What is Lagrangian density: Definition and 55 Discussions

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
One motivation for the development of the Lagrangian formalism on fields, and more generally, for classical field theory, is to provide a clean mathematical foundation for quantum field theory, which is infamously beset by formal difficulties that make it unacceptable as a mathematical theory. The Lagrangians presented here are identical to their quantum equivalents, but, in treating the fields as classical fields, instead of being quantized, one can provide definitions and obtain solutions with properties compatible with the conventional formal approach to the mathematics of partial differential equations. This enables the formulation of solutions on spaces with well-characterized properties, such as Sobolev spaces. It enables various theorems to be provided, ranging from proofs of existence to the uniform convergence of formal series to the general settings of potential theory. In addition, insight and clarity is obtained by generalizations to Riemannian manifolds and fiber bundles, allowing the geometric structure to be clearly discerned and disentangled from the corresponding equations of motion. A clearer view of the geometric structure has in turn allowed highly abstract theorems from geometry to be used to gain insight, ranging from the Chern–Gauss–Bonnet theorem and the Riemann–Roch theorem to the Atiyah–Singer index theorem and Chern–Simons theory.

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  1. P

    I Lagrangian density of a linearly elastic string under gravity

    This was inspired by this:Dropping an extended Slinky -- Why does the bottom of the Slinky not fall?. There is that famous demonstration of dropping a slinky, and the bottom of the slinky does not move until the center of mass reaches the bottom. I was trying to figure out how hard are the...
  2. G

    Deriving equations of motion of abelian gauge field coupled to scalar

    I want to compute the equations of motion for this theory in terms of the functions ##f## and ##a##. My plan was to apply the Euler-Lagrange equations, but it got confusing very quickly. Am I right that we'll have 3 sets of equations? One for each of the fields ##\phi,\phi^\dagger, A_\mu## ...
  3. W

    A Lagrangian density for the spinor fields

    hi, i have seen lagrangian density for spin 0 , spin 1/2, spin 1 , but i am not getting from where these langrangian densities comes in at a first place. kindly give me the hint. thanks
  4. StenEdeback

    Deduction of formula for Lagrangian density for a classical relativistic field

    Hi, I am reading Robert D Klauber's book "Student Friendly Quantum Field Theory" volume 1 "Basic...". On page 48, bottom line, there is a formula for the classical Lagrangian density for a free (no forces), real, scalar, relativistic field, see the attached file. I like to understand formulas...
  5. O

    How can negative integers be used in deriving the Hamiltonian for open strings?

    On ***page 38*** of Becker Becker Schwarz, we're given ***equation 2.69*** which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2})$$ Considering the open string we have...
  6. Paulpaulpa

    Spacetime translations and general Lagrangian density for Field Theory

    In Sydney Coleman Lectures on Quantum field Theory (p48), he finds : $$D\mathcal{L} = e^{\mu} \partial _{\mu} \mathcal{L}$$ My calulation, with ##\phi## my field and the variation of the field under space time tranlation ##D\phi = e^{\mu} \frac{\partial \phi}{\partial x^{\mu}}## ...
  7. Another

    Question about Lagrangian density

    this figure form ( https://en.wikipedia.org/wiki/Effective_mass_(spring%E2%80%93mass_system) ) massive spring ; m K.E. of total spring equal to ## K.E. = \frac{1}{2} \sum dm_i v_i^2 = \frac{1}{2} \sum \rho dy (Vy/L)^2## V is the speed at the end of the spring and V are same speed of mass M...
  8. JD_PM

    Get the equation of motion given a Lagrangian density

    a) Alright here we have to use Euler-Lagrange equation $$\partial_{\alpha} \Big( \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} A_{\nu})} \Big) - \frac{\partial \mathcal{L}}{\partial A_{\nu}} = 0$$ Let's focus on the term ##\frac{\partial \mathcal{L}}{\partial (\partial_{\alpha}...
  9. D

    I Why does a Lagrangian matter for the standard model?

    Hi, I don't know much about the standard model but I'm asking out of interest. Why do we actually need a Lagrangian for the standard model? Surely when you apply the relevant Euler-Lagrange equations, you end up with a variety of equations like the Maxwell equations or Dirac equations. Why...
  10. A

    I Einstein-Hilbert Action, Lagrangian Density & Vacuum Fields

    From the action ∫Ldt =∫ι√|g|d4x where |g| is the determinant of the metric .and ι the lagrangian density. For gravitational field why is this ι is replaced by the Ricci scaler R which yield field equations in vaccum.(Rij-1/2Rgij)=0 Is it that the lagrangian density corresponding to vacuum is the...
  11. K

    Energy-momentum tensor from a Lagrangian density?

    Homework Statement I want to be able, for an arbitrary Lagrangian density of some field, to derive the energy-momentum tensor using Noether's theorem for translational symmetry. I want to apply this to a specific instance but I am unsure of the approach. Homework Equations for a field...
  12. Gene Naden

    I Sum Maxwell Lagrangian 1st Term: Use Minus Signs?

    So the first term of the Lagrangian is proportional to ##{F_{\mu \nu}}{F^{\mu \nu}}##. I wrote out the matrices for ##{F_{\mu \nu}}## and ##{F^{\mu \nu}}## and multiplied at the terms together and added them up, but some of the terms didn't cancel like they should have. Should I have used minus...
  13. D

    Gauge invariance of lagrangian density

    The problem: $$\mathcal{L} = F^{\mu \nu} F_{\mu \nu} + m^2 /2 \ A_{\mu} A^{\mu} $$ with: $$ F_{\mu \nu} = \partial_{\nu}A_{\mu} - \partial_{\mu}A_{\nu} $$ 1. Show that this lagrangian density is not gauge invariance 2.Derive the equations of motion, why is the Lorentzcondition still...
  14. R

    Equation of motion of a Lagrangian density

    Homework Statement from the lagrangian density of the form : $$L= -\frac{1}{2} (\partial_m b^m)^2 - \frac{M^2}{2}b^m b_m$$ derive the equation of motion. Then show that the field $$F=\partial_m b^m $$ justify the Klein_Gordon eq.of motion. Homework Equations bm is real. The Attempt at a...
  15. R

    Derivation of the Eqation of Motion from Fermi Lagrangian density

    Homework Statement Hello, I am trying to find the equations of motion that come from the fermi lagrangian density of the covariant formalism of Electeomagnetism.Homework Equations The form of the L. density is: $$L=-\frac{1}{2} (\partial_n A_m)(\partial^n A^m) - \frac{1}{c} J_m A^m$$ where J...
  16. bleist88

    A The Lagrangian Density and Equations of Motion

    Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be...
  17. saadhusayn

    A Confusion regarding the $\partial_{\mu}$ operator

    I'm trying to derive the Klein Gordon equation from the Lagrangian: $$ \mathcal{L} = \frac{1}{2}(\partial_{\mu} \phi)^2 - \frac{1}{2}m^2 \phi^2$$ $$\partial_{\mu}\Bigg(\frac{\partial \mathcal{L}}{\partial (\partial_{\mu} \phi)}\Bigg) = \partial_{t}\Bigg(\frac{\partial \mathcal{L}}{\partial...
  18. B

    I The Lagrangian of a Coherent State

    How does one write a Lagrangian of a coherent state of vector fields (of differing energy levels) in terms of the the individual Lagrangians? I desperately need to know how to know to do this, for a theory of mine to make any progress. Please stick with me, if I didn't make sense just ask...
  19. mertcan

    I Gravitational Lagrangian density

    Hi, in gravitational theory the action integral is: I = ∫( − g ( x))^1/2 L ( x) d 4 x, but I do not know why there is a square root -g . Could you give me the proof of this integral? I mean How is this integral constructed? What is the logic of this? Thanks in advance...
  20. F

    I Why do we require locality in quantum field theory?

    In quantum field theory (QFT) from what I've read locality is the condition that the Lagrangian density ##\mathscr{L}## is a functional of a field (or fields) and a finite number of its (their) spatial and temporal derivatives evaluated at a single spacetime point ##x^{\mu}=(t,\mathbf{x})##...
  21. M

    How to find Lagrangian density from a vertex factor

    Homework Statement This isn't a homework problem, per se, in that it's not part of a specific class. That being said, the question I would like help with is finding a Lagrangian density from the vertex factor, $$-ig_a\gamma^{\mu}\gamma^5.$$ This vertex would be identical to the QED vertex...
  22. I

    I Why is there a -1/2 in the Lagrangian density for ω mesons?

    My question is why there is -1/2 for ω meson?
  23. K

    Derivative of lagrangian density

    i have a mathematical question which is quite similar to one asked before, still a bit different https://www.physicsforums.com/threads/derivative-of-first-term-in-lagrangian-density-for-real-k-g-theory.781472/the first term of KG-Lagrangian is: \frac{1}{2}(\partial^{\mu} \phi)(\partial_{\mu}...
  24. D

    Coincidence of spacetime events & frame independence

    I have been reading these notes: http://isites.harvard.edu/fs/docs/icb.topic455971.files/l10.pdf in which they claim that if two spacetime events are coincident in one frame of reference then they are coincident in all inertial frames of reference, thus spacetime events are absolute i.e. they...
  25. D

    What is the definition of Locality (in QFT)

    I am slightly unsure as to whether I have understood the notion of locality correctly. As far as I understand it locality is the statement that if two events occur simultaneously (i.e. at the same time) then no information can be shared between them (they are causally disconnected). Thus a...
  26. Strangelet

    Problem with Maxwell Lagrangian Density

    Homework Statement I have to expand the following term: $$\dfrac{1}{4} F_{\mu\nu}F^{\mu\nu} = \dfrac{1}{4} \left(\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}\right) \left(\partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}\right)$$ to get in the end this form...
  27. M

    How does one derive the Lagrangian densities used in QFT?

    I've been working through a qft book by Sadovskii (while I wait for my Peskin book to come in) and I've used some later chapters of Griffith's Into to Elementary Particles as an introduction to some qft. My issue with both of these is that, where in classical mechanics we have the Lagrangian...
  28. DOTDO

    Legendre Transformation of Lagrangian density ?

    Hi I began to study the basics of QED. Now I am studying Lagrangian and Hamiltonian densities of Dirac Equation. I'll call them L density and H density for convenience :)Anyway, the derivation of the H density from L density using Legendre transformation confuses me :( I thought because...
  29. teroenza

    Four Tensor Derivatives -- EM Field Lagrangian Density

    Homework Statement Given the Lagrangian density \Lambda = -\frac{1}{c}j^lA_l - \frac{1}{16 \pi} F^{lm}F_{lm} and the Euler-Lagrange equation for it \frac{\partial }{\partial x^k}\left ( \frac{\partial \Lambda}{\partial A_{i,k}} \right )- \frac{\partial \Lambda}{\partial A_{i}} =0...
  30. P

    Three level Feynman diagramas lagrangian density

    Hi, I am trying to figure out how to draw all the three level Feynman diagrams corresponding to this lagrangian density L = \frac{1}{2} \partial _{\mu} \phi \partial^{\mu} \phi - \frac{\mu^2}{2}\phi^2- \frac{\eta}{3!}\phi^3-\frac{\lambda}{4!} \phi^4+i \bar{\psi} \gamma _{\mu} \partial^{\mu}...
  31. D

    Derivative of first term in Lagrangian density for real K-G theory

    Hey guys, This is really confusing me cos its allowing me to create factors of 2 from nowhere! Basically, the first term in the Lagrangian for a real Klein-Gordon theory is \frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi). Now let's say I wana differentiate this by applying the...
  32. H

    Lagrangian density, regular Lagrangian, E&M

    Greetings, I have two semi-related questions. 1. When making the Lagrangian formalism of electrodynamics, why is it that we use the Lagrangian density \mathcal{L}, rather than the plain old regular Lagrangian L? Is this something that is necessary, or is it more that it is just very...
  33. F

    Energy-momentum tensor, lagrangian density

    Homework Statement I try to calculate the energy tensor, but i can't do it like the article, and i don't know, i have a photo but it don't look very good, sorry for my english, i have a problem with a sign in the result Homework Equations The Attempt at a Solution In the photos...
  34. S

    Deriving wave equation from Lagrangian density

    Hi, This is a worked example in the text I'm independently studying. I hope this isn't too much to ask, but I am stupidly having trouble understanding how one step leads to the other, so was hoping someone could give me a little more of an in-depth idea of the derivation. Thanks. Homework...
  35. G

    Connecting Lagrangian Density with Field Theory

    Now this is a bit of a mix of a math and a physics question, but I think it is best asked here. Assume we are given a Lorentzian manifold ##(Q, g)## together with a metric connection ##\nabla##. Naturally we define geodesics ##\gamma## via $$\nabla_{\dot \gamma} \dot \gamma = 0 \quad ,$$...
  36. D

    The Dirac Lagrangian Density: A Brief Overview

    Hi, This is probably a trivial question, but I just wanted to check my understanding. Is the following expression for the Dirac Lagrangian correct...
  37. BruceW

    Lagrangian density of Newtonian gravity

    Hi everyone! I've been thinking about a certain problem for a while now. And that is a Lagrangian formulation of Newtonian gravity. I know there is a Lagrangian formulation for general relativity. But I was hoping to find a Lagrangian for Newtonian gravity instead (for some continuous mass...
  38. D

    Lagrangian density for continuous distribution of matter

    The Lagrangian for a point particle is just L=-m\sqrt{1-v^2}. If instead we had a continuous distribution of matter, what would its Lagrangian density be? I feel that this should be very easy to figure out, but I can't get a scalar Lagrangian density that reduces to the particle Lagrangian in...
  39. J

    Calculating derivatives of a Lagrangian density

    Hey everyone, I wasn't really sure where to post this, since it's kind of classical, kind of relativistic and kind of quantum field theoretical, but essentially mathematical. I'm reading and watching the lectures on Quantum Field Theory by Cambridge's David Tong (which you can find here), and...
  40. S

    Derive Equation of motion using Lagrangian density?

    Homework Statement [/b] The attempt at a solution[/b] I have done the first bit but don't know how to show that phi(r,t) is a solution to the equation of motion.
  41. S

    Why is the Lagrangian density for fields treated as a functional in QFT?

    This is probably a minor point, but I have seen in some QFT texts the Euler-Lagrange equation for a scalar field, \partial_{\mu} \left(\frac{\delta \cal{L}}{\delta (\partial_{\mu}\phi)}\right) - \frac{\delta \cal L}{\delta \phi }=0 i.e. \cal L is treated like a functional (seen from the...
  42. T

    Lagrangian density for a complex scalar field (classical)

    Hi. Let's say we have a complex scalar field \varphi and we separate it into the real and the imaginary parts: \varphi = (\varphi1 + i\varphi2) It's Lagrangian density L is given by: L = L(\varphi1) + L(\varphi1) Can you tell the argument behind the idea that in summing the densities of...
  43. B

    Understanding the Lagrangian Density Dependence on Field Variables

    Hi, guys, Why do we assume Lagrangian Density only depend on field variables and their first derivative? Currently, I am reading Ashok Das's Lectures on Quantum Field Theory. He says (when he is talking about Klein-Gordon Field Theory): "In general, of course, a Lagrangian density...
  44. D

    Lagrangian Density, Non Linear Schrodinger eq

    Homework Statement Derive the Non-Linear Schrödinger from calculus of variationsHomework Equations Lagrangian Density \mathcal{L} = \text{Im}(u^*\partial_t u)+|\partial_x u|^2 -1/2|u|^4 The functional to be extreme: J = \int\limits_{t_1}^{t_2}\int\limits_{-\infty}^{\infty}\...
  45. N

    Lorentz invariant lagrangian density

    Hi, Would someone know where I can find a derivation of the lorentz-invariant lagrangian density? This lagrangian often pops-up in books and papers and they take it for granted, but I was actually wondering if there's a "simple" derivation somewhere... Or does it take a whole theory and...
  46. J

    What is the difference between two Lagrangian densities in electrodynamics?

    Homework Statement Given the Lagrangian density: L= -\frac{1}{2} \partial_{\mu}A_\nu \partial^{\mu}A^\nu -\frac{1}{c}J_\mu A^\mu (a) find the Euler Lagrange equations of motion. Under what assumptions are they the Maxwell equations of electrodynamics? (b) Show that this Lagrangian...
  47. J

    Lagrangian Density Problem

    Homework Statement Given the Lagrangian density: L= -\frac{1}{2} \partial_{\mu}A_\nu \partial^{\mu}A^\nu -\frac{1}{c}J_\mu A^\mu (a) find the Euler Lagrange equations of motion. Under what assumptions are they the Maxwell equations of electrodynamics? (b) Show that this Lagrangian...
  48. B

    Lagrangian density of linear elastic solid

    I need the general expression for the lagrangian density of a linear elastic solid. I haven't been able to find this anywhere. Thanks.
  49. LarryS

    Definition of Lagrangian Density?

    I understand the definitions of both the classical and relativistic (SR) Lagrangians. But I cannot find a precise mathematical definition of Lagrangian Density. Please assist. Thanks in advance.
  50. R

    Units of Scalar Field \phi & Lagrangian Density

    What are the dimensions of a scalar field \phi ? The Lagrangian density is: \mathcal L= \partial_\mu \phi \partial^\mu \phi - m^2 \phi \phi So in order to make all the terms have the same units, you can try either: \mathcal L=\frac{\hbar^2}{c^2} \partial_\mu \phi \partial^\mu \phi -...
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