Questions tagged [lagrangian-formalism]

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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16 views

Calculation of gravitational Euclidean action of Schwartzchild BH

I am reading the paper of Gibbons and Hawking Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 where they compute the gravitational action of black holes. In ...
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58 views

Thought Experiment: Theoretical Mechanics [closed]

Imagine you are some higher mathematical being, who has access to all of the Math involved in the Theory of Classical Mechanics, I am talking about concepts such as Noether's Theorem, Hamilton's ...
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55 views

How to prove that a certain quantity is conserved?

If one were to consider the following action functional: $$S = \int dt G_{ij}(\textbf{q}) \cdot \overset{.}{q_i} \overset{.}{q_j}. $$ Given that there exists some vector $v_i$ with the following ...
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Coupling Hilbert action to a matter action

When coupling a matter action to the Hilbert action in general relativity, why does one simply add the two actions and not add a coupling action? In electromagnetism, a coupling term is needed between ...
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How is the Weyl lagrangian zero for Weyl spinors?

The action from which Weyl equation can be derived is $$S=i\int{\psi_{L/R}^\dagger\bar{\sigma}^\mu\partial_\mu\psi_{L/R}}$$ where $\bar{\sigma}^\mu=(1,\vec{\sigma})$. Imposing $\delta S=0$ we arrive ...
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Transformation of coordinate in Lagrangian

Lagrangian for a Central force problem is: $$\mathcal{L} = \frac{1}{2}\mu(\dot{r} + r^{2}(\dot{\theta}^{2} + sin^{2}\theta\cdot \dot{\varphi}^{2})) - U(r)$$ We know that angular momentum is defined as:...
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48 views

The definition of generalised momentum in lagrangian mechanics [closed]

Let's assume that no external forces are applied, so the lagrangian is just the kinetic energy. Lagrangian mechanics differentiates it with respect to generalised velocity to get the generalised ...
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68 views

What are the applications of calculus of variations, if any, to the subject of thermodynamics?

If we apply calculus of variations to Newtonian mechanics, we can discuss of functionals such as the lagrangian and how optimizing it leads to the equations of motion. However, does there exist ...
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Gauge-invariance of the Maxwell Lagrangian in Srednicki's book

When it comes to the demonstration of the gauge-invariance of the Lagrangian of the Maxwell-theory Srednicki's book proceeds as follows: $${\cal L} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + J^\mu A_{\mu} \...
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43 views

Does Lagrangian follow superposition principle?

In other words: is the Lagrangian of a composite system the sum of the Lagrangians of its components? It shouldn't be, since it's energy. But then we lost one of the most important principles in ...
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What are the mathematical prerequisites for The Variational Principles of Mechanics by Lanczos? [closed]

I have heard this book takes a rather philosophical approach to mechanics, thus I am interested in knowing the background knowledge needed to understand it.
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Can a Mach number be defined for a hot gas fluid element using its own temperature and velocity?

I know that the ratio of an object's speed to the local sound speed is called the Mach number: $$M = \frac{v}{c_s}$$ where $$c_s = \sqrt{\frac{\gamma kT}{\mu m_p}}$$ I have always thought of the Mach ...
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$S$-matrix computation for QED [closed]

While expanding the s matrix term for qed, we get zero contraction term for the second order term of S matrix as shown. Then how to draw Feynman diagram for the zero contraction term?
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Is the lagrangian density convex if the lagrangian is convex?

Let $L = \dot{q}^T M(q) \dot{q} + V(q)$, i.e., the lagrangian has a quadratic form and hence is convex w.r.t to the velocities, considering that $V(q)$ plays the role of a constant. And now let the ...
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76 views

Deriving the relativistic point particle action from QFT

In principle, the action of a free relativistic particle of mass $m$ and trajectory $x^\mu(\tau)$ $$ S = -m \int d\tau \sqrt{\frac{dx_\mu}{d\tau}\frac{dx^\mu}{d\tau}} $$ should be obtainable as the ...
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Yukawa interaction with modulus

In Standard Model, Yukawa interactions are generally written as $$ {\cal L}_{Y} = y\bar{\psi}\psi \phi + {\rm h.c.} $$ where $\psi$ is some spinor while $\phi$ a scalar (complex in principle), and $y$ ...
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Are Hamilton’s equations time-independent from it’s general derivation?

The general derivation of Hamilton’s equations involve the change in the Hamiltonian and consequently the change in the Lagrangian that is a function of $q$ and $\dot q$, $L(q,\dot q)$. This is shown ...
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Hamiltonian formalism in Goldstein's matrix representation, chap. 8.1

There are several points over which I stumble when studying Goldstein, 3rd ed., chap 8.1, concerning the matrix representation of the hamiltonian formalism. In (8.22) he assumes the lagrangian to be (...
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51 views

S-matrix expansion for the $\phi^4$ theory and the interaction picture

My question is about the perturbative expansion of the S-matrix using Dyson's expansion. Let the Lagrangian density of the $\phi^4$ theory be \begin{equation} \mathcal{L} = \frac{1}{2}\left[\partial_\...
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Why are explicit mass terms allowed for sfermions like higgsinos and gauginos in the MSSM Lagranian if explicit fermionic mass terms are prohibited?

In Martin's SUSY Primer, he claims: For the higgsinos and gauginos, [the ability to have a mass term] follows from the fact that they are fermions in a real representation of the gauge group. As I ...
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101 views

Euler-Lagrange equations for this system

I haven't done this calculations in a long time. Suppose you have a point of mass $m$ constrained to moved to a trajectory described by $y = x^2$ (you have gravity going on the opposite of the $y$ ...
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68 views

Simple pendulum with moving support in a parabola

The problem says that I have to find the equations of motion in the Hamiltonian formulation for a simple pendulum with mass $m$ and lenght $l$, which its support moves with no friction along the ...
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121 views

Equations of motion for a uniform disc in variable gravity field [closed]

I am attempting to simulate rotating disc in a variable gravity field. A uniform disc of radius R and mass M is free to rotate about its axis. a is factor reducing the gravitational field: $0<a\...
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51 views

Lagrangian formalism and collision with hard wall

Assume that a particle with mass $m$ is colliding with a hard wall, making angle $\theta$ before and after the collision. Here are my questions: If we want to use Lagrangian formalism, should we write ...
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64 views

How conserved quantities lead to equations of motion in Lagrangian mechanics

In a classical mechanics exercise, we were asked to derive a system of ODEs from conserved quantities. As we know from Lagrangian mechanics, Euler-Lagrange equations lead to equations of motion. I ...
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Additional term in the Noether current

I've seen this same question before Why is there an extra term in definition of Noether current for spacetime translations? but I didn't understand the answer that was given so I would like to ask ...
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Spin connection equations of motion from Einstein-Palatini action

I am working through "Supergravity" from Freedman and Van Proeyen. In exercise 8.11 one is tasked to vary the Einstein-Palatini action $$ S = \frac{1}{2\kappa^2}\int d^Dx\ e e^\mu{}_a e^\nu{}...
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Equality modulo equations of motion [closed]

What does Qmechanic mean by “equality modulo equations of motion” when talking about Lagrangian formulation/formalism and so on?
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Application of the Cartan Structure Equations seems to imply the Einstein-Palatini action is zero?

The Einstein-Palatini action can be written as $$ S = M_{pl}^2\int\varepsilon_{abcd}\left(e^a\wedge e^b\wedge R^{cd}\right), $$ where $e^a={e^a}_\mu\text{dx}^\mu$ is the basis one-form and $R^{ab}=\...
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Confusion relating to discretizing a Lagrangian and the Fourier transform conventions thereof

My question boils down to how the Fourier transform is discretized when we discretize the field $\phi(x)$ in a Lagrangian $\mathcal{L}$. To put things on a concrete footing, consider the following ...
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56 views

How to evaluate the Euler-Lagrange equation for the electromagnetic Lagrangian? [duplicate]

I'm fascinated with field theories, but have little knowledge about them, so excuse Me if this is a dumb question. We all know, that if we have a Lagrangian in terms of a field $\Phi $, we can just ...
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104 views

Is $ \partial_{\mu} \partial^{\mu} $ the second derivative or derivative squared?

This might be a silly question, but I'm just getting my feet wet with field theories. So far I have assumed that $ \partial_{\mu} \Phi\partial^{\mu}\Phi $ means $ (\Phi_t)^2-(\Phi_x)^2-...$ . I ...
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Lagrangian of a free particle and Galilean invarience

In Landau’s mechanics book, he uses Galilean invariance to show that the Lagrangian of a free particle is $\mathcal{L} = \frac{1}{2}mv^2$. Why does he use Galilean invariance? If the particle is under ...
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50 views

Derivation of Euler-Lagrange equations from Newton's second law [duplicate]

I was thinking about this for some time and I wanted to clarify my question. Can the Euler-Lagrange equation be somehow derived from Newton's second law? Here's a possible way to do it: We start with ...
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First order quantum string action

Considering this post: Quantum String action the action given is of the lowest order but the effective action, for low energies, is given by: $$ S_{ef.}= -\frac{1}{2k^2} \left( S^{(0)}+ \alpha S^{(1)} ...
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Non-linear QED gauge fixing, writing the effective Lagrangian

I have several questions about this problem. I have been given a non-linear gauge condition for a QED theory: $$F = \partial_{\mu}A^{\mu} + \frac{\lambda}{2}A_{\mu}A^{\mu}.$$ Then I need to write the ...
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86 views

Spontaneous symmetry breaking with 1 massive scalar but three unbroken generators?

Consider some theory of four real scalar fields, $\phi_1$, $\phi_2$, $\phi_3$ and $\phi_4$, that is invariant under a global $$SO(4)\cong SU(2)_L\times SU(2)_R$$ symmetry. We can rewrite the real ...
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79 views

How to find the Hamiltonian from the energy-momentum tensor for a free electromagnetic field?

This question is related to a previous question that I have asked before titled: Energy-Momentum Tensor for the Electromagnetic Feild asking why the energy-momentum tensor had the following form $$T^{...
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One-loop approximation in chiral sigma models

Consider a principal chiral model $$S[g] = \frac{1}{4\pi\lambda^2}\int(|g^{-1}dg|^2) = \frac{1}{4\pi\lambda^2}\int(g^{-1}dg\wedge\star g^{-1}dg) ,\tag{2.1}$$ where $g:\Sigma \rightarrow G $ is the so ...
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How to calculate functional derivative correctly?

Let $\phi$ be a real scalar field and $J$ an arbitrary source function. Consider $$S_{E}[\phi, J]=\int d^{4} x\left[\frac{1}{2}(\partial_{\mu} \phi)(\partial^{\mu}\phi)+\frac{1}{2} m^{2} \phi^{2}+V(\...
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27 views

Conceptual Question About Gyroscopic Precession of body about Two Axes

I've been trying to glean some insight into the motion of a body that is rotating about an axis through the COM of said body whilst traveling in a orbital-like path about a perpendicular axis outside ...
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49 views

Dirichlet boundary condtions in Nambu-Goto string action

The Nambu-Goto action for an open string with parameter domain $[0,\tau_1]\times[0,\sigma_1]$ is given by \begin{equation} S_{NG} = \int_{0}^{\tau_1} d\tau \int_{0}^{\sigma_1} \ d\sigma \ \mathcal{L}(\...
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153 views

Justification of the $U(1)$ gauge for electromagnetism?

Why should we expect or require that there is a $U(1)$-gauge symmetry in the theory of a charged particle (such as QED), namely that its physical properties should not change under local changes of ...
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Dimensionality of fluid flow

In some textbooks flow is classified as one, two, or three-dimensional depending on the number of space coordinates (i.e. x,y,z) required to specify the velocity field while according to some other ...
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700 views

Can Lagrangian have a potential term proportional to the quadratic or higher of velocity?

In general, classical Lagrangian $L(q,\dot q)=\frac{m}{2} \dot q^2-U(q,\dot q)$ has a $\dot q$-dependence. For example, potential term $U(q,\dot q)$ of the charged particle is given as follows:$U(q,\...
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2answers
665 views

Does a non-lagrangian field theory have a stress-energy tensor?

In classical field theory, the stress-energy tensor can be defined in terms of the variation of the action with respect to the metric field, or with respect to a frame field if spinors are involved. ...
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112 views

Maxwell equations as Euler-Lagrange equation without electromagnetic potential

The standard way to write the Maxwell equations (say in vacuum in absence of charges) as Euler-Lagrange (EL) equations is to take the first pair of the Maxwell equaitons and to deduce from it ...
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34 views

“Equations of motions” and direction of maximal entropy

Say one has a system of statistical physics whose entropy is given as a function of one or multiple variables; for example as $S(x)$. An example of such a system could be a osmosis system, or a ...
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70 views

Understanding the QED Lagrangian using Yang Mills formalism

In QED the Lagrangian is $$ \mathcal{L} = \bar{\psi}(i \not \partial - m ) \psi - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} - e \bar{ \psi} \gamma^\mu \psi A_\mu $$ which is the sum of a Dirac term, the ...
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2answers
76 views

Inverted pendulum on a cart - Lagrangian without moment of inertia?

I am modeling the inverted pendulum on a moving cart using Lagrangian methods. I see most examples model the pendulum's kinetic energy as a sum of translational and rotational components (using a $I\...

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