Lagrangian Definition and 1000 Threads

Homework Statement I am given the Hamiltonian of the relativistic free particle. H(q,p)=sqrt(p^2c^2+m^2c^4) Assume c=1 1: Find Ham-1 and Ham-2 for m=0 2: Show L(q,q(dot))=-msqrt(1-(q(dot))^2/c^2) 3: Consider m=0, what does it mean? Homework Equations Ham-1: q(dot)=dH/dp Ham-2: p(dot)=-dH/dq...

Homework Statement Consider an inertial laboratory frame S with coordinates (##\lambda##; ##x##). The Lagrangian for the relativistic harmonic oscillator in that frame is given by ##L =-mc\sqrt{\dot x^{\mu} \dot x_{\mu}} -\frac {1}{2} k(\Delta x)^2 \frac{\dot x^{0}}{c}## where ##x^0...

Homework Statement B is 10kg C is 20kg can I find a lagrangian for this system? If so how? Diagram: http://imgur.com/j811rzw Homework Equations L=T-V Kinetic = .5mv^2 Potential = mgh The Attempt at a Solution I know the kinetic energy must be 0 right? How could I find the potential?

Homework Statement The carbon dioxide molecule can be considered a linear molecule with a central carbon atom, bound to two oxygen atoms with a pair of identical springs in opposing directions. Study the longitudinal motion of the molecule. If three coordinates are used, one of the normal...

Homework Statement Take the x-axis to be pointing perpendicularly upwards. Mass ##m_1## slides freely along the x-axis. Mass ##m_2## slides freely along the y-axis. The masses are connected by a spring, with spring constant ##k## and relaxed length ##l_0##. The whole system rotates with...

I have a very simple question. Let's consider the theta term of Lagrangian: $$L = \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu}$$ Investigate parity of this term: $$P(G_{\mu \nu}^a)=+G_{\mu \nu}^a$$ $$P( \tilde{G}^{a, \mu \nu} ) =-G_{\mu \nu}^a$$ It is obvious. But what about...

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...

Homework Statement Hello all ! Over the past few day's, I've been trying to understand how Sean Carroll comes to the conclusion that he does on equation 1.153. I've tried to look for various resources online but I still have trouble understanding how he is able to add both partial derivatives...

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...

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...

What does it means for a physical theory to have hamiltonian, if it is formulated in lagrangian form? Why doesn't someone just apply the lagrangian transformation to the theory, and therefore its hamiltonian is automatically gotten?

Is the following logic correct?: If you have an hamiltonian, that has time has a variable explicitly, and you get the lagrangian,L, from it, and then you get an equivalent L', since L has the total time derivate of a function, both lagrangians will lead to the same equations euler-lagrange...

This T-shirt I bought at a physics conference displays the following equation. It looks like the Lagrangian of the Standard Model of particle physics but I only recognise lines 1 (electroweak) and 3 (Higgs mechanism). What are lines 2 and 4 and what is/isn't included? eg. are quarks, gluons...

We've just been introduced to Langrangians, and my lecturer has told us that the Lagrangian density ##\mathcal{L} = \frac{1}{2} (\partial ^{\mu}) (\partial_{\mu}) -\frac{1}{2} m^2\phi^2## is obviously Lorentz invariant. Why? Yes it's a scalar, but I can't see why it obviously has to be a Lorentz...

Homework Statement Attached: Homework Equations Euler-Lagrange equations to find the EoM The Attempt at a Solution [/B] Solution attached: I follow, up to where the sum over ##\mu## reduces to sum over ##\mu=i## only, why are there no ##\mu=0## terms? I don't understand at all. Many...

Homework Statement Hello! I have some problems with constructing Lagrangian for the given system: (Attached files) Homework Equations The answer should be given in the following form: L=T-U=... The Attempt at a Solution I tried to construct the Lagrangian, but I'm not sure if I did it...

Homework Statement [/B] In this example, I know that I can define the horizontal contribution of kinetic energy to the ball as ##\frac{1}{2}m(\dot{x} + \dot{X})^2##. In the following example, Mass ##M_{x1}##'s horizontal contribution to KE is defined as ##\frac{1}{2}m(\dot{X} -...

Homework Statement I have the answer for part a, which is: $$\theta '' + \frac{a}{r} \cos \theta + \frac{g}{r} \sin \theta$$ My issue lies with getting the following equation of motion for part b, $$\theta '' + \frac{g}{r} \cos \alpha \sin \theta = 0$$ Homework EquationsThe Attempt at a...

Hello, I am trying to "integrate into my understanding" the difference between Hamiltonian and Lagrangian mechanics. In a nutshell: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to the minimization principle? YES; I KNOW about Hamilton's Second...

I’m a bit confused about what exactly lagranigian mechanics is. I know that L = Ke - Pe I also know the equation d/dt(∂L/∂x’) - ∂L/∂x = 0 1.) Apparentaly solving this equation gives the equations of motion. What exactly does that mean though? I solved a very simple problem and got the...

Hello. I have some problems with making Lagrangian. I need your advice. 1. Homework Statement I have this situation: Consider the circular path is intangible and without friction. I have to find Lagrangian for coordinates x and θ. Homework Equations [/B] L = U - V The Attempt at a...

Hello. I solve this problem: 1. Homework Statement The particles of mass m moves without friction on the inner wall of the axially symmetric vessel with the equation of the rotational paraboloid: where b>0. a) The particle moves along the circular trajectory at a height of z = z(0)...

In a thesis, I found double sided arrow notation in the lagrangian of a Dirac field (lepton, quark etc) as follows. \begin{equation} L=\frac{1}{2}i\overline{\psi}\gamma^{\mu}\overset{\leftrightarrow}{D_{\mu}}\psi \end{equation} In the thesis, Double sided arrow is defined as follows...

Homework Statement Hello. I have this problem: I have a ring which is sliding along a wire in the shape of a spiral because of gravity. Spiral (helix) is given as the intersection of two surfaces: x = a*cos(kz), y = a*sin(kz). The gravity field has the z axis direction. I have to find motion...

Homework Statement A sphere of mass m2 and radius R rolls down a perfectly rough wedge of mass m1. The wedge sits on a frictionless surface so as the sphere rolls down, the wedge moves in opposite direction. Obtain the Lagrangian. Homework EquationsThe Attempt at a Solution Here's my diagram...

What books include the theory of lagrangian mechanics? And where can i also find some problems?Could lagrangian mechanics help me in solving problems with oscillations?

Hi, i know that The homogeneity of space and time implies that the Lagrangian cannot contain explicitly either the radius vector r of the particle or the time t, i.e. L must be a function of v only but the lagrangian definition is ##L=\int L(\dot q,q,t)##, so velocity appears in the definition...

I'm trying to follow the calculations in this paper. But I have a weird problem in section 2. To calculate the entanglement entropy using the Ryu-Takayanagi prescription, you have to extremize the area of a surface. So you have to use Euler-Lagrange equations for some kind of an action. The...

I was talking to a friend about Lagrangian mechanics and this question came out. Suppose a particle under a potential ##U(r)## and whose mass is ##m=m(t)##. So the question is: the Lagrangian of the particle can be expressed by ##L = \frac{1}{2} m(t) \dot{\vec{r}} ^2 -U(r)## or I need to...

Homework Statement Show that the classical path satisfying ##\bar{x}(t_a) = x_a##, ##\bar{x}(t_b) = x_b## and ##T = t_b-t_a## is $$\bar{x}(t) = x_b\frac{\sin\omega (t-t_a)}{\sin\omega T} + x_a\frac{\sin\omega (t_b-t)}{\sin\omega T}$$ Homework Equations The Lagrangian: ##L =...

Homework Statement Hi, I'm doing the double pendulum problem in free space and I've noticed that I get two different conserved values depending on how I define my angles. Obviously, this should not be the case, so I'm wondering where I've gone wrong. Homework EquationsThe Attempt at a Solution...

Hi, here i see that the energy of a single particle is calculated by deriving the lagrangian to the speed. I obtain something similar to a linear momentum. and then i see that the total energy is this momentum multiplied by speed and then subtracting lagrangian. could you explain to me these things?

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...

Homework Statement A point mass m is fixed inside a hollow cylinder of radius R, mass M and moment of inertia I = MR^2. The cylinder rolls without slipping i) express the position (x2, y2) of the point mass in terms of the cylinders centre x. Choose x = 0 to be when the point mass is at the...

I've recently read in a textbook that a geodesic can be defined as the stationary point of the action \begin{align} I(\gamma)=\frac{1}{2}\int_a^b \underbrace{g(\dot{\gamma},\dot{\gamma})(s)}_{=:\mathcal{L}(\gamma,\dot{\gamma})} \mathrm{d}s \text{,} \end{align} where ##\gamma:[a,b]\rightarrow...

Hello all, I'm a bit baffled by the fact that the various quite different components of the SM Lagrangian (or other systems, btw) are simply summed up, without even one ponderation coefficient, in the total Lagrangian. I know one reason it is like that is that... it works in practice, but I...

In the paper http://physics.unipune.ernet.in/~phyed/26.2/File5.pdf, the author solves the LC-circuit using Euler-Lagrange equation. She assumes that the Lagrangian function for the circuit is $$L=T-V$$ where $$T=L\dot q^2/2$$ is the kinetic energy part $$V=q^2 / 2C$$ is the potential energy.She...

In Lancaster & Burnell book, "QFT for the gifted amateur", chapter 48, it is explained that, with a special set of ##\gamma## matrices, the Majorana ones, the Dirac equation may describe a fermion which is its own antiparticle. Then, a Majorana Lagrangian is considered...

I note the following: \begin{equation} \begin{split} \langle\vec{x}_n|e^{-i \frac{\mathcal{H}_n}{\hbar} (t_n-t_0)}|\vec{x}_{0}\rangle &=\delta(\vec{x}_n-\vec{x}_0)e^{-i \frac{\mathcal{H}_n}{\hbar} (t_n-t_0)} \end{split} \end{equation}I divide the time interval as follows...

I note the following: \begin{equation} \begin{split} \langle \vec{x}| \hat{U}(t-t_0) | \vec{x}_0 \rangle&=\langle \vec{x}| e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} | \vec{x}_0 \rangle \\ &=e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} \delta(\vec{x}-\vec{x}_0)...

Hi Everyone! I have done three years in my undergrad in physics/math and this summer I'm doing a research project in general relativity. I generally use a computer to do my GR computations, but there is a proof that I want to do by hand and I've been having some trouble. I want to show that...

Hi everyone. So I'm going through Landau/Lifshitz book on Mechanics and I read through a topic on inertial frames. So, because we are in an inertial frame, the Lagrangian ends up only being a function of the magnitude of the velocity only (v2) Now my question to you is, how does one prove that...

currently working on format.. sor i was not preparedHi I think this question would be much related to calculus more than physics cause it seems I'd lost my way cause of calculus... but anyway! it says, Q=- \frac{\partial{U}}{\partial{q}}+\frac{d}{dt}(\frac{\partial{U}}{ \partial{ \dot{q}}} )...

Consider the Dirac Lagrangian, L =\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi, where \overline{\psi}=\psi^{\dagger}\gamma^{0} , and show that, for \alpha\in\mathbb{R} and in the limit m\rightarrow0 , it is invariant under the chiral transformation...

In Hawking-Ellis Book(1973) "The large scale structure of space-time" p69-p70, they derive the energy-momentum tensor for perfect fluid by lagrangian formulation. They imply if ##D## is a sufficiently small compact region, one can represent a congruence by a diffeomorphism ##\gamma: [a,b]\times...

Homework Statement Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to...

Homework Statement A particle of mass m is on top of a frictionless hemisphere centered at the origin with radius a" Set up the lagrange equatinos determine the constraint force and the point at which the particle detaches from the hemisphere Homework Equations L=T-U The Attempt at a...

Hello! I started reading stuff on QFT and it seems that one of the main points is for the Lagrangian to be Lorentz invariant, so that the equations of motion remain the same in all inertial reference frames. I am not sure however i understand how do non inertial reference frames come into play...

Homework Statement A point like particle of mass m moves under gravity along a cycloid given in parametric form by $$x=R(\phi+\sin\phi),$$ $$y=R(1-cos\phi),$$ where R is the radius of the circle generating the cycloid and ##\phi## is the parameter (angle). The particle is released at the point...

We have a car accelerating at a uniform rate ## a ## and a pendulum of length ## l ## hanging from the ceiling ,inclined at an angle ## \phi ## to the vertical . I need to find ##\omega## for small oscillations. From the Lagrangian and Euler-Lagrange equations, the equation of motion is given...