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.
5,756 questions
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Where do experiments enter the renormalization procedure?
I'm studying the renormalization of scalar quantum field theories ($\lambda\phi^4$ in particular). I'm considering renormalization by counterterms with the old non-Wilsonian interpretation of ...
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Sign confusion in Relativistic Lagrangian and Lorentz Force Derivation [duplicate]
This is a rewrite of the original question and calculation, which should be now correct and focusses on the core issues of possible confusion:
I was confronted with some confusion regarding the ...
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Deriving the full crystal Hamiltonian from a Lagrangian density
I'm trying to understand the connection between field-theoretic Lagrangians and the standard Hamiltonians used in solid-state physics. In particular, consider a full crystal Hamiltonian of the form:
$ ...
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How does the relativistic action logically follow from the nonrelativistic action, and why is proper time involved? [closed]
In nonrelativistic mechanics, the action for a particle of mass $m$ moving in a potential $V(\mathbf{x})$ is
$$
S_{\text{classical}} = \int \left(\frac{1}{2} m \mathbf{v}^2 - V(\mathbf{x}) \right) dt.\...
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Accounting for magnetic forces in Lagrangian mechanics [duplicate]
For conservative forces the Euler-Lagrange equation is used to find the relevant details about the system. However magnetic forces are not conservative do not perform any work on an moving charged ...
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Doubt about Lagrangian transformation between reference frames (Susskind, Classical Mechanics, The Theoretical Minimum, pag.117)
I'm working through Susskind's Classical Mechanics book and I reached the point where he explains how to transform the action (and Lagrangian) when changing reference frames. However, I believe there ...
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How is using the principal axis frame in the Lagrangian allowed?
The kinetic energy of a fixed, rotating rigid body is
$$
T =\frac{1}{2}\mathbf{\omega}\mathbf{I}\mathbf{\omega}=\frac{1}{2}I_{xx}\omega_x^2 +\frac{1}{2}I_{yy}\omega_y^2 + \frac{1}{2}I_{zz}\omega_z^2 + ...
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How can we read Feynman rules from the equations of motion?
Reading Feynman rules from a Lagrangian is a quite standard procedure. However I have seen papers (for example Appendix A of arXiv:2412.14858) where this is done from the Equations of Motion instead ...
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Why does wave function renormalization result in a term in front of $(\nabla \phi)^2$?
I have asked previous questions about wavefunction renormalization before (e.g. What exactly is field strength renormalization? or What is the difference between wavefunction renormalization and field ...
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Dimensional Analysis Implications of Momentum and Force as Covectors?
Question: Does the point of view of Lagrangian mechanics, where (canonical) momentum is (re)defined as a covector, have implications for dimensional analysis, which seems to be based on the "...
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Harmonic oscillators and bead in a parabolic wire
I have a question about what in the abstract derivation we cannot assume for the system of a bead in a parabolic wire when we do not consider small displacements.
(1) We want to consider a system in ...
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What exactly is meant by Invariance of the Lagrangian?
So I've learnt that the Lagrangian is invariant under point transformations as well as under gauge transformations. Essentially meaning that if
$$
\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}...
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Lagrangian of a classical field, unique and not
I know that with the Lagrangian of a single particle, we can add a total time derivative to the Lagrangian without changing the equations of motion. But what about the Lagrangian of some field, maybe ...
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How can renormalization by counterterms possibly work?
I'm following Cheng-Li to understand conventional renormalization of $-\lambda \phi^4 / 4!$ theory in 4D.
Context
I understand that the only divergent 1PI Green's functions are $\Gamma^{(2)}(p^2)$ and ...
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Why do we need curvature in classical mechanics?
I am currently taking a course in theoretical mechanics, which is essentially classical mechanics revisited with the aim of formulating it as a geometrical theory. Therefore, after introducing the ...
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Why does the total differential include $q$ but not $t$ in Landau & Lifshitz’s treatment of the Hamiltonian formalism?
In Landau & Lifshitz, Classical Mechanics, §40 (Hamilton’s equations), p.132-133, it is said that when writing the total differential of the Lagrangian and of the Hamiltonian, one ignores the time ...
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How to deal with Equation of Motion with singularities?
I was trying to describe the movement of a ball rolling on bowl.
The degrees of freedom of the system are the following:
The Position of the Center Of Mass (where $r$ is the distance from the origin ...
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What does $\frac{\partial \mathscr{L}}{\partial A^\mu}$ mean?
Suppose we define a variation as
$$
\delta F \equiv F(x,a)-F(x,a=0)=\frac{\partial F}{\partial a}\bigg|_{a=0} a +\mathcal{O}(a^2),
$$
where $a$ is some continuous paramter and $x$ is a spacetime ...
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Equation of motion, 4th order
I have the following term in my Lagrangian:
$$
L=V(r)((\Delta \phi )^2-5(\partial_i\partial_j \phi)^2).
$$
I am kind of confused about computing the equation of motion, I would say (is there a ...
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What is the correct action for an anharmonic oscillator?
Why is the action of a quantum anharmonic oscillator equal to $$S = \int_{-\infty}^{\infty}m\frac{\dot{x}(t)^2}{2}-k\frac{x(t)^2}{2}-λ\frac{x(t)^4}{4!}dt$$
rather than
$$S = \int_{-\infty}^{\infty}\...
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Conventions for Wick rotations of the Dirac Lagrangian
I have some questions about Wick rotations of the Dirac Lagrangian in different signatures. I have seen similar questions, but none of them explain things in the way I need. In fact, I was trying to ...
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The Smoothness Postulate in Lagrangian Mechanics [duplicate]
The Euler-Lagrange equation is central to Lagrangian mechanics:
$$
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0
$$
This equation's structure ...
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The Klein-Gordon equation as a limit of continuous coupled harmonic oscillators?
I've seen espoused at the very least a metaphor between the discrete system of $n$ coupled harmonic oscillators
$$L = \sum_i^n \frac{1}{2}m_i \dot{x}_i^2 - \frac{1}{2}k_i(x_{i+1}-x_i)^2 -\frac{1}{2}\...
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Action for quintessence
Kislev developed black hole metric in quintessence field by considering $T^t_t=T^r_r$ for energy momentum tesor corresponding to quintessence. Is there any action for the quintessence fild that gives ...
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On the interpretation of time dependency of generalized coordinates [closed]
I'm studying theoretical mechanics. I'm having troubles understanding how to interpret the time dependence of generalized coordinates. Let's consider a certain (let's say mechanical, for the sake of ...
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