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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415 views

What is Maupertuis' principle good for?

The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \delta \int p dq =0$$ and I am not sure ...
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71 views

Least action principle — numerical simulation strangeness

I'm trying to get some experience with the least action principle, and for this I chose a simple 1-dimensional problem of a particle moving in some field. The least action principle would then look ...
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2answers
320 views

Kugo and Ojima's Canonical Formulation of Yang-Mills using BRST

I am trying to study the canonical formulation of Yang-Mills theories so that I have direct access to the $n$-particle of the theory (i.e. the Hilbert Space). To that end, I am following Kugo and ...
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1answer
104 views

Variational form of Euler's incompressible fluid equations?

I am trying to derive Euler's incompressible fluid equations in terms of a variational stationary principle. Given Euler's flow equations: $$\frac{\partial v}{\partial t} = -\nabla p$$ $$\nabla\cdot ...
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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
680 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 ...
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2answers
436 views

Landau Mechanics: why does adding Lagrangians remove the indefiniteness of multiplying each Lagrangian by a different constant?

In Landau Mechanics (third edition page 4), why does adding Lagrangians of two non interacting parts remove the indefiniteness of multiplying each Lagrangian by a different constant? If both systems ...
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1answer
485 views

Lagrangian density for a Piano String

So I'm trying to do this problem where I'm given the Lagrangian density for a piano string which can vibrate both transversely and longitudinally. $\eta(x,t)$ is the transverse displacement and ...
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64 views

Hamiltonian Operator for nonrenormalizable Effective Field Theories?

Assuming we have a Effective Field Theory, for example a Real Scalar Field Theory, defined through a Lagrangian density of the form $\mathcal{L}_{eff} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi - ...
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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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81 views

Optimal tunnel shape for travelling inside the earth [duplicate]

Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other ...
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1answer
245 views

Determinant for a coupled fluctuation Lagrangian

Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the ...
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483 views

General equation of motion for elementary particles

Elementary particles can be grouped into spin-classes and described by specific equations, see below: Is there a general Lagrangian density from which all these equations can be derived? A ...
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286 views

For a particle to have physical mass, is it always necessary to have a mass term in the lagrangian?

Since the self-energy adds to the bare mass defined in the Lagrangian, is it possible to create a physical particle mass from the self-energy alone, with no mass terms occuring in the Lagrangian? On ...
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2answers
597 views

How to apply Noether's theorem

Say I have a point transformation: $$x' ~=~ (1 +\epsilon)x,$$ $$t' ~=~ (1 +\epsilon)^2t,$$ and Lagrangian $$ L ~=~ \frac{1}{2}m\dot{x}^2 - \frac{\alpha}{x^2}.$$ How do I go out about showing ...
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928 views

Why are generalized positions and generalized velocities considered as independent of each other?

I'm confused how $$\dot{\mathbf{r}}_{j}=\sum_{k}\frac{\partial\mathbf{r}_{j}}{\partial q_{k}}\dot{q}_k+\frac{\partial\mathbf{r}_{j}}{\partial t}$$ leads to the relation, ...
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3answers
341 views

Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation

In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
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1k views

Lorentz invariance of the integration measure

This is regards to the lorentz invariance of a classical scalar field theory. We assume that the action which is $S= \int d^4 x \mathcal{L}$, is invariant under a Lorentz transformation. How do you ...
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276 views

Euler Lagrange equation in different frames

Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways: ...
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2answers
135 views

Variation of Action with time coordinate variations

I was trying to derive equation (65) in the following review: http://relativity.livingreviews.org/open?pubNo=lrr-2004-4&page=articlesu23.html This slightly unusual then usual classical mechanics ...
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1answer
533 views

Why does Quantum Field Theory use Lagrangians rather than Hamiltonains? [duplicate]

Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains? I heard many reasons, but I'm not sure which is true. Some say it's just a matter of beauty, so Lagrangians are more ...
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3answers
801 views

Hanging chain in a planet's gravitational field

The curve for a chain hanging between two poles in a uniform gravitational field is known as the catenary. Is there known an expression for the curve of a hanging chain on a planet of mass $M$ which ...
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2answers
133 views

Getting the Lagrangian from the action in curved spacetime

Suppose I have this action: $$ S = \int \mathrm d^4 x\sqrt{-g}\times \text{something}$$ where $g$ is the determinant of the metric. Should I take the Lagrangian to be: $$ \mathcal L = \sqrt{-g} ...
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2answers
142 views

Landau's argument for dependence of Lagrangian on magnitude of velocity

In chapter 1, of Landau-Lifshitz Mechanics' book, Landau through isotropy and homogeneity of space and homogeneity of time proves that the Lagrangian must depend of magnitude of velocity of the ...
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1answer
504 views

Lagrangian dynamics with initial conditions: motion of free particle

I am very new to Lagrangian dynamics so I am trying to get my head around the practical usage. So far on here all I could find were proofs and they did not make much sense to me, especially when time ...
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416 views

How do I show that there exists variational/action principle for a given classical system?

We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
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3answers
82 views

Geodesic Equation from variation: Is the squared lagrangian equivalent?

It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, ...
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155 views

When is the principle of stationary action not the principle of least action?

I've only had a very brief introduction to Lagrangian mechanics. In a physics course I took last year, we briefly covered the principle of stationary action --- we looked at it, derived some equations ...
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190 views

Several stationary points of the action functional

In QFT the principle of stationary action states that we choose fields that will make the action stationary but what if the action has many stationary points (for a fixed choice of boundary ...
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1answer
293 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
208 views

Lagrange Multipliers Versus Generalized Coordinates

When forced to explain to someone why one could either set up a general Lagrangian & then incorporate constraints using Lagrange multipliers, as opposed to just setting up a Lagrangian with ...
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2answers
289 views

The “stationary potential energy” condition for static equilibrium in mechanical systems

I've often read that, for a mechanical system which can be described by $n$ generalized coordinates $q_1,...,q_n$, a point $\mathbf{Q}=(Q_1,...,Q_n)$ is a point of equilibrium if and only if the ...
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2answers
586 views

Conversion of the Nambo-Goto action into the Polyakov action?

I`ve read that the Nambo-Goto action containing the induced metric $\gamma_{\alpha\beta}$ $$\tag{1} S_{NG} ~=~ -T\int_{\tau_i}^{\tau_f} d\tau \int_0^{\ell} d\sigma \sqrt{-\gamma}$$ can be converted ...
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1answer
266 views

Non-integrability of the 2D double pendulum

Context: For a system with $n$ degrees of freedom (DOF), one has to deal with $2n$ independent coordinates ($2n$ dimensional phase space), of position $q$ and $\dot{q}$ in Lagrangian formulation, ...
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1answer
120 views

Why is the Hamiltonian the Legendre transform of the Lagrangian?

So, as the title says, why is the Hamiltonian the Legendre transform of the Lagrangian? I know that from quantum mechanics, one can start with the Hamiltonian defined as the generator of time ...
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2answers
133 views

When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?

The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time ...
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2answers
178 views

Lagrange multiplier and constraint force

The Lagrangian with Lagrange multiplier in the form $$L= T- V + \lambda f(q, \dot{q},t).$$ But there are different ways of writing the constraint $f = 0$. Will that lead to different EOMs? Let me ...
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651 views

Symmetry of Euler-Lagrange equations and conservation laws

Continuous symmetry of the action implies a conservation law, but what if equations of motion have a continuous symmetry? Does it imply a conservation law? Also is symmetry of equations of motion ...
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2answers
107 views

Independent systems and Lagrangians

Definition 1: The notion of independent systems has a precise meaning in probabilities. It states that the (joint) probability or finding the system ($S_1S_2$) in the configuration ($C_1C_2$) is ...
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238 views

Virtual differentials approach to Euler-Lagrange equation - necessary?

I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for the Euler-Lagrange equation. The whole notion of, and ...
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3answers
458 views

Generalizing Heisenberg Uncertainty Priniciple

Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$ $$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$ ...
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1answer
332 views

Differentiating inside an integral sign

I'm reading John Taylor's Classical Mechanics book and I'm at the part where he's deriving the Euler-Lagrange equation. Here is the part of the derivation that I didn't follow: I don't get how ...
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1answer
76 views

Total derivative in action of the field theory

Consider a classical field theory. When applying the least action I see that a term is considered total derivative. We say that $$\int \partial_\mu (\frac {\partial L}{\partial(\partial_\mu ...
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1answer
224 views

The BRST construction for YM with or without auxiliary field

I'm learning BRST symmetry for Yang-Mills theory and I see that there are two ways of writing BRST differential. In some books (for example Ryder's and Ramond's textbooks) BRST differential acts as ...
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1answer
125 views

Is it possible to project a problem of mechanics in a lower dimensionality?

I had the intuition that, in classical mechanics, when the trajectory of a body is known, then analysis of its motion can be done in the linear space of that trajectory, if all forces are projected on ...
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2k views

Lagrangian mechanics vs Hamiltonian mechanics [duplicate]

First of all, what are the differences between these two: Lagrangian mechanics and Hamiltonian mechanics? And secondly, do I need to learn both in order to study quantum mechanics and quantum field ...
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1answer
523 views

A Question about Virtual Work related to Newton's Third Law

In describing D'Alembert's principle, the lecture note I was provided with states that the total force $\mathbb F_l$ acting on a particle can be taken as, $$\mathbb F_l=F_l+\sum_mf_{ml}+C_l,$$ ...
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1answer
66 views

Why is the Legendre transformation an application of the duality relationship between points and lines?

When I read the Wiki about Legendre transformation, there is a statement The Legendre transformation is an application of the duality relationship between points and lines. What's the meaning of ...
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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
582 views

Noether current for the Yang-mills-higgs lagrangian

I am trying to calculate the Noether's current, more specifically, the energy density of the Yang-mills-Higgs Lagrangian. Please refer to the equations in the Harvey lectures on Magnetic Monopoles, ...