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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questions with no upvoted or accepted answers
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What properties make the Legendre transform so useful in physics?
The Legendre transform plays a pivotal role in physics in its connecting Lagrangian and Hamiltonian formalisms. This is well-known and has been discussed at length in this site (related threads are e....
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How to show the Gauss-Bonnet term is a total derivative?
It is well-known that the Gauss-Bonnet term
$$\mathcal L_G =R^2 -4 R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\tag 1$$
do not contribute to equations of motion when adding it to the ...
user146681
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Equation of motion for cyclic model of the universe
I recently started to study the cyclic universe. I came across this article [1]. My question is about the action used for describing the cyclic model:
$$S = \int d^{4}x\sqrt{-g}(\frac{1}{16\pi G}R-\...
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Systematically constructing a Lagrangian with given Poincare representations for particles
In one approach to constructing field theories, we begin with some desired particle content (e.g. a massive spin-1 particle), and then we construct a corresponding classical Lagrangian (e.g. the Proca ...
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Bosonization and supersymmetry
In 2D (time + space) there is no notion of statistic. So particles can be described in terms of bosonic and fermionic fields.
Well-known example is Thirring/Sine-Gordon duality. There are also some ...
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What is Noether's Second Theorem?
I have been unable to find a short statement of Noether's second theorem. It would be helpful to have the following:
A short mathematical statement of the theorem. Does it imply a conservation law ...
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Euler-Lagrange equations for chain fountain
Most of us are familiar with chain fountains.
I was wondering how this phenomenon is explained in the Lagrangian mechanics. I mean do we know how the Euler-Lagrange equations look like for this system?...
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Variational principle with $\delta I \neq 0$
In Covariant Phase Space with Boundaries D. Harlow allows boundary terms in the variation of the action. If we have some action $I[\Phi]$ on some spacetime $M$ with boundary $\partial M = \Gamma \cup \...
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Lagrangians in field theory and ignorance
The thing that has always bothered me while taking my QFT course was the seemingly arbitrary nature of Lagrangians. For the Klein Gordon equation we just wrote down the simplest Lorentz invariant ...
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Writing the EL equations in the language of differential geometry
I want to explore generalised Noether currents obtained from $q$-form symmetries in an action.
The regular way we obtain Noether currents is fairly straightforward: We have a 0-form symmetry $\phi \to ...
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Variations of Boundary Actions and Bulk Physics
In physics, we are often taught that the action principle generates only bulk equations of motion on-shell and that boundary terms can be neglected provided the fields in question fall off ...
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Stueckelberg mechanism in path integrals
Suppose we have some gauge invariant Lagrangian $\mathcal{L}_0$ depending on $A$ and some matter fields $\psi$, and we add a mass term for $A$.
$$\mathcal{L}[A,\psi]=\mathcal{L}_0[A,\psi]+m^2A^2$$
...
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Faddeev-Popov Ghosts in the canonical formalism
In the Lorenz gauge in electrodynamics, the timelike and longitudinal components can be eliminated by prescribing the Gupta-Bleuler condition $\partial^{\mu}A_{\mu}|\Psi\rangle=0$ on physical states. ...
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How can I see where this formula for a general vertex factor comes from?
I have been reading Srednicki from the beginning and doing all the exercises, and I hit a big roadblock at Q10.4, as I can't seem to figure out what Srednicki is doing in his solution. Luckily, I ...
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Is there a modified Least Action Principle for nonholonomic systems?
We know that one can treat nonholonimic (but differential) constraints in the same manner as holonimic constraints. With a given Lagrange Function $L$, the equations of motion for a holonomic ...
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Coupling a spinor field to a preexisting scalar field?
So I'm not a physicist, but I'm thinking about a mathematical problem where I think physical insight might be useful.
We're working on a Riemannian manifold $(M,g)$ (positive definite metric) with a ...
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What is the link between free energy and lagrangian?
Free energy is a generalization of energy when the system exchanges heat with the environment. Energy, in its turn, can be extracted from lagrangian under the symmetry of time $(\frac{\partial L}{\...
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General equation of motion for elementary particles
Elementary particles can be grouped into spin-classes and described by specific equations, see below:
Is there a general Lagrangian density from which all these equations can be derived?
A ...
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Deriving wave equation of string without approximation
When deriving the equation for a standing wave of a string, we often approximate that the tension at all points in the wave is constant. but I want to derive the equation without the approximation. I ...
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Is there a Lagrangian $L$ (equivalently an action functional $S$) which yields the Navier-Stokes equation?
The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws.
However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\...
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What are the symmetries in fermionic quantum mechanics?
Consider a $d=0+1$ theory of fermions, i.e., fermionic QM:
$$
L=i\psi\partial_t\psi-V(\psi)
$$
The Hamiltonian is just $H=V$.
What is the definition of a symmetry here? I can construct transformations ...
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When to use (and when not to use) electromagnetic field conjugates in variational formulations
I found something a little bit confusing about writing variational formulas or Lagrangians for electromagnetic fields. I was looking at the book by Schwinger and Milton (chapter 4), and saw that ...
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Actions for relativistic point-particles of higher spin
To describe the behavior of a relativistic point-particle, we have the standard action
$$S=\int d\tau \bigg[\frac{1}{e} \dot X^\mu\dot X_\mu +m^2 e\bigg], $$
where $e$ is the worldline einbein. Then, ...
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Resources on BRST and BV quantisation for local quantum field theories
This is a reference request, to ideally a textbook, monograph, set of lecture notes or lecture videos, on the topics of BRST quantisation and the Lagrangian BV formalism. My constraints are as follows:...
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Importance of an extra total derivative term in Liouville theory
In this paper on boundary Liouville theory, the authors have introduced an extra term, $-\partial_{\sigma}^2\phi$, (the last term in the equation below) in defining the stress tensor of the Liouville ...
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Is there a general formula to translate from *canonical* to *physical* momentum?
In Peskin and Schroeder, after having derived a conserved tensor $T^{\mu \nu}$ associated with translations in space-time (the stress-energy tensor), it is said that the charges $\int d^3 x T^{0i}$: $$...
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Hamiltonian Operator for nonrenormalizable Effective Field Theories?
Assuming we have a Effective Field Theory, for example a Real Scalar Field Theory, defined through a Lagrangian density of the form
$\mathcal{L}_{eff} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi - ...
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Intuition behind the principle of virtual work
To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset \mathbb{R}^...
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Naive questions on the classical equations of motion from the Chern-Simons Lagrangian
Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial _0a_i-...
- 3,604
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Noether's Theorem in non-conservative systems
In most books on classical mechanics, Noether's Theorem is only formulated in conservative systems with an action principle. Therefore I was wondering if it is possible to also do that in non-...
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Effective field theories in curved spacetime
Loosely speaking, in flat spacetime, one defines the effective Lagrangian by writing down all possible operators compatible with the symmetries and suppressed by some energy scale, and one usually ...
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Classical "bird flocking" Hamiltonian with velocity-velocity interaction
Consider the following classical Lagrangian with an interaction between velocities:
$$\mathcal{L} = \sum_{i} \frac{1}{2}m \mathbf{v}_{i}^{2} + \sum_{i < j} J(r_{ij}) \hat{\mathbf{v}}_{i} \cdot \hat{...
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Torsion-free Einstein-Cartan action and Hilbert-Einstein action equations of motion equivalence
Let $e^\alpha_{\ \mu}$ be the tetrad i.e.
\begin{equation}
g_{\mu\nu} = \eta_{\alpha\beta}e^{\alpha}_{\ \mu}e^{\beta}_{\ \nu}
\end{equation}
I'm denoting the internal indices using greek letters $\...
- 351
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First order quantum string action
Considering this post: Quantum String action the action given is of the lowest order but the effective action, for low energies, is given by:
$$ S_{ef.}= -\frac{1}{2k^2} \left( S^{(0)}+ \alpha S^{(1)} ...
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Infinitesimals and calculating action for an infinitesimal time interval (path integral formalism in QM, Feynman and Hibbs)
I am going through Feynman and Hibb's emended edition of Quantum Mechanics and Path Integrals.
Going through chapter 2 made me realise I didn't fully understand infinitesimals, I was hoping someone ...
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How there can be an explicit coordinate dependence on the Lagrangian, if this arises from a Lagrangian density?
I have a very simple question, but strangely I cannot find any answer on the internet; maybe the answer is too simple that I don't notice. I go straight to the point: if I define a Lagrangian from a ...
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Symmetries in quantum field theory and anomalies
Suppose we have a lagrangian quantum field theory, thus a theory where we can write an action in the form
\begin{equation}
S = \displaystyle \int d^4 x \; \mathcal L \, \left( \partial_{\mu} \phi , \...
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Cases of various time symmetries
Is it possible to cook up three physically relevant examples where the Lagrangian has explicit time dependence but the system still has one of the following?
time-reversal invariance,
time ...
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Field strength renormalization and the energy-momentum tensor
This question is about the connection between the energy-momentum tensor, dilation transformations, and field renormalization.
From a Wilsonian perspective on renormalization we start out with a ...
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Deriving the Lagrangian of a set of interacting particles only from symmetry
In section 5 of Landau and Lifshitz's Mechanics book, they show that the Lagrangian of a free particle must be proportional to its velocity squared, $\mathcal{L} = \alpha\mathbf{v}^2$ using only ...
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Lagrangian for system of particles with statistical distribution $f(x_1, ..., x_N)$
For system of $N$ particles it is known that it is a good model to take Lagrangian to be (ignoring electromagnetism) $$L = \sum \limits_{i=1}^N \frac{1}{2}(m_i \mathbf{v}_i^2) -U(\mathbf{x}_1, ..., \...
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Fixed coordinate frame as limit of rotating coordinate frame
I have a question about a fixed coordinate system as limit of rotating system. Consider for example a pendulum. The Lagrangian in the rotating frame is given by
\begin{equation}
L(\mathbf{r}, \mathbf{\...
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Lagrangian of Phonon-photon
A quite interesting but also hard problem are Polaritons. As far as I have understand the concept it's about phonons coupling to light. The Lagrangian function should therefore have a term for the ...
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Equilibrium points of three masses on a rigid spring ring with gravity
I'm trying to find the equilibrium points of a given system using Lagrangian mechanics (the system is still not rotating at the beginning).
should I find the diagonal matrix for the characteristic ...
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Lagrangian of the Euler equations - why are Lin constraints required?
The following equation describes the motion of a rigid body rotation, such as a gyroscope:
$$
\frac{d\textbf{L}}{dt} ={\bf{\tau}}= \textbf{r}\times m\textbf{g}= {\omega}\times \textbf{L}$$
where $...
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Conserved current of $O(N)$ vector model
I was reading a paper by Klebanov and Polyakov, Phys.Lett. B550 (2002) 213, (hep-th/0210114). In the paper, where they are discussing free O(N) vector models, they state that the theory has a class of ...
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Motivation for Non-Abelian Gauge Invariance
I have a very similar question to the one asked below:
Why are non-Abelian gauge theories Lorentz invariant quantum mechanically?
In particular, the setup to my question is essentially the same: ...
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Decomposition of rank-2 field and local interactions
Any rank-2 tensor can be decomposed in the following way
$$
\phi_{\mu\nu} =\phi_{\mu\nu}^{TT} + \partial_{(\mu}\xi_{\nu)} +\frac{1}{4}T_{\mu\nu}s+\frac{1}{4}L_{\mu\nu}(w-3s)
$$
where $\phi_{\mu\nu}^{...
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2
answers
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What's the path of least action for fermions off-shell?
The Lagrangian of fermions is first order both in space-derivatives and time-derivatives. In the variation of the action one usually fixes both the initial point and end point. I have the following ...
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Derivation of Feynman Rules for a $\frac{1}{\phi}$ potential
The question is more mathematical in nature. If one had a potential $V(\phi) = \frac{\lambda}{\phi}$, where $\lambda$ is a constant, then how does one derive the Feynman rules for this scalar field's ...
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