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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Derivation of equations of motion in Nordstrom's theory of scalar gravity?

Nordstrom's theory of a particle moving in the presence of a scalar field $\varphi (x)$ is given by $$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d ...
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1answer
78 views

Stationary action with maximized action [duplicate]

I would like to ask for an example (a lagrangian) both in classical and quantum level for which the action is maximaized (rather than minimized). What is special in these cases?
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2answers
184 views

Equations of motion from the Standard Model

For some time now I have been wondering if you could not derive any sort of equations of motion from the Standard Model: ...
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1answer
91 views

Transforming a lagrangian to hamiltonian and vice versa

I am not refering to Legendre transform, but to something more simple. In analytical mechanics, the Lagrangian can be described as $L=T-V$, and the Hamiltonian is if the Lagrangian doesn't explicitly ...
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0answers
50 views

What is the “momentum” referred to in the energy-momentum tensor

What is the "momentum" referred to in the energy momentum tensor from GR? Is it $m\dot{x}$ or is it the canonical momentum $\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\right)$ Also, I ...
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2answers
95 views

With respect to what quantities do I vary Lagrangians in field theory?

I have recently been wondering, with respect to which quantities (covariant or contravariant) one should vary QFT Lagrangians and whether there is some rule regarding this. Let me give an example ...
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2answers
396 views

Is the Dirac Lagrangian Hermitian?

I'm wondering of the Dirac Lagrangian density $$\mathcal{L} =\overline{\psi}(-i\gamma^\mu \partial_\mu +m)\psi $$ is an hermitian operator, since upon complex conjugating one gets ...
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1answer
128 views

Hamiltonian conservation

Lagrangian formalism does not involve forces that doesn't come from a potential and Hamiltonian formalism says that even though energy is not conserved due to a force like this, the Hamiltonian is ...
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2answers
201 views

Curved spacetime point particle Lagrangian density

This is probably trivially related to the question: Action for a point particle in a curved spacetime , but am a bit unsure how to write it as a Lagrangian density. In curved spacetime the action is ...
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2answers
249 views

Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field

I am tasked with reading and reproducing all the steps in J.J. Hopfield's 1958 paper "Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals". Embarrassingly I am stuck ...
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1answer
242 views

Derivation of Noether's theorem - A problem with physical significance

My question is about the field theoretic version of Noether's theorem. I am deeply troubled by one of the hypotheses of the theorem. As it is the standard textbook for Lagrange mechanics, I'll follow ...
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1answer
292 views

Constraints of massive relativistic point particle in hamiltonian mechanics

I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: ...
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1answer
226 views

Einstein action and the second derivatives

I have naive question about Einstein action for field-free case: $$ S = -\frac{1}{16 \pi G}\int \sqrt{-g} d^{4}x g^{\mu \nu}R_{\mu \nu}. $$ It contains the second derivatives of metric. When we want ...
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1answer
302 views

Point of Lagrange multipliers

I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
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1answer
169 views

What to do with a $\phi$ term in a Lagrangian?

I am considering a Lagrangian that is of the following form: $$\mathcal{L}=-{1\over 2}\partial_\mu\phi\partial^\mu\phi+2\mu^2\phi^2+2\sqrt{6}{\mu^3\over \lambda}\phi + {9\mu^4\over 2\lambda} + ...
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1answer
170 views

Why can we assume independent variables when using Lagrange multipliers in nonholonomic systems?

I'm studying from Goldstein's Classical Mechanics. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form $f_\alpha(q, \dot{q}, t) =0$, $\alpha = 1 ...
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1answer
246 views

Dirac Lagrangian density in curved spacetime

I'm trying to derive this form of the Dirac Lagrangian density in curved space-time: $$ \mathcal{L}~=~\det\left(e\right)\bar{\Psi}\Bigg ...
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8answers
4k 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 ...
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1answer
420 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 ...
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1answer
395 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 ...
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0answers
208 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 ...
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2answers
379 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 ...
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2answers
297 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?
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1answer
133 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 ...
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2answers
352 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 ...
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1answer
120 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 ...
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1answer
51 views

Integrating out fields from classical systems

Has anyone ever heard of integrating out fields from classical Lagrangians if they are quadratic?
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1answer
75 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 ...
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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 ...
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3answers
411 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 ...
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0answers
94 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: ...
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3answers
326 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
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1answer
165 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 ...
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1answer
689 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 ...
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0answers
78 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. ...
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1answer
339 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 ...
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0answers
261 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?
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1answer
283 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 ...
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0answers
468 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 ...
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1answer
44 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 ...
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5answers
524 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 ...
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1answer
211 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
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1answer
679 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
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1answer
192 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 ...
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2answers
234 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 $\mathbb{R} \times \mathbb{R}^2$. ...
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2answers
282 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.
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2answers
411 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. ...
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1answer
202 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 ...
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1answer
93 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 ...
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1answer
131 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, ...