Lagragian Definition and 24 Threads

In the derivation of Hamiltonian mechanics, the concept of "on-shell" vs "off-shell" is involved in the calculation. I searched it for like off-shelf, however it seems it makes sense in the context of four-momentum in special relativity. What is the meaning of that concept in the context of...

Hi Guys Please refer to the attached document for my derivation. The image presents the system in plan view, I know one my think that it is unstable structure based on a single pinned connection however this is a simplification of a complex structure sitting on a slew bearing. Gravity does...

My question is about the general relationship between the constraint functions and the constraint forces, but I found it easier to explain my problem over the example of a double pendulum: Consider a double pendulum with the generalized coordinates ##q=\{l_1,\theta_1,l_2,\theta_2\}##,: The...

One of the first things Landau does in his Mechanics book is give an argument as to why the Lagrangian of a free particle must be our conventional kinetic energy. Heuristically, he justifies it, but leaves out the details, perhaps being too obvious. They aren't obvious to me. While in free space...

The Lagrangian for a massless particle in a potential, using the ##(-,+,+,+)## metric signature, is $$L = \frac{\dot{x}_\mu \dot{x}^\mu}{2e} - V,$$ where ##\dot{x}^\mu := \frac{dx^\mu}{d\lambda}## is the velocity, ##\lambda## is some worldline parameter, ##e## is the auxiliary einbein and...

To calculate the Hamiltonian of a charged particle immersed in an electromagnetic field, one calculates the Lagrangian with Euler's equation obtaining ##L=\frac{1}{2}mv^2-e\phi+e\vec{v}\cdot\vec{A}## where ##\phi## is the scalar potential and ##\vec{A}## the vector potential, and then we go to...

I think I undeerstand Lagrangian mechanics but I have a question that will help to clarify some concepts. Imagine I throw a pencil. For that I have 5 generalised coordinates (x,y,z and 2 rotational). When I express Kinetic Energy (T) as: $$T = 1/2m\dot{x^{2}}+1/2m\dot{y^{2}}+1/2m\dot{z^{2}} +...

Hi all, Consider a system of ##N## noninteracting, identical electric point dipoles (dipole moment ##\vec{\mu}##) subjected to an external field ##\vec{E}=E\hat{z}##. The Lagrangian for this system is...

Hello : i am reading now landau & lifshitz book on mechanics and i have small question : about L(v^2) notation it was not very clear in the book and i couldn't understand it correctly anyone can explain it or provide a link with explanation page (4 - 5) Best regards Hagop

In the first two chapters of Goldstein mechanics, the Lagrange equations are derived from both D'Alembert's principle and Hamilton's principle. I want to know what're the applicability of these two approaches to Lagrangian mechanics? Is one more powerful than the other or are they completely...

It is the first time that I am faced with a complex field, I would not want to be wrong about how to solve this type of problem. Usually to solve the equations of motion I apply the Euler Lagrange equations. $$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...

I must find the Lagrangian for an undamped pendulum using the diagram showed below, I've no idea what to do with the second angle φ2 because is measured from the line that joins the two pivot points. The ecuations I must obtain are as follows I get so many different things but I can't reach...

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...

The final answer should have a negative b^2⋅r(dot)^2⋅r term but I have no idea how that term would become negative. Also I know for a fact that my Lagrangian is correct.

I start out by substituting rcos(Θ) and rsin(Θ) for x and y respectively. This gives me z=(b/2)r^2. The Lagrangian of this system is (1/2)m(rdot^2+r^2⋅Θdot^2+zdot^2)-mgz. (rdot and such is the time derivative of said variable). I then find the time derivative of z, giving me zdot=br⋅rdot and...

Why ##\frac{\partial (0.5*m*\dot{x}^2-m*g*x)}{\partial x}=-mg##? why ##\frac{\partial \dot{x}}{\partial x}=0##? Why ##\frac{\partial (0.5*m*\dot{x}^2-m*g*x)}{\partial \dot{x}}=m*\dot{x}## ? why ##\frac{\partial x}{\partial \dot{x}}=0##? Does it assume that speed is same at every location? I...

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...

Homework Statement Vary the action of a Lagrangian for a scalar field. I kind of just need someone to read over this, I'm not sure if my steps are actually logical (especially the one before we do integration by parts). Since this isn't actually homework, we can move it to the classical...

Is Eulerian view valid when the flow is unsteady? I think Eulerian view is valid only for steady flows because the points in the flow domain should be with constant velocities. Thank you.

Homework Statement A mass m1 slides on a frictionless horizontal table. it is attached by a massless cord passing over a massless pulley to a mass, m2. A cylinder of mass m3, radius r, and moment of inertia 1/2(m3*r^2) rests on m1. (a) Choose and specify generalized coordinates (two are...

Hi Everyone, I'm interested in forming Lagrange's equations of motion using a Lagrangian I made up today. It looks like this: \mathcal{L}(\dot{\psi},\psi)=\sqrt{\langle \dot{\psi}_b|C^{\dagger}_bC_a|\dot{\psi}_a\rangle} where C^{\dagger}_b is a creation operator for a basis b etc... and a...

Hi this is my first time posting on this forum. I have an question about Lagragian points. I was trying to find L2 lagrangian point, a point that lies on the line defined by the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance...

Problem 3 in the continuous systems and fields chapter of (the first edition, 1956 printing) of Goldstein's classical mechanics has the following Lagrangian: L = \frac{h^2}{8 \pi^2 m} \nabla \psi \cdot \nabla \psi^{*} + V \psi \psi^{*} + \frac{h}{2\pi i} ( \psi^{*} \dot{\psi} - \psi...

I set up a Lagragian equation that involves time t. What do I do? I only know how to solve Lagragian equation in the absence of time. Please help.