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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0answers
50 views

Geometric point of view of configuration space and Lagrangian mechanics [on hold]

Even though (I think) I understand the concept of a tangent bundle, I have trouble assimilating the idea of the configuration space being one and in relation to what that is the case. How can I ...
3
votes
1answer
378 views

Reduction of Nambu Goto action to true degrees of freedom

First consider the particle $$S=m\int\sqrt{-\dot{X}^2}d\tau$$ if you choose the static gauge $\tau=X^0$ and replace it in the action you get $$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau$$ So now, you ...
6
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2answers
82 views

Beyond Hamiltonian and Lagrangian mechanics

Lagrangian and Hamiltonian formulations are the bedrock of particle and field theories, produce the same equations of motion, and are related through a Legendre transform. Are there more such ...
3
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2answers
89 views

Are the partial derivatives of Lagrangian in the varied action functional derivatives?

In particle mechanics Lagrangian $L$ depends upon position, velocity (and may be explicitly on time), whereas in field theory the Lagrangian density ${\cal L}$ similarly (or analogously) depends upon ...
3
votes
4answers
843 views

Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
0
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0answers
46 views

Euler-Lagrange Equation in Quantum Field Theory

The quantum fields are operator valued distributions. In some sloppy books like Peskin and Schroeder the Euler-Lagrange equation are used to get the equations of motion. What does it mean to take a ...
2
votes
1answer
33 views

My Hamiltonian for a light ray vanishes

I have the following issue with understanding. A light ray traveling from $q(\tau_1)$ to $q(\tau_2)$ minimizes the integral $\int\limits_{\tau_1}^{\tau_2} n(q(\tau))|\dot{q}(\tau)| d\tau$, so the ...
0
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0answers
46 views

Functional Differetiation of a complex functional

Suppose I have a simple functional $$F=\int{dx\;\phi^{*}(x)\phi(x)}\tag{1}.$$ Assuming $\phi(x)$ and $\phi^{*}(x)$ are independent and I take a functional differential with respect to $\phi(x)$ and $\...
2
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0answers
21 views

Change of energy of a shortening simple pendulum (Ehrenfest Pendulum) [on hold]

I've been going through Lanczos' Variational Principles of Mechanics and have been struggling with a problem that seemed pretty straightforward: A simple pendulum hangs from a fixed pulley. The ...
1
vote
2answers
270 views

Mechanics Landau Galilean Principle

I started reading Landau's Mechanics book and was having some trouble understanding the Galilean Relativity Principle. What does Landau mean by saying space to be homogenous and isotropic and time is ...
1
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0answers
26 views

Functional variation of potential in integral form

I am trying to vary the following action, $$ S=\int_{t_0}^{t_1} \text{d}t\,(v^\mu v^\nu g_{\mu\nu} + V(t)) =\int_{t_0}^{t_1}\text{d}t\,(v^\mu v^\nu g_{\mu\nu} + \int_{t_0}^t\text{d}s T_\mu v^\mu) $$ ...
2
votes
1answer
2k views

Calculating the moment of inertia in bifilar pendulums

I'm an A2 student, and I've been looking into how experimental and theoretical determined mass moments of inertia differ. I came across a method (search Youtube for Measuring Mass Moment of Inertia - ...
1
vote
1answer
108 views

Deriving the equation of motion for a rigid system

I want to derive equation of motion for the system shown in picture. How do I choose a generalized coordinate in order to calculate kinetic and potential energy of the system? I need the ...
0
votes
1answer
25 views

Derivation of photon propagator from EM Lagrangian

I am following Ryder's Quantum Field Theory. In chapter 7, in order to derive the photon propagator, he first derives eq. 7.4 $$\mathcal{L}=\dfrac{1}{2}A^\mu[g_{\mu\nu}\partial^2-\partial_\mu\partial_\...
3
votes
0answers
43 views

Lagrangian of classical electromagnetism without $A_{\mu}$ field [duplicate]

Is there a Lagrangian reproducing Maxwell's equations without the use of the scalar and vector potential? Obviously $\mathcal{L} = -\frac14F_{\mu \nu}F^{\mu \nu}$ doesn't work since in terms of $E$ ...
7
votes
4answers
827 views

Least-action classical electrodynamics without potentials

Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which ...
-1
votes
2answers
35 views

Lagrangian mechanics not relying on time or independent of time [on hold]

If neither the potential energy nor kinetic energy depends on time, then Lagrangian is explicitly independent of time I find this statement a little bit odd because velocity is distance over time or ...
4
votes
1answer
155 views

Locality defined in terms of the Lagrangian density

I've been reading through Matthew Schwartz's book "Quantum Field Theory and the Standard Model" and in chapter 24 there is a section on locality (section 24.4). In it he defines locality in terms of ...
1
vote
1answer
51 views

Connection between “classical” Grassmann variables and Heisenberg Equation of motion

I have been reading di Francesco et al's textbook on Conformal Field theory, and am confused by a particular statement they make on pg 22. Let $\{\psi_i\}$ be a set of Grassmann variables. Starting ...
16
votes
11answers
9k views

Book about classical mechanics

I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...
1
vote
2answers
492 views

Kepler's laws to determine radius of circular orbit

"In nonrelativistic limit of general relativity there is a correction to the Newtonian gravitational potential energy $−h/r^3$ with $h = αL^2/(mc)^2$, where $c$ is the speed of light, $α = GMm$ and $...
-3
votes
0answers
41 views

Kinetic energy of a suspended mass from an overhead crane [closed]

I am using Lagrange's method to formulate the equation of motion for a 3D overhead crane that is shown below: For the formulation of the kinetic energy for the payload (suspended) mass. I have seen ...
3
votes
3answers
63 views

Why do we need $SU(2)\times U(1)$ invariant mass terms if the symmetry will be broken anyway?

In the SM we can not add fermionic mass terms like $m \overline{e}_R e_L$ to the Lagrangian since these terms are not invariant under $SU(2)\times U(1)_Y$. After introducing the Higgs in the unitary ...
4
votes
1answer
62 views

Vary action with respect to velocity

Variation of the action $S$ corresponding to a Lagrangian e.g. $L(x(t),\dot{x}(t))$ gives the Euler-Lagrange equations: $$ \frac{\delta S}{\delta x(t)} = 0 \\ \int du \left ( \frac{\delta L}{\delta x(...
4
votes
1answer
125 views

Is $\phi^4$ theory in 4d conformally invariant at the classial level?

I used to believe the three following statements to be true (at the classical level only): From scale invariance full conformal invariance follows. Scale invariance is present if there are no ...
2
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0answers
42 views

Non-canonical transformation

I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
-1
votes
1answer
77 views

Derivation in Modern Supersymmetry by Terning

I am trying to do some calculations from Modern Supersymmetry by Terning and I am stuck on how he derived a particular term. Specifically, I am looking at 2.67 on page 27. My current work is below. $$...
9
votes
2answers
1k views

Functional derivative in Lagrangian field theory

The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') \end{...
2
votes
1answer
95 views

The difference between the forms of the Euler-Lagrange equations

I'm trying to learn Lagrangian mechanics and have been reading a lot of articles on it. But many of the articles write the equations in different ways, probably for different purposes. The Euler-...
0
votes
1answer
61 views

Checking the basics in QFT notation [closed]

In the very beginning of QFT we face the action (S) as a functional of the Lagrangian. I am still trying to get used to the notation used here, so I would like to check if the following makes sense: $...
0
votes
0answers
19 views

Euler-Lagrange equation no fixed endpoints

The usual way, to show the Euler-Lagrange equation is, to find the minimum of the Integral $$ I = \int_a^b L(q, \dot q, t) dt $$ and argue, that it must satisfy the following equation $$ \frac{d}{dt} \...
0
votes
0answers
47 views

From gauge invariance to charge conservation in covariant electrodynamics

I tried to solve the equations of motion using the action for the electromagnetic field interacting with a current, like $$ L = F_{\mu\nu}F^{\mu\nu} + A_{\nu}j^{\nu} $$ getting the right Maxwell's ...
0
votes
1answer
47 views

Equation of string as hamiltonian field equations

I am using Hamiltonian field theory for the first time and I struggle with some final steps. The task is to derive the equation of vibrating string using Hamilton's field equations. Here is what I ...
5
votes
1answer
67 views

Electron - neutrino scattering effective Lagrangian

The electron and neutrino can interact through an intermediary Z boson, via the Lagrangian: $$ L= \frac{1}{2} \partial_\mu \phi_Z \partial^\mu \phi_Z - \frac{1}{2} m_Z ^2 \phi_Z ^2 -g_{\nu} \phi_Z \...
1
vote
2answers
58 views

Lagrangian of an effective potential

If there is a system, described by an Lagrangian $\mathcal{L}$ of the form $$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$ where $T$ is the kinetic ...
0
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0answers
32 views

A question about Euler-lagrange equations

This question passes my mind so often, why do we stop at the first order of expansion of the action to get the Euler-lagrange equations and it turns out they exactly get us the Newtonian equations. ...
-1
votes
1answer
22 views

Parity tranformation on Lagrangian of free fields

Free lagrangians of scalar, Dirac field and vector fields are always invariant under Parity. I am able to get this result mathematically, but I want to know if there is any obvious reason for it. ...
5
votes
3answers
86 views

The $\dot{q}$ term in the Euler-Lagrange equation

The Euler-Lagrange equation is about the functional $$ \int_{t_1}^{t_2} L(q, \dot{q}, t ) dt . $$ From a mathematical point of view, a simpler functional might be $$ \int_{t_1}^{t_2} L(q, t ) ...
1
vote
0answers
29 views

Generalise Noether's theorem [closed]

I'm not sure how to generalise Noether's theorem. For this L, I think $B\cdot\dot{x}$ is conserved so I tried to relate F and K to this and try to show that that was conserved but got no where. any ...
1
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0answers
42 views

Mass term in Maxwell's Lagragian for Electromagnetism

In the scalar field Lagrangian the mass term is given by $$m^2 \phi^2.$$ But the equivalent term in Maxwell's Lagrangian for electromagnetism is $$m^2A_{\mu}A^{\mu}.$$ But I don't know why the ...
-1
votes
0answers
42 views

Lagrangian of a falling rod that is free to rotate about an axis

So i came along an MIT example involving a tilted rod, length $L$ and tilted by angle $\theta$ from horizontal, whose free to fall and slid, they wrote the Lagrange equation without sweat. Now i am ...
4
votes
1answer
96 views

Lagrangian density for Lorentz force of continuous charge distribution in external field?

It's frequently an exercise to derive the Lorentz force law for a particle with charge $q$ in an external electromagnetic field given by the following Lagrangian: $$L = -mc^2\sqrt{1-\frac{\dot{r}^2}{...
1
vote
0answers
41 views

Reason behind $L = T - V$ (Lagrangian formalism) [duplicate]

I've been learning about the Lagrangian formulation recently, and while I'm with the process, I am still struggling somewhat with the theory behind it. As I (rather poorly) understand it, the ...
1
vote
1answer
76 views

Magnetic monopoles in field theory

In standard QED, we couple the electron to electromagnetism by replacing $$\partial_\mu \to \partial_\mu + i e A_\mu.$$ Upon taking the classical limit, we find that this gives electrons an electric ...
1
vote
1answer
47 views

Lagrangian in a system with a specific velocity dependent potential

I have a system of a particle moving under the generalized central potential $$ V= \frac{1}{r}(1+\dot{r}^2) \tag{1} $$ The general Euler-Lagrange equations for such type of potentials are: $$ \frac{...
3
votes
1answer
382 views

How do I treat the Lagrangian in the case of a rigid body?

Here's Exercise 1.11 from Goldstein's Classical Mechanics 3rd edition (the first one after having derived the Lagrangian basically): Exercise 1.11: Consider a uniform thin disk that rolls without ...
0
votes
2answers
53 views

Virtual Work- Is the presentation in Cornelius Lanczos wrong?

Book: The Variational Principles of Mechanics by Cornelius Lanczos Edition: 4th Chapter: 3, The Principle of Virtual Work I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1). ...
0
votes
1answer
224 views

Degrees of freedom in double Atwood machine?

Why the degree of freedom in double Atwood machine (one block on one side and a pulley with one block in its each side on other side) is 2 and not 1? According to the formula $s=3*n-m$; where $n=$...
7
votes
1answer
114 views

Action of a massive free point-particle in relativistic mechanics

I was reading about the formulation of mechanics in special relativity and found that the action for a massive free point-particle as $$ S = -mc\int_a^b ds $$ So, I did a few observations, ie. $$ S =...
0
votes
1answer
80 views

Why the Lagrangian $L$ is KE - PE? Why not KE + PE!

With Lagrangian, is there any way to intuitively grasp why total energy equals the difference between the kinetic and potential energy? Seems counter-intuitive - whereas Hamiltonian calculation (sum ...