Lagrangian Definition and 1000 Threads

Would the following: $$ L = m(\vec{\dot r + \vec v})$$ (constant velocity added to above eq.) lead to equivalent euler-lagrange equations due to the fact that the ratio of T and V is unaffected by an increase in constant velocity? And would this be an example of energy conservation?

I know that from the given problem, I need to find the expression for Kinetic energy, KE = 1/2 m [r(dot)]^2 and Potential energy, PE = 1/2 k r^2 So L = 1/2 m [r(dot)]^2 - 1/2 k r^2 Hence H = 1/2 m [r(dot)]^2 + 1/2 k r^2 I assume that the fixed length r0 is provided to find the value of end...

Hi, I'm trying to check that the QED Lagrangian $$\mathscr{L}=\bar{\psi}\left(i\!\!\not{\!\partial}-m\right)\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu} - J^\mu A_\mu$$ is parity invariant, I'm using the general transformations under parity given by $$\psi(x) \rightarrow...

Hi, if I want to construct the most general Lagrangian of a single scalar field up to two fields and two derivatives, I usually see that is $$\mathscr{L} = \phi \square \phi + c_2 \phi^2$$ i.e. the Klein-Gordon Lagrangian. My question is, would be valid the Lagrangian $$\mathscr{L} = \phi...

a) Alright, I think that the trick here is to consider ##\phi^{\dagger}## and ##\phi## as independent scalar fields. I've read that the unitary matrices read as follows $$U = e^{i \epsilon}$$ Thus here we have to consider two separate transformations $$\phi \rightarrow \phi' = e^{i...

Summary:: not constant spin How could I calculate the system lagrangian in function of the generalised coordinates and the conserved quantities associated to the system symmetries? I've been struggling for the case with not constant angular velocity, but I don't realize what I have to do...

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}...

REMARK: First of all I have to say that this Lagrangian reminds me of the Lagrangian from which we can derive Maxwell's equations, which is (reference: Tong QFT lecture notes, equation 1.18; I have attached the PDF). $$\mathcal{L} = -\frac 1 2 (\partial_{\mu} A_{\nu} )(\partial^{\mu} A^{\nu}) +...

The given Lagrangian is: ##L = \frac 1 2 m_1 ( \dot x_1^2 + \dot y_1^2 + \dot z_1^2) + \frac 1 2 m_2 ( \dot x_2^2 + \dot y_2^2 + \dot z_2^2) + G \frac{m_1 m_2}{\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}}##Please note: I have been inspired by the post...

Last day in class, a professor told us that, for a Lagrangian to be Lorentz Invariant, the Lagrangian density cannot have second or higher derivatives. Is this true? Because one can write the KG lagrangian as $$\mathscr{L}=\phi(\square + m^2)\phi,$$ which have second derivatives. And, where...

Hi, I am an undergraduate student in the 3rd sem, we have Lagrangian Mechanics in our course but I am unable to follow it properly. Can you please suggest me a book that will introduce me to Lagrangian and Hamiltonian Mechanics and slowly teach me how to do problems. I am beginner, so please...

When a constraint is expressed as F(x)=0, I am quite comfortable in putting such constraints into the Lagrangian. Just add the function with an undetermined multiplier, then treat the multiplier as an additional coordinate, and proceed as before. ##L = T - V + \lambda F ## For example, you...

I'm reading a book on Classical Mechanics (No Nonsense Classical Mechanics) and one particular section has me a bit puzzled. The author is using the Euler-Lagrange equation to calculate the equation of motion for a system which has the Lagrangian shown in figure 1. The process can be seen in...

The equations is of the form of Differential algebraic Equation, of index 3.

In Lagrangians we often take derivatives (##\frac{\partial}{\partial (\partial_{\mu}\phi)}##) of terms like ##(\partial_{\nu}\phi \partial^{\nu}\phi)##. We lower the ##\partial^{\nu}## term with the metric and do the usual product rule. My question is why do we do this? Isn't...

Homework Statement: finding equation of motion for Born-Infeld lagrangian Homework Equations: born-infelf lagrangian i do not know where I'm going wrong. i'll be really grateful for any advice.

My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...

Hello all, I understand the formation of the Lagrangian is: Kinetic Energy minus the potential energy. (I realize one cannot prove this: it is a "principle" and it provides a verifiable equation of motion. Moving on... One inserts the Lagrangian into the form of the "Action" and minimizes it...

Summary: Since L = T - V, and T equals the kinetic energy (KE) of a particle whose trajectory is to be calculated, how is KE defined since some of its motion will be due to the expanding universe? My understanding may well be wrong, but it is the following. if a particle is stationary at...

Hi, I'm trying to solve a differential geometry problem, and maybe someone can give me a hand, at least with the set up of it. There is a particle in a 3-dimensional manifold, and the problem is to find the trajectory with the smallest distance for a time interval ##\Delta t=t_{1}-t_{0}##...

Hi! I am given the lagrangian: ## L = \dot q_1 \dot q_2 - \omega q_1 q_2 ## (Which corresponds to a 2D harmonic oscillator) And I am given two transformations and I am asked to say if there is a constant of motion associated to each transformation and to find it (if that's the case). I am...

Assuming a Lagrangian proportional to the following terms: ##L \sim ( \partial_\mu \sigma) (\partial^\mu \sigma) - g^{m\bar{n}} g^{r\bar{p}} (\partial_\mu g_{mr} ) ( \partial^\mu g_{\bar{n}\bar{p}} ) ~~~~~ \to (1) ## Where ##\mu =0,1,2,3,4## and m, n,r, p and ##\bar{n}, \bar{p}, \bar{m}## and...

Hello everyone ! I recently read an article about Standard Model's Lagrangian. And it made me remember another article (that I read a long time ago) which said that a theory's Lagrangian "represent" the theory. But How ? Maybe I didn't get the sense of "represent". Also I know that there is...

Hello. In a chapter of a book I just read it is given that ##\frac {d} {d\epsilon}\left. L(q+\epsilon \psi) \right|_{\epsilon = 0} = \frac {\partial L} {\partial q} \psi ## While trying to get to this conclusion myself I've stumbled over some problem. First I apply the chain rule...

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...

I am currently studying QFT from this book. I have progressed to the chapter of QED. In the course, the authors have been writing the Lagrangian for different fields as and when necessary. For example, the Lagrangian for the complex scalar field is $$\mathcal{L} \ = \ (\partial ^\mu...

In Landau mechanics it's been given that multiple Lagrangians can be defined for a system which differ by a total derivative of a function. This statement is further used for the following discussion. I understand that the term for L has been expanded as a Taylor series but I can't understand...

In "The Theoretical Minimum" (the one on classical mechanics), on page 218, the authors write a Lagrangian $$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$ They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be...

I have been reading a book on classical theoretical physics and it claims: -------------- If a Lagrange function depends on a continuous parameter ##\lambda##, then also the generalized momentum ##p_i = \frac{\partial L}{\partial\dot{q}_i}## depends on ##\lambda##, also the velocity...

I think my confusion on this is where the best origin for polar coordinates is. I've tried the center of the circle, and note the triangle made from the r coordinate reaching out to ##m, a,## and ##l+a\theta##. Then $$r=\sqrt{a^2+(l+a\theta)^2}$$ $$\dot r = \frac {a(l+a\theta)}...

I like using the Euler–Lagrange equations to solve simple mechanical systems, but I'm not perfectly clear on the theory behind it. Is it derived by assuming that action is minimized/stationary? Or does one define a system's Lagrangian according to what makes the Euler–Lagrange equations...

First of all, disclaimer: This isn't an official assignment or anything, so I'm not even sure if there is a resonably simple solution. Consider the following sketch. (Forgive me if it isn't completely clear, I didn't want to fiddle around for too long with tikz...) Let us assume that we can...

Here is a picture of the problem. I have chosen the origin to lie in the middle of the circle around which the mass moves. I have also chosen the z axis to pass through the origin and through the vertex of the right circular cone. The x-axis and y-axis are so that one when curls his or her...

It's known that the Lagragian is invariant when one adds to it a total time derivative of a function. I was thinking about the possible forms of this function. Could it be a function of ##x,\dot x, \lambda##? Or it should be a function of ##x## only? Here ##x: \ \{x^\mu (\lambda) \}## are the...

If a Lagrangian has explicit time dependence due to the potential changing, or thrust being applied to the object in question, how does calculus of variations handle this? It's easy to get the Lagrange equations from: δL = ∂L/∂x δx + ∂L/∂ẋ δẋ What is not clear is how this works when t is an...

Homework Statement I am trying to reproduce MTW's ADM version of the field Lagrangian for a source free electromagnetic field: ##4π\mathcal {L} = -\mathcal {E}^i∂A_i/∂t - ∅\mathcal {E}^i{}_{,i} - \frac{1}{2}Nγ^{-\frac{1}{2}}g_{ij}(\mathcal {E}^i\mathcal {E}^i + \mathcal {B}^i\mathcal {B}^i) +...

Homework Statement While solving equation of rocket motion with Newton's law in 1-d,I pondered to apply Lagrangian method on this. However, I didn't get correct result. Because I can eliminate last 2nd equation using last equation and get some other equation which is certainly not rockets...

The Standard Model Lagrangian contains terms like these: ##-\partial_\mu \phi^+ \partial_\mu\phi^-## ##-\frac{1}{2}\partial_\nu Z^0_\mu\partial_\nu Z^0_\mu## ##-igc_w\partial_\nu Z^0(W^+_\mu W^-_\nu-W^+_\nu W^-_\mu)## How should one interpret the "derivative particle fields" like...

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...

Alright so I was just messing around with Lagrangian equation, I just learned about it, and I had gotten to this equation of motion: Mg*sin{α} - 1.5m*x(double dot)=0 I am trying to get velocity, and my first thought was to integrate with dt, but I didn't know how to. And I'm not even sure it's...

Homework Statement There's the following problem (the task is to construct the Lagrangian) in the first Landau (part a): My problem is that I don't understand what did we omit exactly and why. Homework EquationsThe Attempt at a Solution I did the calculation myself (even checked with...

Why Lagrangian not depend of higher derivatives of generalised coordinates ?

Q)6.10 in Morin's Classical Mechanic. All that I have to do is find Lagrangian here. After that only simple motion. Finding Lagrangian: Choose the coordinate system at the centre of hoop as shown in 2nd attachment. Then,I found out Lagrangian of the system. Invoking E-L Equations to find the...

I am sorry for asking this stupid question, but in the Yang-Mills lagrangian, there is a term ##Tr(F^{\mu \nu}F_{\mu \nu})##. Isn't ##F^{\mu \nu}F_{\mu \nu}## a number?

I'm reading up a series of papers on hydrodynamical simulations for galaxies and cosmology. They keep mentioning things like "Lagrangian" or "pseudo-lagrangian" or "Eulerian". I tried looking it up on the internet, but the answers are either too broad and could mean a huge number of things in...

From the invariance of space time interval the metric dΓ2=dt2-dx2-dy2-dz2 dΓ2=gμνdxμdxν dΓ=√(gμνvμvμ)dt dΓ=proper time. Can someone please help me in sort out why the term √(gμνdxμdxν) is taken as the Lagrangian,as geodesic equation is solved by taking this to be the Lagrangian.

I'm confused (what else is new) about L2. While watching a video from PBS Digital Spacetime about the latest data drop from Gaia Space Telescope, Matt O'Dowd showed a CGI animation of the telescope leaving Earth then circling/orbiting L2 perpendicular to the Earth/sun plane. I thought that the...

I started studying Lagrangian mechanics, and the movement equation is like this: d/dt (d/dz') L - d/dz L = 0 if the movement is on the z axis. Now the problem is, let's say L = M(z')2/2 - Mgz. How do we derivate an expression depending of z with respect to z' and also , an expression depending...

Consider two arbitrary scalar multiplets ##\Phi## and ##\Psi## invariant under ##SU(2)\times U(1)##. When writing the potential for this model, in addition to the usual terms like ##\Phi^\dagger \Phi + (\Phi^\dagger \Phi)^2##, I often see in the literature, less usual terms like: $$\Phi^\dagger...