For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.

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42
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8answers
6k views

What's the point of Hamiltonian mechanics?

I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I ...
1
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1answer
534 views

Using Lagrange's Equations with Generalized forces

I am a bit confused on how this works. For instance if I wanted to look at an object moving in 2 dimensions only subject to gravity (and assuming that the potential is just mgy), I get that my ...
10
votes
1answer
507 views

What's the physical intuition for symplectic structures?

I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector ...
6
votes
0answers
215 views

Equation of motion for cyclic model of the universe

I recently started to study about cyclic universe. I came across this article [1]. My question is about the action that used for describing the cyclic model: $$S = \int d^{4}x\sqrt{-g}(\frac{1}{16\pi ...
2
votes
2answers
485 views

First integral of relativistic Euler-Lagrange equations

Connsider a pseudo-Riemannian ($4$-dimensional) manifold $M$ with a pseudometric $g_{ab}$. The Lagrangian of a free particle in $M$ (in analogy to the flat case) is $$\mathcal ...
0
votes
2answers
389 views

Math required for learning Lagrangian mechanics [duplicate]

How much knowledge of maths is required for learning Lagrangian mechanics? Also from where can I learn this math?
3
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1answer
153 views

Hamilton-Jacobi formalism and on-shell actions

My question is essentially how to extract the canonical momentum out of an on-shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which ...
1
vote
2answers
514 views

Einstein equation and scalar field stress-energy tensor

Let's have interaction between gravitational and scalar real fields. For an action of gravitational field in vacuum I add term $S_{m} = \int d^{4}x\sqrt{-g}L_{m}$, where $$ L_{m} = \frac{1}{2}g^{\mu ...
1
vote
1answer
131 views

Something between Lagrangian and Hamiltonian called Routhian

So, in my mechanics class, the teacher mentioned there is a special function which is kind of a midpoint between the Lagrangian and the Hamiltonian, called the Routhian. Now, I wanted to give it a ...
1
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1answer
57 views

Integrating out fields from classical systems

Has anyone ever heard of integrating out fields from classical Lagrangians if they are quadratic?
1
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1answer
81 views

Why does a particle fall in a straight line?

In Lagrangian Mechanics we choose the path of least action. Given a uniform gravitational field, and a particle of finite mass; and fixing two points the start & end-point we consider all paths ...
1
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0answers
58 views

Some strange transformation [closed]

In a lecture (look at the chapter "The fermion determinant in a constant field", p. 5) I found some strange transformation, which is given by eq. 18. How to prove it? Exactly, I don't understand the ...
2
votes
3answers
555 views

Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu ...
3
votes
0answers
108 views

Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: ...
7
votes
3answers
363 views

Why not formulate Quantum Mechanics using Lagrangians? [duplicate]

As the title implies, why is it that the most common formalisms we use in quantum mechanics prefer to describe systems in the terms of a Hamiltionian instead of a Lagrangian? Is there some ...
2
votes
1answer
178 views

Derive non-linear $\sigma$ model from a theory of SU(2) matirx

It's said in Chapter VI.4 of A. Zee's book Quantum Field Theory in a Nutshell, a theory defined as $L(U(x))=\frac{f^2}{4}Tr(\partial_{\mu}U^{\dagger}\cdot\partial^{\mu}U)$, can be write in the form of ...
2
votes
1answer
960 views

Equations of motion for a pendulum in 3D?

I am trying to solve for the equations of motion to simulate a pendulum. I decided to use the spherical coordinates. The Lagrange equation is: where L = length of the rope ϕ= angle of the ...
1
vote
0answers
79 views

How to analyze this constraint question

Let $\gamma$ be a smooth curve in the plane, and introduce curvilinear coordinates $q_1,q_2$ on a neighborhood of $\gamma$; $q_1$ is the direction of $\gamma$ and $q_2$ is distance from the curve. ...
6
votes
1answer
405 views

Peskin & Schroeder Chapter 3.1 EoM Lorentz Invariant under Lorentz Invariant Lagrangian

From Peskin & Schroeder QFT page 35: The Lagrangian formulation of field theory makes it especially easy to discuss Lorentz invariance. And equation of motion is automatically Lorentz ...
3
votes
0answers
284 views

How to understand the QED, QCD and standard model Lagrangians? [closed]

How do you read the QED, QCD and standard model Lagrangians? What do all the symbols and tensors represent? And, how can you derive them by yourselves?
2
votes
1answer
346 views

Ricci scalar in Scalar Field in Curved Space-time

I was recently looking at a Lagrangian of a scalar field in curved space-time at http://www.unc.edu/~mgood/research/Carroll_QFT_CS.pdf on page 8. I am not a physicist, and I am currently studying ...
5
votes
0answers
528 views

Is Feynman talking about the Zeroth Law of Thermodynamics?

In Volume 1 Chapter 39 of the Feynman Lectures on Physics, Feynman derives the ideal gas law from Newton's laws of motion. But then on page 41-1, he puts a caveat to the derivation he has just ...
1
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1answer
56 views

Double variation of Schwinger action principle

The Schwinger action principle is given by $$\delta_{1}\big\langle b\big|a\big\rangle= i\int_{t_{a}}^{t_{b}}\text{d}t\,\sum_{c,d}\big\langle b\big|c\big\rangle\big\langle ...
4
votes
5answers
751 views

Motivation for form of Lagrangian

This question (in lagrangian mechanics) might be silly, but why is the Lagrangian L defined as: $L = T - V$? I understand that the total mechanical energy of an isolated system is conserved, and that ...
-4
votes
1answer
286 views

Classical Mechanics - Equation of motion, Lagrangian, Newtons 2nd Law [closed]

I really don't even know where to start with this question any help would go very very far. http://imgur.com/g4KxNY5
5
votes
1answer
886 views

How to find the Stress-Energy tensor?

I am a bit at loss about how to proceed to find the stress-energy tensor given some distribution of matter. The Wikipedia page gives some examples, and some (inequivalent) definitions for it: Using ...
2
votes
1answer
225 views

Can the Lagrange Multipliers depend on the coordinates?

When dealing with Lagrange multipliers to solve systems with constraints we usually have two ways if the constraints are holonomic: Differentiate the constraint and add the appropiate term to the ...
2
votes
2answers
288 views

Exercise about Lagrange-Euler equations

I'm solving an exercise about the Lagrange-Euler equations, that states the following: Let $\gamma (t) = \{ (t,q) : q = q(t), t_0 \leq t \leq t_1\}$ be a curve in $\mathbb{R} \times \mathbb{R}^2$. ...
1
vote
2answers
362 views

Lagrangian and hamiltonian of interaction

How to prove that lagrangian of interaction is equal to hamiltonian of interaction with minus sign? For example, I can't prove it for special case - quantum electrodynamics.
5
votes
2answers
448 views

How can you solve this “paradox”? Central potential

A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one. ...
2
votes
1answer
249 views

Diagonalize mass matrix term for fermions and “doubling trick” in m(atrix) theory

Can someone help me understand the "Doubling trick" at page 36 in http://inspirehep.net/record/887513/files/sis-2002-060.pdf (named "Scattering in Supersymmetric M(atrix) Models" by Robert Helling) or ...
4
votes
1answer
106 views

Can I really take the classical field equations at face value in QFT?

To be concrete, let's say I have a relativistic $\phi^4$ theory [with Minkowski signature $(+,-,-,-)$] $$ \tag{1} \mathcal{L} ~=~ \frac{1}{2} \left ( \partial_{\mu} \phi \partial^{\mu} \phi - m^2 ...
2
votes
1answer
186 views

Expansion of a function

In Landau-Lifschitz, following expansion is given, We have, $$L(v'^2)~=~L(v^2+2\textbf{v}\cdot\epsilon+\epsilon ^2)$$ expanding this in powers of $\epsilon$ and neglecting powers of higher order, ...
3
votes
1answer
1k views

Deriving D'Alembert's Principle

The wiki article states that D'Alembert's Principle cannot derived from Newton's Laws alone and must stated as a postulate. Can someone explain why this is? It seems to me a rather obvious principle.
1
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0answers
146 views

A discrete approach to the catenary

I'm trying to work out a model for the system above, that is, $N$ particles of unitary mass subject to the constraints: $$1=\varphi _i(\mathbf r _1,\mathbf {r}_2,...,\mathbf r _n)=|\mathbf ...
1
vote
1answer
98 views

Classical mechanical problem

I have two planes, one characterized by equation $$\phi_1=f(x)-z=0$$ and another $$\phi_2=\alpha y-z=0$$ where $\alpha$ is arbitrary. In their line of intersection(we assume it exist and is continous) ...
1
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1answer
1k views

Explicit time dependence of the Lagrangian and Energy Conservation

Why is energy(or in more general terms,the Hamiltonian) not conserved when the Lagrangian has an explicit time dependence? I know that we can derive the identity: $\frac{\partial ...
2
votes
0answers
47 views

Acceleration of 2 bodies tied with a string [closed]

Find the acceleration of the block of mass M shown in the figure . The co-efficient of friction between the 2 blocks is μ1 and that between the bigger block and ground is μ2. Could someone help ...
2
votes
1answer
155 views

Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

The question is about: (1) whether giving a Lagrangian is sufficient enough to (uniquely) well-define a Hamiltonianian quantum theory with a Hilbert space? The answer should be Yes, or No. If ...
9
votes
4answers
804 views

D'Alembert's Principle: Necesssity of virtual displacements

Why is the D'Alembert's Principle $$\sum_{i} ( {F}_{i} - m_i \bf{a}_i )\cdot \delta \bf r_i = 0$$ stated in terms of "virtual" displacements instead of actual displacements? Why is it so necessary ...
7
votes
1answer
522 views

In Path Integrals, lagrangian or hamiltonian are fundamental?

When studying path-integrals one question arose to my mind... Which presentation is more fundamental to calculate the propagator? The one based on the Hamiltonian (phase space)? $$K(B|A) = \int ...
1
vote
0answers
327 views

Lagrangian Dynamics Question [closed]

Two equal masses of mass M are glued to a massless hoop of radius R is free to rotate about its center in a vertical plane. The angle between the masses is 2$\theta$. Find the frequency of ...
0
votes
1answer
404 views

Canonical momentum Velocity dependent Lagrangian

I have a homework problem wich I think I'm on the verge of solving but need help with some relations: Show that if the potential $U$ in the Lagrangian contains velocity-dependent terms, the ...
7
votes
2answers
790 views

Functional derivative in Lagrangian field theory

The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') ...
2
votes
1answer
689 views

Lorentz force from velocity-dependent potential and Lagrangian

There is something i'm missing. I am at page 22-23 of Goldstein Classical Mechanics 3rd ed. Lorentz force can be derived from a potential $$U=q\phi-q\mathbf{A}\cdot\mathbf{v}$$ Where $\phi(t,x,y,z)$ ...
1
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1answer
131 views

Lagrange multiplier dependent on time

At the moment I am following a course on variational methods for mathematicians. Last week we derived the Euler-Lagrange equations for a functional under a constraint. In this derivation we found that ...
0
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1answer
351 views

Working with a Routhian for a specific system

I asked a more general question earlier about the Routhian, but I'm still having trouble working with it. Here's my specific case. Given the following Lagrangian: ...
3
votes
1answer
537 views

How do you derive Lagrange's equation of motion from a Routhian?

Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\frac{d}{dt}\frac{∂R}{∂\dot{\phi}}-\frac{∂R}{∂\phi}=0$$ And ...
1
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1answer
423 views

Virtual displacement and generalized coordinates

I have a doubt regarding the expression of a virtual displacement using generalized coordinates. I will state the definitions I'm taking and the problem. The system is composed by $n$ points with ...
1
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1answer
121 views

Landau Lifshitz energy for uniform rotation

Landau Lifshitz claim in their Mechanics book (39.11) that for a uniform rotation we have $ E = \frac{mv^2}{2} - \frac{m}{2} (\omega \times r)^2 + U,$ where the rotation is given by $v' = v + \omega ...