Lagrangian Definition and 1000 Threads

Homework Statement Consider a point mass of mass m suspended from an ideal, massless spring. Let ##\theta ## be measured from the vertical. Find the displacement of the mass as a function of time if the spring is initially stretched/compressed a distance ## l_0 ## and has an initial velocity...

Hi. I don't understand the meaning of "up to total derivatives". It was used during a lecture on superfluid. It says as follows: --------------------------------------------------------------------- Lagrangian for complex scalar field ##\phi## is $$ \mathcal{L}=\frac12 (\partial_\mu \phi)^*...

Lagrangian is defined by ##L=L(q_i,\dot{q}_i,t)## and hamiltonian is defined by ##H=H(q_i,p_i,t)##. Why there is relation H=\sum_i p_i\dot{q}_i-L end no H=L-\sum_i p_i\dot{q}_i or why ##H## is Legendre transform of ##-L##?

What is the intuitive reasoning for requiring that a Lagrangian describing a free-field contains terms that are at most quadratic in the field? Is it simply because this ensures that the EOM for the field are linear and hence the solutions satisfy the superposition principle implying (at least...

Homework Statement A homogeneous hollow cylinder (mass M, radius R) is in the gravitational field and a horizontal axis through the center P rotatably mounted (central axis of the cylinder is fixed and can be rotated). A small, homogeneous solid cylinder (mass m, radius r) is rolling inside...

Homework Statement Homework Equations The last part of this question is an example of this result: The Attempt at a Solution Here is the solution I think L' is missing a term: If we take the Earth as your frame of reference.(i.e. You are stationary, watching the movement of the railway...

I'm just in need of some clearing up of how to differentiate the lagrangian with respect to the covariant derivatives when solving the E-L equation: Say we have a lagrangian density field \begin{equation} \mathcal{L}=\frac{1}{2}(\partial_{\mu}\hat{\phi})(\partial^{\mu}\hat{\phi}) \end{equation}...

The Yang-MIlls Lagrangian is given by ##\mathcal{L}_{\text{gauge}} = F_{\mu\nu}^{a}F^{\mu\nu a} + j_{\mu}^{a}A^{\mu a}.## We can rescale ##A_{\mu}^{a} \to \frac{1}{g}A_{\mu}^{a}## and then we have ##\frac{1}{g^{2}}F_{\mu\nu}^{a}F^{\mu\nu a}.## How does the second term change? Does the...

Homework Statement Let's say that I have a potential ##U(x) = \beta (x^2-\alpha ^2)^2## with minima at ##x=\pm \alpha##. I need to find the normal modes and vibrational frequencies. How do I do this? Homework Equations ##U(x) = \beta (x^2-\alpha ^2)^2## ##F=-kx=-m\omega ^2 x## ##\omega =...

Homework Statement Homework EquationsThe Attempt at a Solution This is the problem in Goldstein's classical mechanics exercise 1.11 I wonder why the solution doesn't consider rotational kinetic energy (1/2 I w^2) So, L= T = 1/2mx^2 + 1/2 I w^2 .

Consider the following tree-level Feynman diagrams for the ##W^{+}W^{-} \to W^{+}W^{-}## scattering process. The matrix element for this diagram can be read off from the associated quartic term ##\mathcal{L}_{WWWW}## in the electroweak boson self-interactions, where ##\mathcal{L}_{WWWW} =...

In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian...

To find the geodesics of a space(time), what we need to do is extremizing the functional ##\displaystyle \int_{\lambda_1}^{\lambda_2}\sqrt{g_{\mu \nu} \frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}} d\lambda ##. But sometimes the presence of the square root makes the equation of motion too...

Let me begin by saying I know I'm doing something wrong here, but I'm having trouble seeing what it is. This is a reformulation of https://www.physicsforums.com/threads/plugging-eom-into-lagrangian.905099/, where I've reduced the issue to a much simpler problem. Moderators, feel free to close...

I know that in general plugging the EOM into the Lagrangian is tricky, but it should be perfectly valid if done correctly. Can someone help me see what I'm doing wrong here? I know I'm doing something dumb but I've been staring at it for too long Start with the E&M Lagrangian: L =...

(L=T-V) In the Lagrangian function we saw to different type of energy conservation's. That is kinetic energy and potential energy. And I have doubt in one topic. How to define potential energy?

The decay processes of the ##W## bosons are completely governed by the charged current interaction terms of the Standard model: $$\mathcal{L}_{cc} = ie_{W}\big[W_{\mu}^{+}(\bar{\nu}_{m}\gamma^{\mu}(1-\gamma_{5})e_{m} + V_{mn}\bar{u}_{m}\gamma^{\mu}(1-\gamma_{5})d_{n})\\...

Hello, I need some help to find the correct symmetrized Lagrangian for the field operators. After some work I guess that $$\mathcal{L} = i[\overline{\psi}_a,({\partial_\mu}\gamma^\mu \psi)^a] -m[\overline{\psi}_a,\psi^a ]$$ should be the correct Lagrangian but I'm not sure with this. I'm...

Hi PF! I am reading about Eulerian vs Lagrangian perspectives. To me, it seems that Eulerian considers a volume and follows that volume (which may deform) through space. A Lagrangian frame of reference doesn't track volume, but instead specific particle matters. Am I correct? If so, what are...

Homework Statement derive the equation of motion of a mass-spring-pulley system using lagrange's equations. A mass m is connected to a spring of stiffness k, through a string wrapped around a rigid pulley of radius R and mass moment of inertia, I. Homework Equations kinetic energey T = 1/2...

Homework Statement Show that $$ \mathcal{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu} = - \frac{1}{2}\partial^{\mu}A^{\nu}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}) $$ Where $$ F^{\mu \nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu} $$ Homework EquationsThe Attempt at a Solution $$...

Has somebody a paper, text, (whatever), which explains the Lagrangian Coherent Structures ? I want an explanation by words and not all the mathematical stuff. All I can find about LCS is 1. They organize flow 2. you can detect them with the Lyapunov exponents. But I want to understand what LCS...

Homework Statement An ideal spring of relaxed length l and spring constant k is attached to two blocks, A and B of mass M and m respectively. A velocity u is imparted to block B. Find the length of the spring when B comes to rest. Homework Equations ∆K + ∆U = 0 U = \frac {kx^2}{2} ∆p = 0...

We can write the Born-Infeld Lagrangian as: L_{BI}=1 - \sqrt{ 1+\frac{1}{2}F_{\mu\nu }F^{\mu\nu}-\frac{1}{16}\left(F_{\mu\nu}\widetilde{F}^{\mu\nu} \right)^{2}} with G^{\mu\nu}=\frac{\partial L}{\partial F_{\mu\nu}} how can we show that in empty space the equations of motion take the form...

Can someone help me understand how the following two actions are related? S_1 = \int \left(-\dfrac{1}{2}mg_{\mu\nu}\dot{x}^\mu\dot{x}^\nu - U\right) d\tau S_2 = \int \left(-m\sqrt{g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu} - U\right) d\tau Both of them lead to the correct geodesic equation as the...

Hi all, I've tried to figure this out for some time without luck. Hope you might be able to give some input. I've implemented a model-based dynamics software in MATLAB based on the works of Roy Featherstone's Springer book "Rigid Body Dynamics Algorithms". So, I have the EoM of an...

Using the Lagrangian : L = ½mv^2 - qφ + qAv What is the physical intuition of Av ? I know that A is the magnetic vector potential and that v is the velocity of the charged particle. I just don't know what their dot product means physically .

I'm currently studying Quantum Field Theory and I have a confusion about some mathematics in page 30 of Mandl's Quantum Field Theory (Wiley 2010). Here is a screenshot of the relevant part: https://www.dropbox.com/s/fsjnb3kmvmgc9p2/Screenshot%202017-01-24%2018.10.10.png?dl=0 My issue is in...

" We can put the Lorentz force law into this form by being clever. First, we write $$\frac{dA_j}{dt}=\frac{d}{dt}(\frac{\partial{}}{\partial{v_j}}(v.A)),$$ since the partial derivative will pick out only the jth component of the dot product. Now, since the scalar potential is independent of the...

To include slonczewski-like torque in the Lagrangian there enter as dissipation by Rayleig function? ST= σ m x (m x mp) where m is the magnetization in free layer and mp current direction in the pinned layer (-z). The Rayleig function is: RF=(dm/dt+σ m x mp)2 Then L= ∫ (RF+E)dx Thanks

Hi, I am trying to model the position of the suspended mass at the end on a boom crane. This is basically a spherical pendulum, however further complicated by the fact that the mass can be hoisted up and down and also the pivot is connected to an arm (boom) which can be rotated up and down and...

Homework Statement NOTE - When I post the thread my embedded images aren't showing up on my web browser, but they do show up when I bring it up to edit, so I don't know if other users can see the pictures or not... If not, they're here: Problem outline: http://tinypic.com/r/34jeihj/9 Solution...

Homework Statement 2 masses are connected by a spring. They are on a frictionless plane inclined relative to the horizontal by ##\alpha##. The masses are free to slide, rotate about their center of mass, and oscillate. 1. Find the Lagrangian as a sum of the Lagrangian for the COM motion and a...

Lagrangians that include a particle field and its corresponding antiparticle field always have the particle field and the antiparticle field in the same terms. For example, in the theory of a complex scalar boson ##\phi##, the Lagrangian is a function of ##\phi^{*}\phi##, and not of ##\phi##...

Why can we always drop any constant term in a Lagrangian density in quantum field theory? This issue is somehow related to the constant term being some kind of cosmological constant? Can you please explain this issue?

Homework Statement A particle of mass m slides without rolling down on a inclined plane, Find the generalized force and the Lagrangian equation of motion of mass m. Homework Equations T = (mx'^2)/2 Generalized force Q=-d/dx(V) The Attempt at a Solution To find the generalized force first I...

Homework Statement Uniform chain of length L is kept of a horizontal table in such a way that l of its length keeps hanging from the table. If the whole system is in equilibrium, find the Lagrangian of the system. Homework Equations Lagrangian of the system = Kinetic energy (T) - Potential...

Homework Statement I'm reading through A. Zee's "Quantum Field Theory in a nutshell" for personal learning and am a bit confused about a passage he goes through when discussing field theory for the electromagnetic field. I am well versed in non relativistic quantum mechanics but have no...

I have a question about the use of trace in QFT in general - more specifically the use of trace in the lagrangian in the effective theory concerning chiral symmetry in QCD. I am slowly trying to get a hang of everything, and most things i am able to calculate, but i still have som very specific...

Homework Statement String is wrapped around two identical disks of mass m and radius R. One disk is fixed to the ceiling but is free to rotate. The other is free to fall, unwinding the string as it falls. Find the acceleration of the falling disk by finding the lagrangian and lagrange's...

Homework Statement Two blocks of equal mass, m, are connected by a light string that passes over a massless pulley. One block hangs below the pulley, while the other sits on a frictionless horizontal table and is attached to a spring of constant k. Let x=0 be the equilibrium position of the...

The (classical, relativistic) Lagrangian for electrodynamics contains the field energy density -FμνFμν/4 and the interaction term -Aμjμ. I understand the maths of that - for one thing, the equations of motion turn out right if you plug this into the Euler Lagrange equantion. Now I recall having...

Two masses, m1 and m2, are attached by a light string of length D. Mass m1 starts at rest on an inclined plane and mass m2 hangs as shown. The pulley is frictionless but has a moment of inertia I and radius R. Find the Lagrangian of the system and determine the acceleration of the masses using...

Homework Statement Two masses, m1 and m2, are attached by a light string of length D. Mass m1 starts at rest on an inclined plane and mass m2 hangs as shown. The pulley is frictionless but has a moment of inertia I and radius R. Find the Lagrangian of the system and determine the acceleration...

for a disck rolling on a horizontal plane the kinetic energy should be the kinetic energy of the CM of the disk with respect to the origin plus the kinetic energy due to the rotation of the disc about his CM so T= 1/2 (M V^2) +1/2(I ω^2) where M is the mass of the disk and V is the velocity of...

There is waterpipe with smooth interior wall as shown in the figure.The curve of waterpipe could be describe as function ##q_y=f(q_x)##. A ball going through this pipe was put at ##O## point with zero initial speed.At the point ##(q_x,q_y)##,the speed of the ball could be decomposed as...

I have read in different places that an up to date definition of energy refers to the Lagrangian and Noether. But isn't the Lagrangian too limited because it refers to an ideal situation involving translational KE and to PE only? I would have thought that a good definition of energy would be...

A sphere is rolling inclined wall (θ radian). and the momentum of that sphere is L = 1/2 mv^2 + 1/5 mv^2 + mgx sin(θ) = 7/10 m v^2 + mgx sin(θ) ∂L/∂v = p 7/5 m v = p but I can't understand why the factor of mv is 7/5. p is the linear momentum of sphere. which means the factor of mv must be...

Homework Statement Find the Lagrangian for the double pendulum system given below, where the length of the massless, frictionless and non-extendable wire attaching m_1 is l. m_2 is attached to m_1 through a massless spring of constant k and length r. The spring may only stretch in the m_1-m_2...

Hi folks, I am looking to learn the Lagrangian and Hamiltonian approach to celestial mechanics - I have previous experience in Newtonian numerical solutions for orbital motion but am looking to achieve similar things but through the use of Hamiltonian formulations. After having a poke around...