Lagrangian Definition and 1000 Threads

  1. beowulf.geata

    I Cartesian velocity and generalized velocity

    Hi, I'm reading A Student's Guide to Lagrangians and Hamiltonians by Patrick Hamill and, in the following section on generalized velocity, I'm wondering if we should have ##\delta_{kj}## rather than ##\delta_{ij}##? Many thanks.
  2. binbagsss

    A Show Lagrangian is invariant under a Lorentz transformation without using generators

    This is probably a stupid question but, I want to show that a Lagrangian written in field theory is Lorentz invariant WITHOUT using the Lorentz transformation representation / generators. I know we know that a Lorentz scalar is automatically Lorentz invariant, but, I want to show this by...
  3. A

    I Metal Ball Rolling in a Parabolic Bowl in the presence of a magnet

    I found this equation as a solution to the original problem of a point-like ball rolling without friction in a parabolic cavity from Lagrangian mechanics: Is it possible to add an extra force to this equation? I want to add a "magnet" at a certain position that will affect the ball. The...
  4. arivero

    I The components of Dirac Equation -- Bosonic Lagrangian?

    The four components of Dirac equations obey the Klein-Gordon equation for a particle of mass m. This is always explained when introducing Dirac equation, but it is never exploited further. I am wondering: Can we then write a bosonic lagrangian for these four "particles"? Is this related to the...
  5. M

    Two masses connected by Pulley - Lagrangian problem

    For this problem, My solution to (a) is, We have constraint ##x + y = L##. There are many places we could define our (x,y) Cartesian coordinate system. However, the most easiest I think for the problem would be to attach a ##x^*## and ##y^*## coordinate system at the COM of ##m_1##. We define...
  6. L

    Spring pendulum Lagrangian has any ignorable coordinates?

    Are there any ignorable coordinates in this scenario? I don’t think so right, because the lagrangian has explicit dependency on both x and theta. Ignorable coordinates means there is no explicit dependence of that coordinate right? If there are no ignorable coordinates that also means there are...
  7. D

    Covariant Euler-Lagrange Computation

    Does anybody know of a software (or software package) that can solve the Euler-Lagrange field equations for a manifestly-covariant Lagrangian density in full tensor form? Mathematica has a "Variational Methods" package, but none of the examples given are in manifestly-covariant form. I am not...
  8. M

    Double pendulum Lagrangian using small angle approximation formula

    For this part (b) of this problem, From (a), we know that ##\mathcal{L}\left(\phi_{1}, \phi_{2}, \dot{\phi}_{1}, \dot{\phi}_{2}\right)=\frac{1}{2} m \ell^{2}\left[2 \dot{\phi}_{1}^{2}+\dot{\phi}_{2}^{2}+2 \cos \left(\phi_{1}-\phi_{2}\right) \dot{\phi}_{1} \dot{\phi}_{2}\right]+m g \ell\left(2...
  9. Melkor77

    Solving projectile motion problems using Lagrangian mechanics

    TL;DR Summary: Lagrangian for projectile motion in an inclined plane with negative slope. I am a bit unsure on how to find the Lagrangian for projectile motion in an inclined plane with negative slope. I can solve it using Newton Mechanics, but am a bit new to lagrangian mechanics. Also could...
  10. M

    Lagrangian of a block on an inclined plane

    For this problem, I am confused where they get these two terms from Can someone please explain? I get all the other terms expect those. Thanks alot!
  11. M

    Spherical pendulum confusion [Issue resolved]

    For this problem, I am confused my what they mean by ##\phi##. I have looked at the figure, but it is confusing. Makes it look like the x-axis and y-axis are not perpendicular, even thought I'm assuming they are since this is a right handed coordinate system. Does someone please know what...
  12. B

    Equations of motion for Lagrangian of scalar QED

    Well, I started with the first equation of motion for the scalar field, but I'm really not sure if I'm doing it the right way. \begin{equation} \begin{split} \frac{\partial \mathcal{L}}{\partial \varphi} &= \frac{\partial}{\partial \varphi} [(\partial_\mu \varphi^* -...
  13. Bling Fizikst

    How Does Coriolis Force Influence Particle Motion in Rotating Systems?

    Source : JEE Advanced , Physics Sir JEE YT I tried to attempt it using Lagrangian , so according to the coordinate axes given in the diagram , the position of the particle is let's say ##(0,d,-z)## Let ##r## be the distance between the particle and the axis of rotation such that it subtends...
  14. G

    Finding slip-off angle for mass off of sphere?

    [Rewriting this as per the suggestions. Thanks once again.] I won't be using the Lagrangian because it was never explicitly stated that I have to so I'll just use conservation of energy. $$ T = \frac{1}{2}mv^2 = \frac{1}{2}m(R\dot{\theta})^2 = \frac{1}{2}mR^2\dot{\theta}^2 $$ $$ V = mgy =...
  15. H

    I What does "transforms covariantly" mean here?

    The Lagrangian, $$\mathcal L(x)= \frac 1 2 \partial^{\mu} \phi (x) \partial_{\mu} \phi (x) - \frac 1 2 m^2 \phi (x)^2$$ for a scalar field ##\phi (x)## is said to be Lorentz invariant and to transform covariantly under translation. What does it mean that it transforms covariantly under translation?
  16. K

    A Renomalizabilty and triple/quadruple vertices

    I read that a Lagrangian is renormalizable if it contains only triple and quadruple vertices, or at most four powers of the fields. Where can I read more about the precise mathematical conditions?
  17. Hennessy

    I Calculus Question within Lagrangian mechanics

    Hi all currently got a lagrangian function which i've found to be : \begin{equation}\mathcal{L}=\frac{1}{2}m(\dot{x}^2+\dot{y}^2+4x^2\dot{x}^2+4y^2\dot{y}^2+8xy\dot{x}\dot{y})- mg(x^2+y^2) \end{equation} Let us first calculate $$(\frac{\partial L}{\partial \dot{x}})$$ which leads us to...
  18. G

    I Taylor expansion about lagrangian in noether

    I was studying a derivation of noether's theorem mathematically and something struck my eyes. Suppose you have ##L(q, \dot q, t)## and you transform it and get ##L' = L(\sigma(q, a), \frac{d}{dt}\sigma(q,a), t)##. ##\sigma## is a transformation function for ##q## Let's represent ##L'## by...
  19. Z

    A Symmetry in terms of Lagrangian

    Hi, as we know SM is symmetric under SU(3) X SU(2) X U(1) , But my question is , how can we check the invariance of terms in Lagrangian under these symmetries - thanks
  20. ronniegertman

    Analytical mechanics- working with Lagrangian and holonomic constraints

    The tutor solved the problem using kinetic spinning energy though I find it very difficult and confusing to do so, therefore, I would like to know if there is a way to solve the problem using effective potential energy, Veff = J2/(2mr2 below is a sketch of the problem
  21. Z

    A Lagrangian density , for scalar field , vector field and Spinor field

    hi, I have go through many books - they derive Dirac equation from Dirac Lagrangian, KG equation from scalar Lagrangian - but my question is how do we get Dirac or scalar Lagrangian at first place as our starting point - kindly help in this regard or refer some book - which clearly elaborate...
  22. Z

    A Dimension of terms in Lagrangian

    hi, when we say in SM , we can add terms having dimension 4 or less than that- in this to what dimension we are refering ? kindly help how do you measure the dimension of terms in Lagrangian. thanks
  23. H

    I Relativistic Lagrangian of a particle in EM field

    I'm following the derivation in Lancaster and Blundell. First, the Lagrangian for the free particle is ##L=-\frac {mc^2} {\gamma}## and the action ##S=\int -\frac {mc^2} {\gamma} \, dt##. Then, EM is "turned on" with the potential energy ##-qA_{\mu}dx^{\mu}##. Then, they say, the action becomes...
  24. H

    A Discretisation of a PDE in Lagrangian coordinates

    I am writing a 2D hydrocode in Lagrangian co-ordinates. I have never done this before, so I am completely clueless as to what I'm doing. I have a route as to what I want to do, but I don't know if this makes sense or not. I've gone from Eulerian to Lagrangian co-ordinates using the Piola...
  25. Precious Adegbite

    I Restoring S.I. units to a Lagrangian in natural units

    If we have a natural unit Lagrangian, where some fundamental quantities have been excluded to ease calculations...and aim to restore it's S.I units back, do we just have to plug back the fundamental quantities that were initially excluded Into the Lagrangian...or we use some specific scaling...
  26. LCSphysicist

    What is the Fermion's mass in this Lagrangian?

    We have a Lagrangian of the form: $$ \mathcal{L} = \overline{\psi} i \gamma_{\mu} \partial^{\mu} \psi - g \left( \overline{\psi}_L \psi_R \phi + \overline{\psi}_R \psi_L \phi^* \right) + \mathcal{L}_{\phi} - V(|\phi|^2) $$ Essentially, what we are studying is spontaneous symmetry breaking...
  27. G

    A Lagrangian: kinetic matrix Z_ij and mass matrix k_ij

    Can somebody explain why the kinetic term for the fluctuations was already diagonal and why to normalize it, the sqrt(m) is added? Any why here Z_ij = delta_ij? Quite confused about understanding this paragraph, can anybody explain it more easily?
  28. deuteron

    Bead moving down a Helical Wire subject to Constraints

    One of the constraints is given as ##r=R##, which is very obvious. The second constraint is however given as $$\phi - \frac {2\pi} h z=0$$ where ##h## is the increase of ##z## in one turn of the helix. Physically, I can't see where this constraint comes from and how ##\phi=\frac {2\pi}h z##.
  29. G

    I Using reference frames for derivation of Lagrangian

    I had an interesting thought and it just might be me, but I'm looking forward to hearing your thoughts, but not like the thoughts - "yeah, Landau is messy, complicated, don't read that, e.t.c". Just think about what you think about my thoughts. Landau first starts to mention the variational...
  30. Y

    A Lagrangian in the Path Integral

    Using free scalar field for simplicity. Hi all, I have a question which is pretty simple, we have the path integral in QFT in the presence of a source term: $$ Z[J] = \int \mathcal{D}\phi \, e^{i \int d^4x \left( \frac{1}{2} \phi(x) A \phi(x) + J(x) \phi(x) \right)} $$ So far so good. Now...
  31. G

    I Why Lagrangian includes 1/2m by Landau?

    I feel like I might be posting many questions, so I hope you're not angry with me on this. I'm following Landau's book and try to understand why L = K - U but first we need to figure out why it contains K. Landau discusses two inertial frames where the speed of one frame relative to another is...
  32. G

    I Lagrangian doesn't change when adding total time derivative

    When we have ##L(q, \dot q, t)##, The change in action is given by: ##\int_{t_1}^{t_2} dt (\frac{\partial L}{\partial x} f(t) + \frac{\partial L}{\partial \dot x} \dot f(t))## when we change our true path ##x(t)## by ##x(t) + \epsilon f(t)##. Now, attaching the image, check what Landau says...
  33. O

    Deriving Maxwell's equations from the Lagrangian

    This isn't a homework problem (it's an example from David Tong's QFT notes where I didn't understand the steps he took), but I am confused as to how exactly to take the partial derivative of the Lagrangian with respect to ##\partial(\partial_\mu \mathcal{A}_\nu)##. (Note the answer is...
  34. G

    I Lagrangian symmetry for the bullet and gun

    I have been learning emma noether’s theorem where every symmetry results in conservstion law of something, sometimes momentum, sometimes angular momentum, sometimes energy or sometimes some kind of quantity. Every explanation that I have listened to or learned from talks about homogeneity of...
  35. G

    I Noether’s theorem -- Question about symmetry coordinate transformation

    Let's say we got one particle in x direction only and we got some motion x(t) which we figured it out through Lagrangian. In Noether's theorem for coordinate transformation symmetry, we start with the following: x(t)' = x(t) +εf(t) (ε - some number) - I denoted new path with x(t)' I'm...
  36. M

    I Confused about applying the Euler–Lagrange equation

    Hello! I have a Lagrangian of the form: $$L = \frac{mv^2}{2}+f(v)v$$ where ##f(v)## is a function of the velocity. I would like to derive the equation of motion in general, without writing down an expression for ##f(v)## yet. I have that ##\frac{\partial L}{\partial x} = 0##. However, what is...
  37. Matthew_

    I Help with Canonical Poisson Brackets & EM Field

    We were introduced the lagrangian for a particle moving in an eletromagnetic field (for context, this was a brief introduction before dealing with Zeeman effect) as $$\mathcal{L}=\dfrac{m}{2}(\dot{x}^2_1+\dot{x}^2_2+\dot{x}^2_3)-q\varphi+\dfrac{q}{c}\vec{A}\cdot\dot{\vec{x}}.$$ A...
  38. PhysicsRock

    I Why isn't the Lagrangian invariant under ##\theta## rotations?

    I just calculated the Lagrangian of a particle of mass ##m## in a radially symmetric potential ##V(r)##. I thought it would be a good idea to switch to spherical coordinates for that matter. What I get is $$ L = \frac{1}{2} m \left( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \dot{\varphi}^2...
  39. Rick16

    I Lagrangian approach for the inclined plane

    I want to use the Lagrangian approach to find the equation of motion for a mass sliding down a frictionless inclined plane. I call the length of the incline a and the angle that the incline makes with the horizontal b. Then the mass has kinetic energy 1/2m(da/dt)2 and the potential energy should...
  40. PhysicsRock

    Square of orthogonal matrix vanishes

    I found a the answer in a script from a couple years ago. It says the kinetic energy is $$ T = \frac{1}{2} m (\dot{\vec{x}}^\prime)^2 = \frac{1}{2} m \left[ \dot{\vec{x}} + \vec{\omega} \times (\vec{a} + \vec{x}) \right]^2 $$ However, it doesn't show the rotation matrix ##R##. This would imply...
  41. L

    Determine the Lagrangian for the particle moving in this 3-D cos^2 well

    Hi, I am not quite sure whether I have solved the following problem correctly: I have now set up Lagrangian in general, i.e. $$L=T-V=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)-mgz$$ After that I imagined how ##x##,##y## and ##z## must look like and got the following: $$x=\beta \cos^2(\alpha r)...
  42. B

    I Checking if a stationary point is a minimum using Lagrangian Mechanics

    I'm having trouble understanding how to find out whether or not a stationary point is a minimum and I'm hoping for some clarification. In my class, we were shown that, using Euler's equation, the straight-line path: with constants a and b results in a stationary point of the integral: A...
  43. James1238765

    I What is the significance of the T - V Lagrangian of a system?

    Let E be a fixed immutable quantity. E can be freely exchanged between T and V, as long as $$T + V = E$$ 1. What does the quantity $$\int_x T - V $$ signify? What is the importance of this quantity? -------------------- Let E now be the budget of a factory. E can either be spent on T or V in...
  44. D

    I Equation of motion for a simple mechanical system

    The system is shown below. It consists of a rod of length ##L## and mass ##m_b## connecting a disk of radius ##R## and mass ##m_d## to a collar of mass ##m_c## which is in turn free to slide without friction on a vertical and rigid pole. The disk rolls without slipping on the floor. The ends...
  45. M

    I Effective mass from the Lagrangian

    Hello! I have the following Lagrangian: $$L = \frac{1}{2}mv^2+fv$$ where ##v = \dot{x}##, where x is my coordinate and f is a function of v only (no explicit dependence on t or x). What I get by solving the Euler-Lagrange equations is: $$\frac{d}{dt}(mv+f+\frac{\partial f}{\partial v} v) =...
  46. BiGyElLoWhAt

    A How can you tell the spin of a particle by looking at the Lagrangian?

    I'm just starting to get into QFT as some self study. I've watched some lectures and videos, read some notes, and am trying to piece some things together. Take ##U(1)_{EM}: L = \bar{\psi}[i\gamma^{\mu}(\partial_{\mu} - ieA_{\mu}) - m]\psi - 1/4 F_{\mu\nu}F^{\mu\nu}## This allegedly governs spin...
  47. Pironman

    I How to find the equation of motion using Lagrange's equation?

    Good morning, I'm not a student but I'm curious about physics. I would like to calculate the equation of motion of a system using the Lagrangian mechanics. Suppose a particle subjected to some external forces. From Wikipedia, I found two method: 1. using kinetic energy and generalized forces...
  48. andris0110

    Vertical Spring System (Lagrangian)

    I am trying to solve this and get the equations of motion using the Lagrangian method. I could do all the steps but the equations (especially the third one) seems..weird. What am I doing wrong? Sorry if the equations aren't in their simplest form, they are pulled straight from Wolfram...
  49. 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
  50. Ahmed1029

    I Time dependence of kinetic energy in Lagrangian formulation

    Could kinetic energy possibly depend explicitly on time in the lagrangian for some arbitrary set of generalized coordinates?
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