Questions tagged [lagrangian-formalism]
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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Derivation of Einstein-Cartan (EC) action for parametrized connection $A$ & introduction of torsion
I have some trouble with one missing step when I want to get the teleparallel action from general EC theory, which I am not fully understanding.
The starting form of action is:
$$
S_{EC}=-\frac{1}{8\...
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Cubic 1st-order Lagrangian for nonlinear harmonic oscillation [closed]
I am considering the nonlinear oscillators and dedcide to work for the higher order form. And some books writes the 3rd order of Lagrangian of the nonlinear harmonic oscillator in this way:
$$L=\frac{...
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Centrifugal Governor Question [closed]
I've been working through Hand and Finch's Analytical Mechanics and have just attempted this question:
My attempt at a solution is as follows:
First, find the kinetic energy of the two masses $m$ by ...
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Find curve minimizing energy loss due to friction [closed]
I am looking for an ansatz of the following problem:
Given a mass $m$ moving in a constant gravitational field along curves $C$ connecting two fixed points I want to find the curve $C_0$ that ...
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Energy-momentum tensor and equation of motion in Einstein-Dilaton theory
I am following this paper (see eq. 19-22) and trying to derive the equation of corresponding to Einstein-Dilaton gravity (ignoring the Maxwell part for now)
\begin{align}
S_{\text{E-D}} = \int d^4 ...
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QFT introduction: From point mechanics to the continuum
In any introductory quantum field theory course, one gets introduced with the modification of the classical Lagrangian and the conjugate momentum to the field theory lagrangian (density) and conjugate ...
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Independence of the equations resulting from the action principle $\delta (I_{\text{gravity}} + I_{\text{other fields}}) = 0$
In Dirac's "GTR", Chap. $30$, he discusses the "comprehensive action principle" and shows that variation of the combined action of the Hilbert-Einstein action plus all other matter-...
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How do I obtain the low energy supergravity actions from the 5 superstring theories?
In Domain-Walls and Gauged Supergravities by T.C. de Witt, there is a small table giving the 5 string theories and each of their effective sugras. I am looking for detailed reviews of how these sugras ...
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Derivation of the Noether current (Gauss law operator) in anomalous chiral gauge theory
I am reading Fujikawa-Suzuki's Path Integrals and Quantum Anomalies, §6.3. The Lagrangian I am looking at is
\begin{equation}
\mathcal{L}=-\frac{1}{4g^2}\left(\partial_\mu L_\nu^a-\partial_{\nu}L_\mu^...
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
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Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
In Lagrangian mechanics the momentum is defined as:
$$p=\frac{\partial \mathcal{L}}{\partial \dot q}$$
Also we can define it as:
$$p=\frac{\partial S}{\partial q}$$
where $S$ is Hamilton's principal ...
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Prerequisites for studying Lev Landau Mechanics vol. 1 [closed]
Lev Landau Mechanics vol. 1 dives directly into Lagrangians and Hamiltonians. What do you think are the prerequisites in order to study and grasp it?
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Unitary Gauge Removing Goldstone Bosons
The Lagrangian in a spontaneously broken gauge theory at low energies looks like
$$ \frac{1}{2} m^2 ( \partial_\mu \theta - A_\mu )^2 $$
and the gauge transformations look like $\theta \rightarrow \...
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How can the stress-energy tensor $T^{μν}$ be unique when the Lagrangian $L$ is not?
A relation (or definition) between the stress-energy tensor and the Lagrangian in GR is routinely seen:
$$T^{μν} = -2 \frac{∂L}{∂g_{μν}} - g^{μν} L \quad\quad(*)$$
(or some variation of this, ...
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Volume preserving transformation in the Circular Restricted Three-Body problem
the Lagrangian of the planar circular restricted three-body problem in the rotating coordinate frame is:
$$\mathcal{L}(x,y,\dot{x},\dot{y})=\frac{1}{2}(\dot{x}-\Omega y)^2 + \frac{1}{2}(\dot{y}+\Omega ...
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Robin conditions from action principle
Consider the Lagrangian density
$$L(\tilde{\phi}, \nabla \tilde{\phi}, \tilde{g}) = \tilde{g}^{\mu \nu} \nabla_{\mu} \tilde{\phi} \nabla_{\nu} \tilde{\phi} + \xi \tilde{R} \tilde{\phi}^2$$
with $\...
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How to add a non-chiral lepton doublet to the Standard Model?
How would the Standard Model Lagrangian (before symmetry breaking) change if we were to add a non-chiral lepton doublet $\ell_{L,R}$ with weak hypercharge $y=-\frac{1}{2}$ to the $SU(2)\times U(1)$ ...
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Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?
This is from Analytical Mechanics by Louis Hand et al. The proof is about Maupertuis' principle. The author seems to say that Hamilton's principle allow a path to have both a process of time forward ...
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Stress-energy tensor in terms of the Lagrangian [closed]
In Dirac's "General Theory of Relativity" (Chap 30) he gets
$$T^{μν} = -\frac{2}{√} \frac{∂\mathscr{L}}{∂g_{μν}}$$
where $\mathscr{L}$ is the Lagrangian density and $√$ means $\sqrt{-g}$.
$\...
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How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
I want to experiment with this relation (from Dirac's "General Theory of Relativity"):
$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$
using the electromagnetic Lagrangian $L = -(...
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Action-angle variables for three-dimensional harmonic oscillator using cylindrical coordinates
I am solving problem 19 of ch 10 of Goldstein mechanics. The problem is:
A three-dimensional harmonic oscillator has the force constant k1 in the x- and y- directions and k3 in the z-direction. Using ...
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Possible boundary conditions in derivation of Euler-Lagrange equations
Given a Lagrange density
$$\mathcal{L} = g^{ij} \phi_{,i} \phi_{,j} - V(\phi)\tag{1}$$
I have read (e.g. here) that the boundary term that occurs through variation of the action
$$ \delta I = \int_V ...
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2
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Which higher-order terms require 4th-order integration of quadratically-constrained dynamics?
I was interested in demonstrating the notion of geodesics in constrained motion and prepared the calculation of force-free motion on the unit sphere, following Hertz' take on mechanics. Since the ...
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1
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Derivation of Noether Current in Condensed Matter Field Theory by Altland and Simons
In Section 1.6 of Condensed Matter Field Theory by Altland and Simons, they prove Noether's theorem. In order to do so, they consider an infinitesimal transformation of the coordinates and the field:
$...
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How to do Variational Principle in QFT? ($SU(2)$-Yang-Mills)
I am currently familiarizing myself with QFT and have a question about this article. My understanding is that the Lagrangian density in (2) couples my gauge fields to the Higgs field. And with ...
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Vanishing virtual work done by non-holonomic constraints
I was reading classical mechanics by NC Rana. I was reading a topic on vanishing virtual work done due to constraint forces. How do you prove that the virtual work done by non-holonomic constraint ...
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Second-order equations of motion for higher derivative gravity?
We know that Lovelock gravity is the most general theory of gravity possible for Lagrangians which depend only on the metric tensor and the Riemann tensor
\begin{equation*}
L = L \left(g_{\alpha\beta},...
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Uniqueness of Maxwell Lagrangian: Why does it not include the term $c_3 (\partial_\mu j_\nu)F^{\mu \nu}$?
In the textbook Condensed Matter Field Theory by Altland and Simons, it is said that the Maxwell Lagrangian $\mathcal{L}$ coupled to a four-current $j^\mu$ satisfying $\partial_\mu j^\mu = 0$ is the ...
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Plugging solution into Lagrangian [duplicate]
Consider a simple ODE in $x \equiv x(t)$, e.g.
$$\ddot{x} = -k x \quad \Longrightarrow \quad x(t) = e^{ik t}.$$
This system's Lagrangian is
$$L = \frac{1}{2} \dot{x}^2 - \frac{k}{2} x^2.$$
Knowing the ...
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Generalized momentum
I am studying Hamiltonian Mechanics and I was questioning about some laws of conservation:
in an isolate system, the Lagrangian $\mathcal{L}=\mathcal{L}(q,\dot q)$ is a function of the generalized ...
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