} P_r ...
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88 views

How do I obtain the Lagrangian in standard for using action? [closed]

I have action as shown below $$S=\int \mathrm{d}t \int \mathrm{d}x^3 \bar\psi\left(i\partial_t\psi +\frac1{2m}\bar\nabla^2\psi-V(x)\psi\right)$$ How do I manipulate it to obtain the Lagrangian ...
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65 views

Converse of the Lagrangian form-invariance

The form-invariance of the Lagrange equations implies the existence of a function $\ A( q_k, t)$ so that $\ \begin{equation} L' (q_k, v_k, t) -L(q_k, v_k, t) = \frac d {dt} A( q_k, t) ...
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69 views

Transforming a lagrangian to hamiltonian and vice versa

I am not refering to Legendre transform, but to something more simple. In analytical mechanics, the Lagrangian can be described as $L=T-V$, and the Hamiltonian is if the Lagrangian doesn't explicitly ...
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128 views

Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field

I am tasked with reading and reproducing all the steps in J.J. Hopfield's 1958 paper "Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals". Embarrassingly I am stuck ...
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200 views

Using Lagrange's Equations with Generalized forces

I am a bit confused on how this works. For instance if I wanted to look at an object moving in 2 dimensions only subject to gravity (and assuming that the potential is just mgy), I get that my ...
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1answer
180 views

Proper time of circular motion under Schwarzschild metric

I'm trying to calculate the proper time of a massive particle circulating Schwarzschild black hole, using EL equation of the following Lagrangian: ...
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2answers
143 views

Relativistic Lagrangian transformations

I need to study the relativistic lagrangian of a free particle. It's $\ L= - m c^2 \sqrt[2]{1- \frac{|u|^2}{c^2}} $ I need to study the translation, boost and rotation symmetry. I say it doesn't ...
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168 views

Potential in Relativistic Scalar Field Theory

My intention is to establish a Soliton equation. I have cropped a page from Mark Srednicki page no 576. I have understand the equation (92.1) but don't understand that how they guessed the ...
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293 views

How would you use the Euler-Lagrange equation to predict the motion of projectiles with linear (Stokes) drag (but no wind)?

My first instinct would be to use the force $$\vec{F} =- \alpha \vec{v}$$ and therefore $$V(\vec{r}) = \alpha \int_C \vec{v}\cdot d\vec{s} = \alpha \int_C \vec{v}\cdot \vec{v} dt = \alpha \int_C ...
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251 views

What's the motivation behind the action principle? [closed]

What's the motivation behind the action principle? Why does the action principle lead to Newtonian law? If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and ...
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143 views

Cyclic co-ordinates implying the constant velocity motion of center of mass of a system of particles

I'm reading the section on Central Force in my textbook (Goldstein's Classical Mechanics has a similar argument in the chapter titled "The Central Force Problem", first section), where we have the ...
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78 views

Smooth trajectory on a smooth manifold

Physicists talk about a smooth trajectory of a particle on a smooth manifold and they label it as q(t) where q_1(t)....q_n(t) are component functions coming from the homeomorphism. I don't see how we ...
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659 views

Problems that Lagranges equations of the 1st kind can solve whereas the 2nd kind can't?

Can anyone give examples of mechanics problems which can be solved by Lagrange equations of the first kind, but not the second kind?
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32 views

Defining quantum effective action (Legendre transformation), existence of inverse (field - source)?

Given a Quantum field theory, for a scalar field $\phi$ with generic Action $S[\phi]$, we have the generating functional $$Z[J] = e^{iW[J]} = \frac{\int \mathcal{D}\phi e^{i(S[\phi]+\int d^4x ...
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80 views

How do I recover the 1D wave equation from the Lagrangian?

Consider small displacements, $y(x,t)$, of an element of a string (circled in red and shown below) from equilibrium. The force balance in the vertical direction yields: $$+\uparrow \Sigma F: ...
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60 views

Minimal vs. Non-minimal coupling

What is the difference between Minimal vs. Non-minimal coupling in General Relativity? A brief introduction to Minimal Coupling in General Relativity could be useful too.
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193 views

Optimal selection of generalized coordinates in Lagrangian system

EDITED: The number of bonds is actually 2, not 1 (look at edit history). Fixed for archiving purposes. Problem: The edge A of an homogeneous rod (of length $\ell$ and mass $m$) is performing a smooth ...
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76 views

What is the neatest way to describe a “non-autonomous” (lagrangian) system?

The configuration space of a system of particles $(m_i,x_i)$, $i=1,\dots,n$, subject to constraints $$\Phi (x)=0,\qquad \Phi\colon \mathbb R^{3n}\to \mathbb R ^{3n-k},\qquad x=(x_1,...,x_n),$$ if the ...
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29 views

Lagrangian Oscilattor

I want to know how to calculate the normal modes from a Lagrangian. I make the T (kinetic energy matrix) and U (potential energy matrix), and then I calculate the determinant of $|T-\omega ^2 U|$ ...
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128 views

Calculate Hamiltonian from Lagrangian for electromagnetic field

I am unable to derive the Hamiltonian for the electromagnetic field, starting out with the Lagrangian $$ \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\partial_\nu A^\nu \partial_\mu A^\mu ...
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117 views

Hamiltonian conservation

Lagrangian formalism does not involve forces that doesn't come from a potential and Hamiltonian formalism says that even though energy is not conserved due to a force like this, the Hamiltonian is ...
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143 views

What to do with a $\phi$ term in a Lagrangian?

I am considering a Lagrangian that is of the following form: $$\mathcal{L}=-{1\over 2}\partial_\mu\phi\partial^\mu\phi+2\mu^2\phi^2+2\sqrt{6}{\mu^3\over \lambda}\phi + {9\mu^4\over 2\lambda} + ...
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38 views

Integrating out fields from classical systems

Has anyone ever heard of integrating out fields from classical Lagrangians if they are quadratic?
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68 views

Why does a particle fall in a straight line?

In Lagrangian Mechanics we choose the path of least action. Given a uniform gravitational field, and a particle of finite mass; and fixing two points the start & end-point we consider all paths ...
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34 views

Double variation of Schwinger action principle

The Schwinger action principle is given by $$\delta_{1}\big\langle b\big|a\big\rangle= i\int_{t_{a}}^{t_{b}}\text{d}t\,\sum_{c,d}\big\langle b\big|c\big\rangle\big\langle ...
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81 views

Classical mechanical problem

I have two planes, one characterized by equation $$\phi_1=f(x)-z=0$$ and another $$\phi_2=\alpha y-z=0$$ where $\alpha$ is arbitrary. In their line of intersection(we assume it exist and is continous) ...
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263 views

Explicit time dependence of the Lagrangian and Energy Conservation

Why is energy(or in more general terms,the Hamiltonian) not conserved when the Lagrangian has an explicit time dependence? I know that we can derive the identity: $\frac{\partial ...
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289 views

Lorentz force from velocity-dependent potential and Lagrangian

There is something i'm missing. I am at page 22-23 of Goldstein Classical Mechanics 3rd ed. Lorentz force can be derived from a potential $$U=q\phi-q\mathbf{A}\cdot\mathbf{v}$$ Where $\phi(t,x,y,z)$ ...
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76 views

Lagrange multiplier dependent on time

At the moment I am following a course on variational methods for mathematicians. Last week we derived the Euler-Lagrange equations for a functional under a constraint. In this derivation we found that ...
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1answer
227 views

Virtual displacement and generalized coordinates

I have a doubt regarding the expression of a virtual displacement using generalized coordinates. I will state the definitions I'm taking and the problem. The system is composed by $n$ points with ...
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93 views

Landau Lifshitz energy for uniform rotation

Landau Lifshitz claim in their Mechanics book (39.11) that for a uniform rotation we have $ E = \frac{mv^2}{2} - \frac{m}{2} (\omega \times r)^2 + U,$ where the rotation is given by $v' = v + \omega ...
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74 views

Finding the EOM for a charged relativistic particle

For an exercise sheet of a course in general relativity I'm asked to derive the equations of motion for a charged particle in an EM-field given by a potential $A^\mu$. I am give the action: $$S = ...
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134 views

Hamiltonian from Euclidean lagrangian?

Can somebody help me in deriving the Hamiltonian of system starting from Euclidean Lagrangian? Say we are given the Minkowski Lagrangian $$L_m = \frac{\dot{\phi}^2}{2} - V(\phi).$$ The Hamiltonian ...
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231 views

Non-relativistic limit of complex scalar field

In page 42 of David Tong's lectures on Quantum Field Theory, he says that one can also derive the Schrödinger Lagrangian by taking the non-relativistic limit of the (complex?) scalar field Lagrangian. ...
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164 views

Beads sliding on a hoop

Two particles, $P_1$ and $P_2$, of equal masses $m$ are linked by a spring of stiffness $k$ and natural length $a$. They are sliding freely without friction along a horizontal fixed hoop of radius ...
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125 views

Applying Lagrange's equations ignoring normal reaction

A small bead is sliding on a smooth vertical circular hoop of radius $a$, which is constrained to rotate with constant angular velocity $\omega$ about its vertical diameter. $\theta$ is an angle ...