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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Non-inertial frames in Lagrangian mechanics?

Building on this Phys.SE post I am interested in how non-inertial frames can be considered in Lagrangian mechanics. My understanding is that changing the reference frame causes a transformation of the ...
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2answers
108 views

What exactly is the Action? (Learning lagrangian)

I have been trying to wrap my head around lagrangian mechanics but I find some parts confusing. For example, what exactly is action and why is it defined by the Kinetic energy minus the potential ...
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495 views

Why don't all free particles lose their kinetic energy?

I'm currently studying Action. I've been reading about how a particle has particular probabilities of ending at an infinite number of events. Say I have a free particle that isn't experiencing any ...
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53 views

On the connection between forces and the principle of stationary action

Feynman tries to account for the relation between the principle of stationary action, which is a statement about the whole path of a particle, and Newton's second law, which is a statement about the ...
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43 views

What is meant by invariant under change of coordinates **to first order**?

I am studying elementary Lagrangian mechanics, and I'm a bit confused about the what's meant by invariance of the Lagrangian under change of coordinates to first order. More specifically, Noether's ...
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Using Lagrangian to get the equations of motion [closed]

The thin hollow cylinder rolls on the plane without slipping. My attempt is summarised in the sketch below. X and Theta are my coordinates. I got stuck while writing the T
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89 views

How to write the Lagrangian in terms of a projection

We know that $$ L=\frac{1}{2}\left(\partial_{\mu} A_{\nu} \partial^{\mu} A^{\nu}-\partial_{\mu} A_{\nu} \partial^{\nu} A^{\mu}\right) $$ But how do we write the Lagrangian in the following way: ...
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288 views

Are there other less famous yet accepted formalisms of Classical Mechanics?

I was lately studying about the Lagrange and Hamiltonian Mechanics. This gave me a perspective of looking at classical mechanics different from that of Newton's. I would like to know if there are ...
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Determination of the ground state of a field theory

Consider the Spontaneous symmetry breaking in the theory $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{\mu^2}{2}\phi^2+\frac{\lambda}{4!}\phi^4.$$ By the ground state of a ...
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Can the sign of metric change physics?

Consider the Lagrangian of a massless real scalar (classical field) in $\phi(\textbf{x},t)$: $$\mathcal{L}=\frac{1}{2}\partial^\mu\phi\partial_\mu\phi$$ The Hamiltonian density in two different ...
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30 views

Virtual work (generalized forces) for rotation with Euler angles

Here is what we know from virtual work: \begin{equation} \delta W=\sum_{i=1}^N{\vec F_i\cdot\delta\vec r_{i}} \end{equation} Where $N$ is the number of bodies in the system. I am considering a ...
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About turbulence modeling

I have some questions about this paper: Lagrangian/Hamiltonian formalism for description of Navier-Stokes fluids. R. J. Becker. Phys. Rev. Lett. 58 no. 14 (1987), pp. 1419-1422. After reading ...
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Formulating the Lagrangian in terms of invariant quantities

Consider a closed system consisting of $N$ point particles, whose Lagrangian is given in the standard way, by the total kinetic energy minus the potential energy: $\mathcal{L}(\dot{q},q):= T(\dot{q}) ...
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1answer
282 views

Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
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1answer
118 views

Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I ...
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2answers
96 views

Scalar and vector defined by transformation properties

In Classical Mechanics, we are defining scalars as objects that are invariant under any coordinate transformation. Vectors are defined as objects that can be transformed by some transformation matrix ...
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Total time derivative of magnetic vector potential $A$

I am looking at this document, which tries to establish the Lagrangian of the Lorentz force. Everything is fine, but I don't see why: $$\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+\frac{\partial ...
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63 views

Finding the action of a discretized Lagrangian

I am trying to find the action associated with the Lagrangian density $$ \mathcal{L} = \frac{1}{2}\left( \frac{\partial\phi}{\partial x} \right)^2 + \frac{1}{2}m^2\phi^2. \tag{1} $$ I am supposed to ...
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Deriving Newton's first law from the principle of least action

Newton's first law states that if the net force on an object is zero, then this object moves with constant velocity. I'm interested in the derivation of this law from the principle of least action. ...
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39 views

Physical significance of omitting a purely time dependent term from a Lagrangian

For a simple pendulum whose point of support moves on a vertical circle of radius $a$ with constant frequency $\gamma$, you can write the Lagrangian down. The potential energy can be written as ...
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230 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, $α = GMm$ ...
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41 views

3-cylinder surface element (Poisson's “A Relativist's Toolkit”)

From Poisson's "A Relativist's Toolkit": he introduces the non-dynamical term $$ S_0=\frac{1}{8\pi}\int_{\partial\Omega}\epsilon K\sqrt{\lvert h\rvert}d^3x $$ in the GR action, where $h$ is the ...
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A question on elementary Lagrangian mechanics(perhaps a bit of maths) [duplicate]

I am stuck with this question Consider the action, from $t=0$ to $t=1$, of a ball dropped from rest. From the Euler-Lagrange equation, we know that $y(t)={-gt^2 \over 2}$ yields a stationary value ...
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2answers
186 views

Lagrange's equation implying Newton's 2nd law? [duplicate]

The typical first application of Lagrange's equation is showing that it implies Newton's law for a particle whose Lagrangian is $L=\frac{1}{2}mv^2-V(x)$. Plugging this Lagrangian into Lagrange's ...
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1answer
53 views

Landau's Ideal fluild one dimentional flow equation

There is a similar Phys.SE question here, but I still didn't get the idea. The problem is: Write down the equations for one-dimensional motion of an ideal fluid in terms of the variables $a$ and ...
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Noether's theorem: meaning of transformation of coordinates

I have a question regarding Noether's theorem. In our introductory QFT class (which is based on the book by Michele Maggiore) we have derived the Noether currents in the same form as displayed in this ...
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Particle slides on incline where incline angle increases with rate $\omega$: why does kinetic energy have a term $(1/2)m(\omega^2 x^2)$?

A particle slides on a smooth inclined plane whose inclination is $\theta$ is increasing at a constant rate $w$. If $\theta = 0$, at time t = 0 at which time the particle start from rest, Find the ...
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2answers
73 views

What can be inferred about this particle from a Lagrangian?

If Lagrangian, $\mathscr L = \dot{q}^2 - q \dot{q}$. Then what can be inferred about the particle? Simply that it is a free particle or something else?
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Lagrangian of non-linear 3 mass, 2 spring system

Given 3 masses connected by 2 springs with the angle of intersection constant, but the springs themselves bending. Young's modulus, which is a variation of Hook's Law, applies to the flexing that ...
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1answer
311 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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1answer
408 views

Sufficient conditions for the energy to be not conserved?

I'm almost embarrased to ask this question because I thought I was by now very confident with classical mechanics. Someone has stated that given a mechanical system with a Lagrangian $L$ s.t. ...
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195 views

Energy-Momentum Tensor for Electromagnetism in Curved Space

$\newcommand{\l}{\mathcal L} \newcommand{\g}{\sqrt{-g}}$$\newcommand{\fdv}[2]{\frac{\delta #1}{\delta #2}}$I want to calculate the energy-momentum tensor in curved free space by functional ...
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1answer
59 views

From Noether's theorem to canonical Energy-Momentum tensor using translations

In this text that I am reading it says that the transformation $\delta \phi(x)$ is a symmetry if the Lagrangian changes by a total derivative: $$\delta \mathcal{L}= \partial_{\mu}F^{\mu} . $$ From ...
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Where does the $(\ell + x)^2\dot\theta^2$ term come from in the Lagrangian of a spring pendulum?

I am reading some notes about Lagrangian mechanics. I don't understand equation 6.9, which gives the Lagrangian for a spring pendulum (a massive particle on one end a spring). $$T = ...
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168 views

What are the equations of motion for the scalar field in the tetrad formalism?

The action of a massless scalar field in curved spacetime is given by: \begin{equation} S(\phi)=\int d^{4}x \sqrt{-g}\left(g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right) \end{equation} Now the action can ...
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The Nambu-Goto action how do we know the Hamilton's principle applies?

I am reading 'A first course in string theory' by Barton Zwiebach (2ed) on page 112 he comes up (after a small derivation) the action formula: $$S=-\frac{T_0}{c} \int d\tau d \sigma \sqrt{-\gamma}.$$ ...
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Generalised velocities enough to be deterministic in Lagrangian mechanics?

In classical determinism we need to know $2n$ quantities of our system and the equation of motion to predict it's future. In Lagrangian mechanics this is equivalent to knowing $q$ and $\dot q$, the ...
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Beyond the third time derivative [duplicate]

Why do texts on classical mechanics never mention any derivative of position beyond the jerk, while at the same time being general in the sense of using of generalized coordinates?
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What is an effective potential in classical mechanics?

What is an effective potential in classical mechanics? I have read the wikipedia article and David Tong's lectures notes, but I didn't understand how an effective potential simplifies a situation or ...
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How to write the Lagrangian for a body that exhibits gravitation?

Can anyone tell me how one actually goes about writing the Lagrangian for a mass that exhibits gravity? If I wanted to write the Lagrangian for, say, a spherical mass in space that curves the ...
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1answer
93 views

Problem to find field equations with Euler-Lagrange in field theory [closed]

I have the Euler-Lagrange equations, as stated in field-theory: $$\partial_\nu \left(\frac{\partial L}{\partial (\partial_\nu \phi_\rho)}\right) - \frac{\partial L}{\partial \phi_\rho}=0$$ However ...
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1answer
80 views

Redefinitions of Lagrangians using EOM

I am trying to understand an statement of this paper. In section 2 this Lagrangian is introduced ...
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342 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 - ...
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Inclined plane - constraint - equation of motion

A mass point of mass m moves frictionlessly down an inclide slope under influence of gravity. Solve the equations of motion and determine the constraint with the use of the lagrange equation of ...
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2answers
83 views

Problem with Lagrangian density [closed]

Ok, I tried sometimes already, however I can't see my mistake. What I need to do is to find the field equation of $$ L = -(\partial_\mu A^\nu)(\partial_\nu A^\mu) + \frac{m^2}{2} A_\mu A^\mu + ...
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102 views

Independence of position and velocity in Lagrangian from the point of view of physics?

I would like to continue discussion from my previous post on time dependence of lagrangian Time dependence of the Lagrangian of a free particle?. I have also read this old post Why does calculus of ...
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180 views

Invariance of the QED Lagrangian under charge conjugation

Is it true that the QED Lagrangian $$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu-m) \psi $$ is invariant under charge conjugation? $$\begin{align} \psi &\mapsto -i(\gamma^0 \gamma^2 \psi)^T\\ ...
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222 views

Why do we consider Lagrangian densities in QFT?

My question is: Why do we consider Lagrangian densities in QFT (as opposed to Lagrangians as in classical mechanics)? Is it simply because of the following? We wish the theories to be Lorentz ...
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1answer
47 views

Exact meaning of locality and its implications on the formulation of a QFT

As far as I understand it, locality in physics is the statement that interactions can only occur between physical objects if the spacetime interval separating them is null or time-like. Thus, if the ...
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1answer
140 views

Null geodesic equation

For a null geodesic curve $X^i$, $$0=g_{ij}V^iV^j.$$ When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the ...