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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201 views

Hookes Law and Objective Stress Rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
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61 views

Local symmetry and General Relativity

First I want to consider an example of 1D motion. Lagrange equation: $$ \frac{d}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0 $$ If we transform $ L \rightarrow L+a $ ...
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1answer
199 views

Landau & Lifshitz - Euler's equation for one-dimensional flow

One page 5 in Landau & Lifshitz Fluid Mechanics (2nd edition), the authors pose the following problem: Write down the equations for one-dimensional motion of an ideal fluid in terms > of the ...
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1answer
105 views

Interacting Lagrangian - Coupling constant and cutoff factor

I have a general question concerning a given interacting Lagrangian: $$\mathfrak{L}_I = \frac{g}{\Lambda^2} \bar{\chi} \ \gamma^\mu \gamma_5 \ \chi \ \partial^\nu F_{\mu\nu}$$ where $F_{\mu\nu}$ is ...
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51 views

Double pendulum find first integral [closed]

Consider the following situation of a double pendulum in 2D. We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\ ...
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113 views

Feynman rules of a theory in non-standard form

I am currently studying lecture notes by Akhmedov on interacting scalar field theory in de Sitter space. In these notes, he considers a scalar field theory of the form ...
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199 views

Can you give example of some problems with solutions in each of Newtonian, Lagrangian and Hamiltonian method? [closed]

I am a student from information system and just want to know about classical mechanics. I know Newtonian mechanics from high school and I have read about Lagrangian and Hamiltonian mechanics in ...
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55 views

Translation symmetry and the non-conserved momentum in Viscous fluids

Even though a viscous fluid has a translation symmetry (invariance) for its Lagrangian , it still 'waste' Linear momentum. How come ?, isn't the rule that every symmetry yields a conservation law ?
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1answer
267 views

Lagrangian approach to spinning thread reel

I am trying to better understand Lagrangian dynamics and am struggling to complete the following question: A reel of thread of mass $m$ and radius $r$ is allowed to unwind under gravity, the upper ...
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1answer
395 views

Is there a better choice of coordinates for a bead on a rotating helical wire?

A bead of mass $m$ is threaded around a smooth spiral wire and slides downwards without friction due to gravity. The $z$-axis points upwards vertically. Suppose the spiral wire is rotated about the ...
2
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1answer
341 views

Ricci scalar in Scalar Field in Curved Space-time

I was recently looking at a Lagrangian of a scalar field in curved space-time at http://www.unc.edu/~mgood/research/Carroll_QFT_CS.pdf on page 8. I am not a physicist, and I am currently studying ...
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47 views

Acceleration of 2 bodies tied with a string [closed]

Find the acceleration of the block of mass M shown in the figure . The co-efficient of friction between the 2 blocks is μ1 and that between the bigger block and ground is μ2. Could someone help ...
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353 views

Trouble with calculating Christoffel symbols of FLRW metric using Lagrangian method

The FLRW metric which I am using is $$ds^2 = dt^2 - \frac{a(t)^2}{c^2} \left( dx^2 + dy^2 + dz^2 \right)$$ where $a(t)$ is the so-called 'scale factor'. I did not want to calculate the Christoffel ...
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1answer
248 views

Euler-Lagrange for constrained system

Suppose we have Euler-Lagrange system with generalized coordinate $r_1$ and $r_2$, and input $u_1$ and $u_2$. I know how to prove this system is indeed Euler-Lagrange system. Suppose now if we have a ...
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1answer
720 views

Does Action in Classical Mechanics have a Interpretation? [duplicate]

Possible Duplicate: Hamilton's Principle The Lagrangian formulation of Classical Mechanics seem to suggest strongly that "action" is more than a mathematical trick. I suspect strongly ...
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2answers
109 views

How would gravitons couple to the Stress-Energy tensor?

How would gravitons couple to the Stress-Energy tensor $T^{\mu\nu}$? How did physicists arrive at this result? I've read that it follows from the analysis of irreducible representations of the ...
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2answers
364 views

Why is ${\partial^i}{\partial_i\phi}$ = ${\partial^i {\phi}}{\partial_i{\phi}}$?

This notation can be found on page 254 of Victor Stenger's Comprehensible Cosmos and in David Tong's Lectures on QFT (Equation 2.4 http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf), and in EDIT: on ...
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175 views

Adding a total time derivative term to the Lagrangian

This is proof that $L'$ represents same equation of motion with $L$ through Lagrange eq. I understand $L'$ satisfies Lagrange eq, but how does this proof mean $L'$ and $L$ describe same motion of ...
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1answer
96 views

Why does $\frac{d\tau}{d\sigma} = L$?

I am given a (3+1)-dimensional spacetime that has the line element \begin{equation} ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1} dr^2 + r^2 d\phi^2 \end{equation} ...
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3answers
69 views

In Orbital Mechanics what is the quantity described below called?

I seem to recall that $r^2 \dot{\theta}$ is a conserved quantity in orbital mechanics, which I just proved using the Euler-Lagrange equations. Namely via: $ \mathcal{L} = \frac{m}{2} (\dot{r}^2+r^2 ...
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1answer
2k views

What is the Lagrangian from which the Klein-Gordon equation is derived in QFT?

Is there a well-known Lagrangian that, writing the corresponding eq of motion, gives the Klein-Gordon Equation in QFT? If so, what is it? What is the canonical conjugate momentum? I derive the same ...
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2answers
377 views

What is the relativistic action of a massive particle?

all Lorentz observers watching a particle move will compute the same value for the quantity $$ds^2 = -(c \, dt)^2 + dx^2 + dy^2 + dz^2,$$ $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu},$$ and ''ds/c'' is then ...
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3answers
187 views

Constructing Supersymmetric Lagrangians

It is a very trivial doubt but somehow I am not able to figure it out. While constructing a supersymmetric lagrangian we always even number of fermionic fields. One reason is of course the product ...
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2answers
460 views

Geodesics and trajectories

I'm a mathematician studying Arnold's Mathematical Methods of Classical Mechanics. On p. 83 the following definition is given. Let $M$ be a differentiable manifold, $TM$ its tangent bundle, and ...
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2answers
87 views

Phase space Lagrangian?

Reading out of this lecture series we define a phase space Lagrangian $\mathcal L$ to be a function of $4n+1$ variables namely $q,\dot q,p,\dot p,t$. My question is, what space is this function ...
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2answers
112 views

Why does the classical electrodynamics Lagrangian density equation have a “field” term and an “interaction” term?

On Wikipedia's page on classical electrodynamics, they state the Lagrangian density equation as follows \begin{equation} \mathcal{L} = \mathcal{L}_{\text{field}} + \mathcal{L}_{\text{int}} = ...
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1answer
255 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? What is the problem that we do not consider such Lagrangian densities?
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2answers
105 views

With respect to what quantities do I vary Lagrangians in field theory?

I have recently been wondering, with respect to which quantities (covariant or contravariant) one should vary QFT Lagrangians and whether there is some rule regarding this. Let me give an example ...
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3answers
331 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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1answer
3k views

Derivation of Dirac equation using the Lagrangian density for Dirac field

How can I derive the Dirac equation from the Lagrangian density for the Dirac field?
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3answers
517 views

Must the Lagrangian always be known for the Euler-Lagrange equations to be of any use?

When studying classical mechanics using the Euler-Lagrange equations for the first time, my initial impression was that the Lagrangian was something that needed to be determined through integration of ...
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1answer
148 views

Gravity in $d$ spacetime dimensions

Given the following action $$S=\frac{1}{16\pi G}\int d^4x \sqrt {-g}(R+aR^2+bR_{\mu\nu}R^{\mu\nu}+cR_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}),$$ which is in 4D. How to we generalise this ...
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1answer
143 views

Subtlety in derivation of Noether's theorem by Di Francesco

In the book 'Conformal Field Theory' by Di Francesco et al, a derivation of Noether's theorem is demonstrated by imposing that, what I believe is said to be a more elegant approach, the parameter ...
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1answer
83 views

$\cos^{2}(\phi)$ in the kinetic energy term of the Lagrangian is one?

I'm doing some homework in Classical Mechanics, and is about to write out the Lagrangian of a system. But, when I check the answer from my teacher, something is missing. The kinetic energy I'm using ...
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2answers
245 views

Equations of motion from the Standard Model

For some time now I have been wondering if you could not derive any sort of equations of motion from the Standard Model: ...
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2answers
312 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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2answers
359 views

Lagrangian and hamiltonian of interaction

How to prove that lagrangian of interaction is equal to hamiltonian of interaction with minus sign? For example, I can't prove it for special case - quantum electrodynamics.
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412 views

Why lagrangian is negative number?

In the special relativistic action for a massive point particle, $$\int_{t_i}^{t_f}\mathcal {L}dt,$$ why is the Lagrangian $$\mathcal {L}=-E_o\gamma^{-1}$$ a negative number?
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2answers
319 views

Can cos(x) or sin(x) be the function of stationary action?

Is there a way to express $\cos(x(t))$ (or $\sin(x(t))$) as the solution to the Euler-Lagrange equation, in other words is there a sense in which this function is the path of stationary action?
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1answer
584 views

Relation 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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1answer
66 views

Classical spin viewed as $SU(2)$

In which sense is the configuration variable of a classical spin $SU(2)$? I can view a classical spin as a unit vector in $\mathbb{S}^2$ (2-dim. sphere), but it seems it is really given by a matrix ...
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1answer
108 views

Differentiating the Lagrangian to find geodesic equations?

I'm stuck pretty much at the first hurdle trying to follow the derivation of the geodesic equations from the Lagrangian ...
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3answers
183 views

Prerequisites for classical mechanics by Susskind

So I am an undergraduate in Electrical Engineering. We had a course on Physics in our freshman year which is equivalent to Classical Mechanics I as taught in MIT. I am interested in studying advanced ...
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1answer
78 views

Practical Book on Hamiltonian and Lagrangians? [duplicate]

Are there any terse, accessible books that are geared specifically at learning these two formalisms and how to effectively use them? So far I've only see either topic introduced as a part of another ...
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1answer
284 views

Classical action of the simple harmonic oscillator

I have been calculating the classical action of the harmonic oscillator, the problem I have is that I am only able to solve it if I set the integration limits of the action integral to be $t=T$ and ...
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1answer
150 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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1answer
80 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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1answer
254 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
511 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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1answer
471 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 ...