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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5
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2answers
821 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 ...
3
votes
1answer
161 views

Are there two types of D-term and two types of F-term in SUSY?

I've noticed that one can obtain D-terms either by integrating a vector superfield (the vector multiplet) over superspace or by integrating a Kahler potential over superspace. In both cases we get ...
1
vote
1answer
307 views

Hamiltonian from Euclidean lagrangian?

Can somebody help me in deriving the Hamiltonian of system starting from Euclidean Lagrangian? Say we are given the Minkowski Lagrangian $$L_m = \frac{\dot{\phi}^2}{2} - V(\phi).$$ The Hamiltonian ...
3
votes
2answers
699 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 ...
3
votes
1answer
410 views

Non-relativistic limit of complex scalar field

In page 42 of David Tong's lectures on Quantum Field Theory, he says that one can also derive the Schrödinger Lagrangian by taking the non-relativistic limit of the (complex?) scalar field Lagrangian. ...
6
votes
2answers
615 views

Derivation of the Polyakov Action

As is usually done when first presenting string theory, the Nambu-Goto Action, $$ S_{\text{NG}}:=-T\int d\tau d\sigma \sqrt{-g} $$ ($g:=\det (g_{\alpha \beta})$ is the induced metric on the ...
9
votes
6answers
840 views

What is Quantization?

In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & ...
3
votes
1answer
139 views

Can the Solar System be assumed a single body concentrated in the Sun?

This question springs from a comment against my question posted on the Space SE My questions may seem inane, or obvious to most of you real physics people too ... Any number of sources put the peg ...
4
votes
1answer
127 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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0answers
132 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
2
votes
1answer
211 views

Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0 = ...
3
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0answers
72 views

Setting Lagrangian [closed]

Can you help me to set Lagrangian? I found that $$\vec r_A=b\sin\theta\vec i+b\cos\theta\vec j$$ $$\dot{\vec r_A}=b\dot\theta\cos\theta\vec i-b\dot\theta\sin\theta\vec j$$ For point $G$ I've got ...
0
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0answers
715 views

Scalar field lagrangian in curved spacetime

I am studying inflation theory for a scalar field $\phi$ in curved spacetime. I want to obtain Euler-Lagrange equations for the action: $$ I\left[\phi\right] = \int ...
3
votes
1answer
1k views

Equations of motion for a spherical pendulum in a non-inertial reference frame

Take a spherical pendulum with bob mass $m$, rod length $\ell$ and physical coordinates $\theta$, $\phi$ (spherical angles) and $h$ (the hinge height with respect to the coordinate origin). The rod is ...
3
votes
1answer
415 views

Euler's equations of rigid body motion from least action principle

I would like to derive Euler's equations of rigid body motion from least action principle. Suppose we are in free space so we have no gravity so Lagrangian is equal to kinetic energy. $$ L = T = ...
5
votes
1answer
493 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 ...
1
vote
1answer
265 views

Beads sliding on a hoop

Two particles, $P_1$ and $P_2$, of equal masses $m$ are linked by a spring of stiffness $k$ and natural length $a$. They are sliding freely without friction along a horizontal fixed hoop of radius ...
1
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0answers
1k views

Equations of motion for a pendulum and spring system

The question is available here: I've modeled the building as a rod on a torsional spring (with a pendulum hanging from the top). $\phi$ is the angle from the centre for the pendulum and $\theta$ ...
3
votes
1answer
407 views

D'Alembert's principle

Actually I have some troubles to understand what this principle is all about, so I want to use the simple pendulum in order to get the idea. Since I have read a few passages that dealt with this ...
3
votes
1answer
349 views

How to introduce the electromagnetic field in Quantum Field Theory?

There are many ways to introduce the electromagnrtic field in Quantum Field Theory(QFT), such as canonical quantization method which introduces the creation and annihilation oprators by treating the ...
4
votes
2answers
823 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 ...
9
votes
2answers
404 views

How do I read the simple, but contradictory, Lagrangian ($\mathcal{L} = x + v$)?

I understand the lagrangian formulation of classical mechanics, to a degree. I can derive the Euler-Lagrange equations from the "least" action principle, and equivalently can determine the equations ...
2
votes
1answer
1k 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 ...
1
vote
1answer
154 views

Applying Lagrange's equations ignoring normal reaction

A small bead is sliding on a smooth vertical circular hoop of radius $a$, which is constrained to rotate with constant angular velocity $\omega$ about its vertical diameter. $\theta$ is an angle ...
4
votes
1answer
228 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 ...
9
votes
5answers
5k views

What is canonical momentum?

What does the canonical momentum $\textbf{p}=m\textbf{v}+e\textbf{A}$ mean? Is it just momentum accounting for electromagnetic effects?
8
votes
3answers
2k views

Maxwells Equation from Electromagnetic Lagrangian

In Heaviside-Lorentz units the Maxwell's equations are: $$\nabla \cdot \vec{E} = \rho $$ $$ \nabla \times \vec{B} - \frac{\partial \vec{E}}{\partial t} = \vec{J}$$ $$ \nabla \times \vec{E} + ...
4
votes
2answers
348 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 ...
2
votes
4answers
515 views

Are Lagrangians and Hamiltonians used by Engineers?

Analytical Mechanics (Lagrangian and Hamiltonian) are useful in Physics (e.g. in Quantum Mechanics) but are they also used in application, by engineers? For example, are they used in designing bridges ...
2
votes
1answer
194 views

Definition of Kinetic energy

In class we had that $ T= \frac{1}{2}T_{ij}v_iv_j$ where we used the Einstein summation convention. Hitherto we only discussed examples where the kinetic energy was dependent of the square of one ...
1
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3answers
332 views

Dimensions in lagrangian potential

According to Mankowski flat space dimensions We can write, $$L= \int \text{dt} \text d^d{x} \left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
2
votes
1answer
116 views

Cyclic integral?

Can anyone explain me what is cyclic integral and give me some instructions how to show that there exist cyclic integral for Lagrangian $$L~=~\frac{1}{2}ma^2{\dot {\theta}}^2+\frac{1}{2}ma^2{\dot ...
0
votes
2answers
299 views

What's an “Action” and what does the Lagrangian equation mean exactly?

How and why would a particle take the shortest path? $L=KE-PE$? What's the $KE-PE$ mean in English? I understand the 'mechanics' but not the idea itself. Please explain simply, I do know Calculus ...
7
votes
0answers
487 views

Mechanical similarity in Landau

I've read this very short paragraph from Landau & Lifshitz's Mechanics (Chap.2, Par.10) (that you can find here) about Mechanical similarity. I was looking for some more detailed explanations of ...
1
vote
1answer
149 views

Finding out Energy value

A Lagrangian is given by, $$L= \left(\frac{\pi}{2}\right)^2 R^d \left[\frac{1}{2}\dot A^2 - V(A_{max})\right]$$ $$E=\left(\frac{\pi}{2}\right)^2R^d V(A_{max}) $$ where V (A) now includes nonlinear ...
3
votes
0answers
196 views

Angular momentum of particle in dipole magnetic field

Basically I'm just trying to find the expression for the angular momentum of a particle of mass $m$ and charge $q$ in a dipole magnetic field. In cylindrical coordinates, ...
3
votes
2answers
276 views

A kind of Noether's theorem for the Hamiltonian

How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law using ...
10
votes
3answers
697 views

Equation of motion of a photon in a given metric

I have this metric: $$ds^2=-dt^2+e^tdx^2$$ and I want to find the equation of motion (of x). for that i thought I have two options: using E.L. with the Lagrangian: $L=-\dot t ^2+e^t\dot x ^2 $. ...
8
votes
5answers
340 views

Why can't we obtain a Hamiltonian by substituting?

This question may sound a bit dumb. Why can't we obtain the Hamiltonian of a system simply by finding $\dot{q}$ in terms of $p$ and then evaluating the Lagrangian with $\dot{q} = \dot{q}(p)$? Wouldn't ...
3
votes
1answer
434 views

Principle of Least Action [duplicate]

Is the principle of least action actually a principle of least action or just one of stationary action? I think I read in Landau/Lifschitz that there are some examples where the action of an actual ...
11
votes
1answer
449 views

What makes a Lagrangian a Lagrangian?

I just wanted to know what the characteristic property of a Lagrangian is? How do you see without referring to Newtonian Mechanics that it has to be $L=T-V$? People constructed a Lagrangian in ...
2
votes
2answers
282 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 ...
20
votes
2answers
442 views

How to combine these equations of constraint?

I want to model a nonholonomic system of an arbitrary rotating disk in 3D, which rolls without slipping, and doesn't have to stay vertical. (think spinning a penny on the table) I want to use the ...
7
votes
1answer
127 views

Is it possible to derive the brane action in pure supergravity?

The branes that source the RR fields of supergravity are described by the DBI action plus a CS term. I know this only from superstring considerations. Is there a way to find this result without ...
6
votes
3answers
503 views

Principle of Least Action via Finite-Difference Method

I am reading Gelfand's Calculus of Variations & mathematically everything makes sense to me, it makes perfect sense to me to set up the mathematics of extremization of functionals & show that ...
1
vote
2answers
223 views

Is this a valid derivation of the Legendre transformation from the Euler-Lagrange condition

E-L condition: $$\frac{d p}{dt}=\frac{\partial L}{\partial q}$$ Where $p=\frac{\partial L}{\partial \dot{q}}$ Are the following steps valid: $$\frac{\partial q}{dt} dp=\partial L$$ $$\dot{q} \: ...
2
votes
1answer
112 views

Question about Lagrangian mechanics

I have a question about the Lagrangian-formalism; they state that the holonomic constraints are expressed as: $$f_\alpha(x_1,y_1,z_1,...,x_n,y_n,z_n,t) = 0 $$ where $\alpha = 1,...,L$ They state ...
2
votes
3answers
4k views

How is a Hamiltonian constructed from a Lagrangian with a Legendre transform

many textbooks tell me that Hamiltonians are constructed from Lagrangians like $$L=L(q,\dot{q})$$ with a Legendre transformation to obtain the Hamiltonian as $$H=\dot{q}\frac{\partial L}{\partial ...
4
votes
2answers
238 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 ...
0
votes
1answer
244 views

Solving differential Equation for the Two-Body Problem

So, I'm following the derivation in D. Morin, Introduction to Classical Mechanics, of the equations for a two-body system. I understand all of it, aside from this one step. When he's talking about ...