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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Noether charge on complex scalar field

For complex scalar field, we write the Lagrangian as: $$ \mathcal{L}=\partial_{\mu}\phi^{*}\partial^{\mu}\phi-m^2 \phi^{*}\phi $$ with the $U(1)$ symmetry, and under infinitesimal transformation: $$ \...
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Rayleigh's Dissipative Function in Lagrangian Mechanics

What exactly is the derivation of Rayleigh's dissipative function? How does one know what to assume the function to be while dealing with a problem in Lagrangian mechanics? These two questions have ...
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Independence and ambiguity of holonomic constraints

I've got a couple of questions concerning constraint equations: Suppose I've got $n$ holonomic constraint equations for a particle, how can I be sure those are all the ones there are and I didn't ...
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Relativistic Euler-Lagrange equation

I am confused from the equation 6, why we get Euler-Lagrange equation from equation 8 but not from equation 6? Why we need to use $\zeta$ as invariant parameter in equation 8 even we already have ...
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Classical Mechanics Lagrangian from Underlying Quantum Field Theory

Does the K - T classical mechanics Lagrangian emerge from some structure of the Lagrangian of the underlying QFT?
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Geodesics Equation from Lagrangian [duplicate]

In the book Introduction to General Relativity Blackholes and Cosmology by Yvonne Choquet-Bruhat, she defines the length of a causal curve as $$\ell\equiv \int_a^b \left( -g_{\alpha \beta} \frac{d \...
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How to get the Magnetic Force from the Electromagnetic Tensor using Hodge Decomposition?

The following notation is used below: d: exterior derivative $\delta$: codifferential (adjoint of d) $\times$: skew-symmetric operator of a $\mathbb{R}^3$-vector $\nabla\times$: Curl operator in ...
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Deducing equation of motion for a free particle using the form of the Lagrangian

This is in reference to the question: Deriving the Lagrangian for a free particle My question is specifically in regards to QMechanic's answer to this question, and I have quoted the relevant part of ...
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Field shift in free Klein-Gordon theory

I am reading Peskin & Schroeder Ch9 and am stuck on a calculation going from equation 9.36. The problem is essentially a change of variable of a Klein-Gordon field. Beginning, we have an integral ...
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How did Noether use the total time derivation to get her conservation of energy? [duplicate]

I was informed by @hft that by combining the total time derivation: $$\frac{dL}{dt} = \frac{\partial L}{\partial x}\dot{x} + \frac{\partial L}{\partial \dot{x}}\ddot{x} + \frac{\partial L}{\partial t}$...
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When is a Variation a *Small* Variation in the Calculus of Variations? [closed]

This question is related to a derivation in the famous book by Gelfand and Fomin, pp. 54-57, see https://archive.org/details/gelfand-fomin-calculus-of-variations/page/54/mode/2up To calculate the ...
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Why do we apply a Legendre transform to the Lagrangian in the first place? [duplicate]

I understand how the legendre transform of the Lagrangian with respect to $\dot{q}$ yields the hamiltonian, but I do not understand why one would think to do this in the first place? what is the ...
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Variation of normal, coordinate and mean curvature with respect to metric

Using the divergence theorem, we can compute volume by integrating along the surface: $$\mathrm{vol}(M)=\int_M\mathrm{d}V=\oint_{\partial M}\mathrm{d}S \,\vec{n}\cdot\vec{v},$$ where $\vec{n}$ is ...
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Why is the action guranteed to have unique extrema in classical mechanics? [duplicate]

Reading any classical mechanics book which introduces the Lagrangian formalism of mechanics, a one particle system is introduced to show that we obtain the euler-lagrange equations from Newton's ...
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Why is the action guranteed to have one unique extrema in classical mechanics? [duplicate]

Reading any classical mechanics book which introduces the Lagrangian formalism of mechanics, a one particle system is introduced to show that we obtain the euler-lagrange equations from Newton's ...
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What is the gauge symmetry of wave equations in curved spacetime?

The lagrangian of electroweak interaction is invariant under $SU(2)\times U(1)$ gauge transformations, QCD lagrangian is invariant under $SU(3)$ gauge transformations. There're lagrangians of ...
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Are the sources in QFT just particles?

I'm reading A. Zee's Quantum Field Theory in a Nutshell, where he introduces QFT using path integral formulation. One thing that I'm not sure I got correctly is this: Zee adds a source term to the ...
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What's the difference between these two lagrangian? [closed]

The lagrangian for scalar field is defined as, $$L=L(\phi,\partial_{\mu}\phi,\partial_{\mu}\partial_{\nu}\phi)\tag{1}$$ $but$ there is also another lagrangian which is defined as, $$L=L(\phi,\nabla_{\...
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Euler-Lagrange Equation, $\dot{P_{\alpha}}\neq\partial_{q_{\alpha}}L$

We know \begin{equation} \frac{d(\partial_{\dot{q}_{\alpha}}T) }{dt}=\partial_{q_{\alpha}}(T-U) \\ \partial_{\dot{q}_{\alpha}}U=0 \rightarrow\dot{P_{\alpha}}=\partial_{q_{\alpha}}L \end{equation} ...
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Variation of functional with area and volume term

Let $M$ be a closed manifold in $\mathbb{R}^3$ and $\partial M$ its surface. I want to find (in general terms) the manifold that minimizes a functional of the form $$I[M]=\int_{\partial M}f\,\mathrm{d}...
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What is the lagrangian for a collection (lattice) of harmonic oscilators?

I am a self-learner so I probably missed some exercises in classical mechanics. Anyhow, I'm learning QFT from A. Zee's book "Quantum Field Theory in a Nutshell", and on page 4, he wrote down ...
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Do Legendre transformation form a group?

In my classical mechanics class, my professor asked if Legendre transformations form a group, and in my little knowledge about groups, I know that a transformation group consists of a set of ...
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Does nature perform optimal linear quadratic control?

Given any linear quadratic control problem $$\min \sum_t c_x(x_t, t) + c_u(u_t, t)$$ where $u_t$ is a "control variable" (think of it as an adjustable velocity) and $x_t$ is a "state ...
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Hamiltonian and Lagrangian for a particle on a ring [duplicate]

In the book Condensed Matter Field Theory (A. Altland & B. Simons)(page 498, 2nd edition) they suggest the following Hamiltonian and Lagrangian for a particle on a ring in the presence of a ...
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Is the action of any interacting 6D CFT known?

"6D super-conformal (2,0) CFT" is predicted to exist as an $AdS_7\times S_4/CFT_6$ dual of M-Theory. My question is do we have any Lagrangians for a 6D CFT? The conformal theories I know ...
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Is there an hamiltonian formulation of $D_p$-brane theory?

I'm aware that the dynamics of $D_p$-branes can be studied using the DBI action: $$S_{DBI}=-T_p\int d^{p+1}\xi\sqrt{-\det(g_{\alpha\beta})}$$ (with external fields set to zero). I'd like to know ...
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Relativistic Particle theory [duplicate]

I have read relativistic field theory and non-relativistic particle theory. My question is How to derive Lagrangian for Relativistic Particle theory (not field).
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Varying the complete electromagnetic action

To obtain the Lorentz force law, it is sufficient to vary: $$S=\int d\tau \sqrt{\eta_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau}}+A_{\rho}(x)\frac{dx^{\rho}}{d\tau}{d\tau}$$ and get: $$\frac{d^...
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Path Integral of Photon

I am having issues recalling how to perform integration by parts for the path integral of the photon, namely the term, $$Z[J] = \int\mathcal{D}[A_\mu]\exp(i\int\mathcal{L}\:dx)$$ where $\mathcal{L} = -...
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What is the relation between the free energy and the action? More generally, what is the relation between Thermodynamics and Lagrangian Mechanics?

My question stems from the sentence said by my professor "The action is the free energy" which I don't understand. Thinking that probably I'm missing some key concepts, I'd like to know: ...
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2 answers
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Why did Noether use the Lagrangian for her conservation of energy theorem?

So I know that for Noether's conservation of energy theorem, the Lagrangian is used. However, I know that the Lagrangian doesn't always equal energy. So why did she use the Lagrangian and not other ...
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Why does the Lagrangian have $O(4)$ symmetry after Wick rotating (previously Lorentz symmetry)?

Pertaining to the answer within link. Why is it the case, that for Lorentz invariant Lagrangian $\mathcal{L}$, after Wick rotation, the $O(4)$ invariance is established, thus manifesting itself as ...
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How to determine these derivatives wrt. matrix-valued fields?

We know strength tensor for $W$-boson field $$W_{\mu\nu}=\partial_\mu W_\nu-\partial_\nu W_\mu+\frac{ig}{2}[W_\mu, W_\nu]$$ Where $[W_\mu, W_\nu]=W_\mu W_\nu-W_\nu W_\mu$ is Lie bracket/commutator and ...
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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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Does a constant in the action always have unobservable consequences in classical mechanics?

Background So in classical mechanics, my understanding is that for the action by using a the principle of least action one can get the equations of motion. Adding a constant to the action does not ...
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2 answers
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Klein-Gordon equation from general relativity?

I am trying to derive the Klein-Gordon equation from Einstein's field equation, since the energy momentum tensor for the Klein-Gordon equation is defined as: $$T^{\mu\nu} =\partial^{\mu}\phi \ \...
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Calculating conjugate momenta for a spin-2 field

Consider a symmetric spin-2 field $h_{\mu \nu}$. I have the following Lagrangian for this field: $$\mathcal{L} = - \frac{1}{4}\left(\partial_{\lambda}h_{\mu \nu} \text{ } \partial_{\phi}h_{\alpha \...
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Is there an infinite amount of conserved currents for a given finite symmetry?

Let's say we have a field $\phi(x)$ that gets transformed to $\phi(x, \epsilon)$ under some finite transformation. We also define $\phi(x,0)=\phi(x)$. If we Taylor expand our transformation we get: $$\...
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4 votes
1 answer
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Kalb-Ramond current fall-offs at future null infinity

I can couple the electromagnetic field to a current generated by the complex scalar field for example: $S=- \int d^4x \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + A_\mu J^\mu$ with $J_\mu = i(\partial_\mu \phi^...
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Expression of a Lagrangian in other form

I'm reading Matthew D. Schwartz, Quantum field theory and standard model and some question arises In his book, p.133, he says that Any vector field can be written as $$ A_{\mu}(x) = A^{T}_{\mu}(x)+ \...
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Equations of motion of $\mathcal{L}= - \frac{1}{4}F^{2}_{\mu \nu} - A_{\mu}J_{\mu}$ in momentum space

I'm reading the Matthew D. Schwartz, Quantum field theory and the standard model, p.128 and some question arises. Consider a lagrangian $\mathcal{L}= - \frac{1}{4}F^{2}_{\mu \nu} - A_{\mu}J_{\mu}$ ($...
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2 votes
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Charge of antifermions in 1+1d QED, and in 1+1 CED

NEW REPRHAISING OF THE QUESTION: Is it possible that in 1+1d QED the negative charge of the $\psi^\dagger$ fields appears only in the quantum model when we introduce the commutator relation for ...
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Should rotation of a rigid body confined to a sphere couse it to divert from a big circle?

Consider an axially symmetric body constrained to a unit sphere in such a way that the symmetry axis of the body is always normal to the sphere. Edit: I guess what I really wanted to ask (although it ...
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How fundamental Physics is constructed? [duplicate]

I was wondering, how do theoretical physicist arrive to such fundamental things like Lagrangians or actions. For example, the QED, action is given by: $$ \mathcal S_{QED} = \int_{\mathcal M} {\mathrm ...
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1 vote
1 answer
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Steps taken to differentiate action in wave equation [closed]

I'm currently reading Blundell and Lancaster's "Quantum Field Theory for the Gifted Amateur." In chapter 1, example 1.4, they talk about how the action and Lagrangian density ideas are super ...
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Solution of the two-body problem

I'm self-studying Lagrangian Mechanics using Goldstein's Classical Mechanics, supplemented with Lemos's Analytical Mechanics, a modern version of the same, and Landau and Lifshitz's Mechanics. ...
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Are self-loop diagrams zero if $L_\text{int}$ has a derivative term?

I was looking at the theory with interaction Lagrangian $L_\text{int}=\phi^3 \cdot \partial_{\mu}\phi$. I was computing the following self-loop diagram \begin{equation} \langle \phi_x \phi_z^3 \cdot \...
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Commutator of gauge field and the scalar field in the Stueckelberg Lagrangian with gauge-fixing terms

I was trying to add a gauge fixing term to Stueckelberg Lagrangian and cancel the mixing term between scalar $\chi$ field and vector $A_\mu$ field. $${\cal L}_{Stueckelberg} = -\frac{1}{4}V_{\mu\nu}V^{...
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How do take non-relativistic limit of this Lagrangian to obtain the Hamiltonian?

In this paper, it is claimed in equation (1) and (2) that when we take non-relativistic limit, the following Lagrangian (the interaction part) $$L=g \partial_{\mu} a \bar{\psi} \gamma^{\mu}\gamma^5\...
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Asymmetry of the indices in a Weyl transformation for the Polyakov action

The Polyakov action $${\mathcal {S}}={T \over 2}\int \mathrm {d} ^{2}\sigma {\sqrt {-h}}h^{ab}g_{\mu \nu }(X)\partial _{a}X^{\mu }(\sigma )\partial _{b}X^{\nu }(\sigma )$$ is invariant under the ...
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