# 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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### Noether's theorem and energy in four-momentum

In Newtonian physics, momentum and energy are often treated as distinct entities, which happen to be separately conserved. In relativity, energy is regarded as the "time" component of the ...
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### Lorentz-invariant Lagrangian for spinor field

Schwartz book on QFT (page 167), Zee book on group (page 461) and Maggiore book on QFT (page 55), prove that $\psi_R^{\dagger}\sigma^{\mu}\psi_R$ is a 4-vector, where $\psi_R$ is a right-handed spinor ...
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### Intuition about non-invariance of the Hamiltonian in canonical transformation

Suppose $q={\{q_i\}}_{i=1}^n$ is the set of generalized coordinates of a dynamical system. $L(q,\dot q,t)$ is the Lagrangian of the system. Now we make coordinate transformation $Q_i=Q_i(q,t)$. Then ...
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### How did Lagrange derive this Hamilton-like equation in Mechanique analytique

I was trying to understand how Hamiltonian formulation was derived in the history, and read Mechanicque Analytique, and found these pages from section V of this book, which derived the Hamiltonian-...
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### Why do we use two ways to write the kinetic term in a Lagrangian?

I have just started reading Schwartz's book on QFT and I see from the first few chapters that he writes the kinetic part of the Lagrangian in a way I find strange. As an example, for the massless ...
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### Lagrangian in parabolic co ordinates given by $x= u*v$ and $y= (u^2+v^2)/2$

I found 2 equations of motion in $v$ and $u$ for $L=m/2*[x^2+a*y^2+w^2(x^2+y^2)]$ where $a,w$ are constants so the phase space has the generalized axes $u, v$. How does the parabolic coordinate work?...
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### High wire artist

We want to derive a Lagrangian equations for a high wire artist which uses a balancing rod for stability purposes. We know the masses and the polar moment of inertia of the artist and the rod.
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### Anharmonic terms of Lagrangian of spring pendulum with free support

I am trying to find the normal modes of a spring pendulum with moving support. The spring has spring constant $k$ and unstretched length $l_0$. Sorry for my bad paint skills. The problem was stated ...
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### How to solve lagrangian equations of motion in momentum space?

How do you find the equations of motion for a field given the Lagrangian by working in momentum space?
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### Some index operation confusion about Maxwell field's Euler-Lagrange e.q

Consider this Lagrangian density $$\mathcal{L}(x)=-\frac{1}{2}[a\partial_{\mu}A^{\nu}\partial^{\mu}A_{\nu}+b\partial_{\mu}A^{\nu}\partial_{\nu}A^{\mu}+c(\partial_{\mu}A^{\mu})^2+dA_{\mu}A^{\mu}]$$ The ...
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### Is nonlocality consistent with scale invariance?

For sure I'm excluding gravity at first step, the question is that if nonlocality is compatible with scale invariance. At the classical and quantum levels for field theory in Minkowski spacetime. Then ...
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### A strange term in the variation of an action

I am varying an action to compute some equations of motion. The starting point is \begin{equation} \delta S \sim \int d^4x\, \log\Box\delta \phi \end{equation} where $\Box$ is the d'Alembertian ...
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### What is the reason behind the stationarity of action? [duplicate]

I am reading Goldstein right now to understand the least action principle. I understood that the action needs to be stationary under small variation and this specifies the equation of motion, but do ...
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### Propagator/phase factor of a free particle

I am reading a literature review on modelling standard model particles wavefunctions. I am struggling with deriving the result Ive attached as a photo: From a quantum mechanics course this year i ...
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### $\phi^4$ theory in 5 dimensions

$\phi^4$ theory is not perturbatively renormalizable in 5 dimensions. I have come across literature where renormalizability is discussed w.r.t $N$, for fields obeying $O(N)$ symmetry. But it is not ...
How do I get the group of symmetries and the constant of motion of $L=\frac{\dot{x}^2}{2}m+V(x+ct)$ where c is a constant? When I tried to solve it, it was to look for shifts in $x$ and $ct$ under ...
I'm studying lagrangian mechanics, and there's a property where you could obtain an equivalent lagrangian $\mathcal{L'}$ from $\mathcal{L}$ by adding a function which satisfies:  \mathcal{L'}\...