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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312 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
121 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

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
57 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
185 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
112 views

How Hamilton's Principle was found?

Hamilton's principle states that the actual path a particle follows from points $p_1$ and $p_2$ in the configuration space between times $t_1$ and $t_2$ is such that the integral $$S = ...
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1answer
90 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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3k views

Lagrangian of two particles connected with a spring, free to rotate

Two particles of different masses $m_1$ and $m_2$ are connected by a massless spring of spring constant $k$ and equilibrium length $d$. The system rests on a frictionless table and may both oscillate ...
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1answer
533 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, ...
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1answer
485 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
65 views

How the lagrangian density is found?

In Classical Mechanics one usually considers the Lagrangian as $L = K - U$ where $K$ is the kinetic energy of the system and $U$ is the potential energy. One then gets the Euler-Lagrange equations and ...
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188 views

Non-local Lagrangian contact interaction

Conside a contact interaction given by a delta function on their worldlines. Use a gauge fixed Lagrangian for two point particles in terms of their proper times $t$ and $t^{\prime}$. Is it possible to ...
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2answers
238 views

Why is the Lagrangian quadratic in $\dot{q}$? [duplicate]

My teacher said we only consider Lagrangians which are quadratic in $\dot{q}$, and we don't take other Lagrangians. I couldn't understand why. Can anyone please explain this?
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1answer
290 views

speed of sound and the potential energy of an ideal gas; Goldstein derivation

I am looking the derivation of the speed of sound in Goldstein's Classical Mechanics (sec. 11-3, pp. 356-358, 1st ed). In order to write down the Lagrangian, he needs the kinetic and potential ...
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2answers
354 views

Derivation of the Lagrangian method using discretized time axis

I'm watching this video lecture by Leonard Susskind of Stanford: http://www.youtube.com/watch?v=3apIZCpmdls After some preliminaries, at 34 minutes he jumps into a discretization of the time axis ...
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219 views

Is the form of the Lagrangian relevant before the renormalization procedure?

In the renormalization procedure, is writing things like $$\varphi=\sqrt{Z_{\varphi}}\ \varphi_R\ ,\ \ m_0^2=Z_m\ m_R^2\ ,\ \ g_0=Z_g \mu^{\epsilon}\ g_R$$ and $$Z_i=1+\sum_{\nu=1}^\infty ...
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3answers
394 views

Calculating lagrangian density from first principle

In most of the field theory text they will start with lagrangian density for spin 1 and spin 1/2 particles. But i could find any text where this lagrangian density is derived from first principle.
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80 views

Gauge Invariance of Yang Mills Lagrangian

I am trying to show the invariance of the following Yang Mills Lagrangian: $$L= -\frac{1}{4} F^a_{\mu \nu} F_a^{\mu\nu} + J_a^\mu A_\mu^a$$ under the following gauge transformation ($\theta$ being a ...
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94 views

Gauge Invariance of the Non-abelian Chern-Simons Term

I'm trying to prove that, under the gauge transformation $$A_{\mu} \rightarrow A_{\mu}^{\prime} = g^{-1} A_{\mu} g + g^{-1} \partial_{\mu} g,$$ the non-abelian Chern-Simons Lagrangian density: ...
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128 views

How do you determine the Lagrangian?

I have always been puzzled by how do you arrive at Lagrangians? That is, how do you know that the functional you need to get Newton's equations is $L$ = $T-V(x)$ Do you derive the Lagrangian first ...
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1answer
98 views

Caldeira-Leggett Dissipation: cannot get it

I am trying to understand the Caldeira-Leggett model. It considers the Lagrangian $$L = \frac{1}{2} (\dot{Q}^2 - (\Omega^2-\Delta \Omega^2)Q^2) - Q \sum_{i} f_iq_i + \sum_{i}\frac{1}{2} (\dot{q}^2 - ...
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1answer
207 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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4answers
447 views

Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system?

I am a Physics undergraduate, so provide references with your responses. Landau & Lifshitz write in page one of their mechanics textbook: If all the co-ordinates and velocities are ...
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579 views

Is there a conserved quantity that enforces planar orbits in central force motion?

From what I remember, one of the first steps in finding the equations of motion for an orbiting body is to argue that the body's motion has to be restricted to a plane, because the central force has ...
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172 views

Are there measurable effects to scaling the action by a constant?

Classically, we obtain the equations of motion by finding a path which has an action that is stationary with respect to small changes in the path. That is the path for which: $\delta S =0$ Scaling ...
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82 views

Higher order derivatives - Equation of motion

One possible starting point to create a physical theory is the Lagrangian $L$. There we assume that the variation of the action $\delta S = \delta \int_{-\infty}^\infty dt \ L = 0$. In classical ...
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2answers
479 views

Lagrangian and conservation of energy

If Lagrangian of the motion is $$\mathcal{L}=\frac{1}{2}m\left(a^2\dot\phi^2+a^2\dot\theta^2\sin^2\phi\right)+mga\cos\phi,$$ how can I show that total mechanical energy is conserved? I've read ...
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100 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
379 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
374 views

Relating generalized momentum, generalized velocity, and kinetic energy: $2T~=~\sum_i p_{i}\dot{q}^{i}$

According to equation (6) on the first page of some lecture notes online, the above equation is used to prove the virial theorem. For rectangular coordinates, the relation $$ 2T~=~\sum_i ...
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1answer
139 views

Definition of Local Function

Now a days I am studying Srednicki's QFT book. In its third chapter it is written that Any local function of φ(x) is a Lorentz scalar, [...] . Now my question is: What is a local function?
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633 views

Good book for Analytical Mechanics

What is a good book for Analytical Mechanics? To be more specific, I would prefer a book that: Is written "for mathematicians", i.e. with high mathematics precision (for example, with less emphasis ...
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3answers
278 views

Is the Lagrangian “math” or “science”?

I've seen in class that we can get from Lagrangian to derive equations of motion (I know its used elsewhere in physics, but I haven't seen it yet). It's not clear to me whether the Lagrangian itself ...
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3answers
515 views

What is the difference between manifest Lorentz invariance and canonical Lorentz invariance?

I often read that the Lorentz symmetry is manifest in the path integral formulation but is not in the canonical quantization - what does this really mean?
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1answer
165 views

Missing terms in Hamiltonian after Legendre transformation of Lagrangian

Short question Given any Lagrangian density of fields one could possibly conceive, is it the case that after one has performed a Legendre transformation, if the Hamiltonian is then expressed in terms ...
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2answers
382 views

Global phase symmetry for complex scalar field theory

I have started to study QFT. And I have some difficulties in such classical situation. Suppose i want to calculate $\frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\phi$ for lagrangian ...
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2answers
241 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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1answer
452 views

The form of Lagrangian for a free particle

I've just registred here, and I'm very glad that finally I have found such a place for questions. I have small question about Classical Mechanics, Lagrangian of a free particle. I just read Deriving ...
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1answer
288 views

What variables does the action $S$ depend on?

Action is defined as, $$S ~=~ \int L(q, q', t) dt,$$ but my question is what variables does $S$ depend on? Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$? In ...
3
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1answer
61 views

Lagrangian description of Brownian motion?

I'm interested in the existence of a Lagrangian field theory description of Bronwnian motion, does such a thing exist? Given a particle of some spin $\sigma$, which has a Lagrangian associated with ...
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2answers
103 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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1answer
149 views

Is there a Lagrangian whose Euler-Lagrange equation is the gradient?

I am trying to recast a problem I am working on in terms of Lagrangian mechanics. I am in the following situation. Suppose I have a function $f:X \rightarrow \mathbb{R}$ (a field). In the its ...
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2answers
176 views

Bertrand's theorem

I found in Goldstein's Classical Mechanics that the condition for closed orbits is given by $\frac{d^2 V_{eff}}{dr^2}>0$.(bertrand's theorem). Can somebody explain to me, how this inequality is ...
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1answer
351 views

Gauge-invariant field strength term in Yang-Mills Lagrangian

I am reading the chapter of non-abelian gauge invariance from Peskin and Schroeder. Why is the term $-\frac{1}{4}(L_{\mu\nu}^i)^{2} $ gauge invariant?
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152 views

Does locality emerge from (classical) Lagrangian mechanics?

Consider a (classical) system of several interacting particles. Can it be shown that, if the Lagrangian of such a system is Lorenz invariant, there cannot be any space-like influences between the ...
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341 views

Variational method applied to brownian motion

It's possible apply the variational method to the brownian motion ? I mean, one of requisites on $y(t)$ is that it must be continuous and $\partial_t{y(t)}$ too, and in this case, $\partial_t{y(t)}$ ...
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1answer
50 views

Lagrangian formulation of the problem: small oscillations around an equilibrium

I'm having trouble understanding how some conclusions are made in my book. I'm studying from a coursebook based on Goldstein's "Classical Mechanics", here's what's written in my book, with my problems ...
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1answer
73 views

Translations and Noether's Theorem

I'm fine with $U(1)$ symmetry and Noether's Theorem, but struggling with the translations of the field; namely $$\phi'(x^{\mu})=\phi(x^{\mu}-a^{\mu}),$$ where $a^{\mu}$ constant four-vector ...
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1answer
226 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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2answers
512 views

Small oscillations of the double pendulum

From the Lagrangian I've got the following equations of motion for the double pendulum in 2D. (The masses are different but the lengths of the two pendula are equal.) Let $m_2$ be the lowest-hanging ...