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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2
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2answers
120 views

Is angular momentum conserved for a mass fixed to a horizontal guide

I have the system shown below. Mass 1 confined to a vertical guide, and mass 2 confined to a horizontal guide joined together by a spring. My question is very simple: is the total angular momentum ...
1
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1answer
93 views

Equations of motion for double spherical pendulum simply?

I am attempting to simulate a double spherical pendulum, i.e. a combination of the spherical pendulum and the double pendulum. I understand that the equations of motion can be derived via the ...
5
votes
2answers
172 views

Derivation of momentum in QFT - from Energy-Momentum Tensor [closed]

The conserved 4-momentum operator for the complex scalar field $\psi = \frac{1}{\sqrt{2}}(\psi_1 + i\psi_2)$ is given in terms of the mode operators in $\psi$ and $\psi^{\dagger}$ as $$P^{\nu} = \int ...
1
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0answers
35 views

Origin of Hamilton's variational principle [duplicate]

My question is what is the theoretical origin of Hamilton's principle. I mean is there any rigorous mathematical proof of this principle from some more basic principles?
2
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1answer
104 views

Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
1
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4answers
228 views

What are the boundary conditions associated to this lagrangian?

Suppose that $L(q^i, \dot{q}^i)$ is a standard and well behaved lagrangian associated to some Dirichlet boundary conditions : $q^i(t_1) = q_1^i$ and $q^i(t_2) = q_2^i$. Now I have this new lagrangian ...
1
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1answer
104 views

Gauge the symmetry $φ \to φ + a(x)$ for a free massless real scalar field

How does one alter the Lagrangian density for a real scalar field $$\frac{∂_μφ∂^μφ}{2}$$ such that is will be invariant under the gauge transformation $φ → φ + a(x)$? For a complex scalar field ...
3
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0answers
82 views

On the surface, is the law of maximum entropy production the same as principle of least action?

I just have read about the law of maximum entropy production. Someone has idolized it enough to make an whole website just for it: http://www.lawofmaximumentropyproduction.com/ A system will ...
4
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1answer
87 views

$SO(N)$ symmetric theory of $N$ real scalar fields, why do charges have correct commutation relations of generators?

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda(\Phi^a ...
0
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0answers
152 views

Deduction of Lagrange's equations for non-conservative systems using Hamilton's Principle

Consider $\vec{F}$, as the total force applied on the system, $U$ the potential energy of the existent field, and $\vec{Q}$ a non-conservative force. We have that: $$F_k=-\frac{\partial U}{\partial ...
1
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0answers
103 views

A bead is threaded on a frictionless vertical wire loop of radius R

The question is the very last sentence at the end of this post. In this post, I'll demonstrate how I reach to a contradiction(the conditions mentioned in conjecture 1 should be satisfied by all ...
0
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1answer
213 views

Bead on a smooth rotating rod

Suppose a bead ($m$) is free to move on a thin rod in the otherwise empty space. The rod is made to rotate at constant angular speed $\omega$. Lets assume the initial position of the bead is $(r_o, ...
3
votes
0answers
37 views

Action for solution of general nth order differential equation [duplicate]

Suppose I want to find solution to a general nth order differential equation. (If I am right about the logic then) one might say that the solution $y\equiv y(x)$ is that function for which the ...
1
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0answers
44 views

Proof of Hamilton's principle [duplicate]

Is there a anything like a proof of Hamilton's principle? Where would I find it?
0
votes
1answer
44 views

How to explain the motion of these pendulums? [duplicate]

Got very interested recently in a video I saw running thru my feed: https://www.facebook.com/PortalAECweb/videos/913996365374257/ Well, I got very intrigued about the physics of it and wanted to ...
0
votes
1answer
65 views

Deriving the motion equation for a minimally-coupled scalar field in general relativity [closed]

From the following Lagrangian $$ L = \sqrt{-g} (R+\epsilon\partial^\mu\phi\partial_\mu\phi) $$ I'm getting the motion equation (for the field $g_{\mu\nu}$ and vaccum)as: $$ R_{\mu\nu} - ...
3
votes
2answers
194 views

Is there something more to Noether's theorem?

From the definition of Lagrangian mechanics, Noether's theorem shows that conservation of momentum and energy comes from invariance vs time and space. Is the reverse true? Are Lagrangian mechanics ...
4
votes
1answer
134 views

The Lagrangian of a Rocket

I am trying to understand how to write the Lagrangian of a system which consists of a rocket losing gas mass in a rate of $\frac{dm}{dt}$, the gas moving in a velocity of $u_0$ in the rocket's view? ...
0
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0answers
35 views

Action functional of Born-Infeld model

I have a Born-Infeld action functional like this $$I[A,\phi]~=~\int b^2(\sqrt{1+(|\bigtriangledown\times A|^2)/b^2}-1)+|D_A\phi|^2 + b^2(1-\sqrt{(1-|\phi|^2)^2/b^2} ).$$ Have any books or notes talk ...
1
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0answers
77 views

Proof of Lagrangian

I'm having some trouble with some math on a problem for a physics class (looking for help with some partial derivatives, not an answer). Let $$L'=L+\dfrac{dF}{dt},$$ where $L$ is a Lagrangian and $F$ ...
1
vote
1answer
102 views

What does it mean for an action to be defined “on-shell”?

Some actions like 11D supergravity are defined "on-shell". What does this mean exactly? Can you give me an example? Say for example the Klein-Gordon action. Can this be defined on-shell too?
0
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1answer
45 views

Fourier transform for $W(J)$ in a free QFT

In chapter I.3 and I.4 of A. Zee's QFT in a Nutshell, he starts with the theory $$ \mathcal{L}(\varphi) = \frac 12[(\partial\phi)^2-m^2\varphi^2]+J\varphi $$ Using the path integral approach, he ...
0
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0answers
102 views

Spring mass on a rotating disc

I demonstrated the problem in the figure below. Edit: Original figure is 90 degrees rotated clockwise. I couldn't rotate the figure here. Disc is massles and horizontal. Vibrating mass only moves ...
2
votes
0answers
105 views

How to get the inverse of the propagator?

For a free EM Lagrangian, the propagator is as below in momentum space: $$ S[A]=\int d^4kA_{\mu}(k)\underbrace{[-k^2g^{\mu\nu}+k^{\mu}k^{\nu}]}_{M}A_{\nu}(k). $$ It is easy to calculate the $\det(M)$ ...
0
votes
1answer
180 views

Acoustical wave equation from Hamilton's principle

It is common to show the features and power of the Hamilton's principle by deriving the equation of vibrating string, membrane etc. using this principle. But I have never seen that used for deriving ...
0
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0answers
81 views

Non-symmetry of a lagrangian

If a transformation $\Phi \rightarrow \Phi + \alpha \partial \Phi/ \partial \alpha$ is not a symmetry of the Lagrangian, then the Noether current is no longer conserved, but rather ...
0
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0answers
92 views

Mass term in field Lagrangian

In the Klein-Gordon or in the Dirac Lagrangian density, the mass term is quadratic in the field. The other way around, I have heard a quadratic term in a general Lagrangian density be referred to as a ...
0
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0answers
19 views

When considering local phase transformations are we forced to use covariant derivatives?

When considering local phase transformations $e^{i\theta(x)}$ of the fields $\phi$ and $\phi^*$ corresponding to \begin{equation} \mathcal{L}=\partial_\mu\phi^*\partial^\mu\phi-m^2\phi^*\phi ...
2
votes
0answers
44 views

Why is the term $m\delta_2\delta_m\bar{\psi}\psi$ ignored in the QED Lagrangian?

Consider the QED Lagrangian $$\mathcal{L}=\bar{\psi}_0(i\gamma^{\mu}\partial_{\mu}-e_0\gamma^{\mu}A_{0\mu}-m_0)\psi_0-\frac{1}{4}(\partial_{\mu}A_{0\nu}-\partial_{\nu}A_{0\mu})^2$$ where the 0 ...
0
votes
1answer
46 views

Avoiding a singularity in the simulation of a spherica pendulum

I didn't know whether to put this here or in StackOverflow - so I open to answers just telling me to go there! I am looking to simulate the motion of a spherical pendulum. The Lagrangian is $$ ...
1
vote
1answer
36 views

Behaviour of action with respect to time

I was wondering if it was possible to say something general on the behaviour of the action : $$ S[x(\tau)]=\int_0^T L(x,\frac{dx}{d\tau},t) dt $$ (where $x(\tau)$ defines a trajectory, with certain ...
6
votes
2answers
171 views

How general are Noether's theorem in classical mechanics?

I'm going through the derivations of Noether's theorems and I have several criticisms as to how they are presented in popular sources (note that I'm only referring to classical mechanics here and not ...
0
votes
3answers
63 views

Examples of non-linear field symmetries?

Consider a Lagrangian theory of fields $\phi^a(x)$. Sometime such a theory posseses a symmetry (let's talk about internal symmetries for simplicity), which means that the Lagrangian is invariant under ...
6
votes
1answer
129 views

Why does the Palatini formalism of GR work? [duplicate]

We can get the Einstein field equations of GR from the Einstein-Hilbert action via two distinct methods: First, by taking the metric as the only degree of freedom, and imposing right away that the ...
5
votes
1answer
123 views

How to find symmetry transformations?

For a given Lagrangian $$ {\cal L} = - \frac{1}{4} F_{\mu \nu} F^{\mu\nu} + |D_{\mu} \phi|^2 -V (\phi) $$ with $\phi = \frac{1}{\sqrt{2}} (\phi^1 + i \phi^2)$, there are the infinitesimal local ...
0
votes
0answers
62 views

Quadratic terms in QED lagrangian density

I recently learned that when we speak about a "free lagrangian", this actually means that the lagrangian is quadratic in the fields. When considering the Lagrangian density describing the coupling to ...
0
votes
2answers
96 views

Textbook for mathematical Lagrangian mechanics [duplicate]

I'm looking for a textbook or online notes or a review article etc on a rigerous formulation of Lagrangian mechanics. I'm well aware of the book by Arnold but I would like something to accompany it. ...
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0answers
42 views

Nabla Terms in the Energy Density of the Lagrangian for the Massive Spin 1 Field (Schwartz QFT 1st Ed. Eqn. 8.19)

The relevant part starts with a Lagrangian guess of, $$\mathcal{L}=-\frac{1}{2}\partial_{\nu}A_{\mu}\partial_{\nu}A_{\mu}+\frac{1}{2}m^2A_{\mu}^2$$ where the EOM's are, $$(\Box+m^2)A_{\mu}=0$$ The ...
0
votes
0answers
17 views

Finding the gauge transformtation of a Lagrangian [duplicate]

I am asked to find the gauge symmetry of the following Lagrangian: $L = -\frac{1}{4} F^2_{\mu \nu} + (\partial_{\mu} \phi_1 - m_1 A_{\mu})^2 + (\partial_{\mu} \phi_2 - m_2 A_{\mu})^2$ Then I have to ...
0
votes
1answer
43 views

Isotropic of Inertial frame?

My understanding of isotropic is the a particular physics law remain same no matter at what direction I look at it? Now suppose in case of inertial frame, we know that its is homogeneous and ...
0
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1answer
62 views

Lagrangian is isotropic in space

In Landau & Lifshitz Mechanics, while deriving the properties of Lagrangian of a free particle in inertial frame, he uses the following points $:$ As space is homogeneous in inertial frame, a ...
2
votes
1answer
62 views

Lagrangian isn't unique [closed]

If $L$ is a Lagrangian for a system of n degrees of freedom satisfying Lagrange's equations, show by direct substitution that $$L' = L + \frac{\mathrm{d}F(q_1,\dots,q_n,t)}{\mathrm{d}t}$$ ...
2
votes
1answer
157 views

What is a Lagrangian of a photon? [duplicate]

In sense of classical mechanics+special relativity what is lagrangian of a photon? Lagrangian of a relativistic massive particle is as follows: $$ L_{massive}= -mc\sqrt{c^2-v^2} $$ So is it a zero?
2
votes
2answers
58 views

Invariance under local diffeomorphisms

In the context of the Polyakov action, the action for a relativistic point particle $$ S_P = \frac{1}{2} \int \mathrm{d}\tau \, e(\tau) \left(\frac{1}{e^2(\tau)}\left(\frac{\mathrm{d} ...
6
votes
1answer
93 views

Feynman propagator for arbitrary values of the gauge parameter $\zeta$

For the choice $\zeta = 1$ the Lagrangian can be brought into a particularly simple form upon integration by parts in the action integral. Equation$$\mathcal{L}' = -{1\over4}F_{\mu\nu}F^{\mu\nu} - ...
0
votes
1answer
94 views

Why does overall action need to have an extremum?

Quoting from Landau's and Lifshitz' Mechanics : The integral ${\int\limits_{t_1}^{t_2}}L(q, \dot{q},t)\,dt$ for the entire path must have an extremum, but not necessarily a minimum. This, ...
1
vote
1answer
122 views

Conserved quantities for the system, classical mechanics

In certain situations, particularly one-dimensional systems, it is possible to incorporate frictional effects without introducing the dissipation function. As an example, find the equations of motion ...
1
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0answers
84 views

Canonical Momentum Conjugate vs. Momentum

I stumbled upon this while reading about Legendre Transforms today. So consider an n-particle system. The Lagrangian is a function of $ q_i$'s and $\dot q_i$'s. If you consider the manifold $M$ where ...
2
votes
1answer
132 views

Noether's Theorem for Hamiltonians and Lagrangians

Looking around I see one version of Noether's Theorem that creates conserved quantities from symmetries that preserve the Lagrangian (e.g. http://math.ucr.edu/home/baez/noether.html), and another ...
0
votes
1answer
62 views

When can I apply Lagrangian mechanics?

I am trying to understand Lagrangian mechanics. I am having trouble capturing all of the nuances in one gulp. I can see the equations, but not necessarily the semantics behind such equations. I ...