# Tagged Questions

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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### Quadratic terms in QED lagrangian density

I recently learned that when we speak about a "free lagrangian", this actually means that the lagrangian is quadratic in the fields. When considering the Lagrangian density describing the coupling to ...
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### Textbook for mathematical Lagrangian mechanics [duplicate]

I'm looking for a textbook or online notes or a review article etc on a rigerous formulation of Lagrangian mechanics. I'm well aware of the book by Arnold but I would like something to accompany it. ...
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### Nabla Terms in the Energy Density of the Lagrangian for the Massive Spin 1 Field (Schwartz QFT 1st Ed. Eqn. 8.19)

The relevant part starts with a Lagrangian guess of, $$\mathcal{L}=-\frac{1}{2}\partial_{\nu}A_{\mu}\partial_{\nu}A_{\mu}+\frac{1}{2}m^2A_{\mu}^2$$ where the EOM's are, $$(\Box+m^2)A_{\mu}=0$$ The ...
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### Finding the gauge transformtation of a Lagrangian [duplicate]

I am asked to find the gauge symmetry of the following Lagrangian: $L = -\frac{1}{4} F^2_{\mu \nu} + (\partial_{\mu} \phi_1 - m_1 A_{\mu})^2 + (\partial_{\mu} \phi_2 - m_2 A_{\mu})^2$ Then I have to ...
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### Isotropic of Inertial frame?

My understanding of isotropic is the a particular physics law remain same no matter at what direction I look at it? Now suppose in case of inertial frame, we know that its is homogeneous and ...
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### Lagrangian is isotropic in space

In Landau & Lifshitz Mechanics, while deriving the properties of Lagrangian of a free particle in inertial frame, he uses the following points $:$ As space is homogeneous in inertial frame, a ...
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### Lagrangian isn't unique [closed]

If $L$ is a Lagrangian for a system of n degrees of freedom satisfying Lagrange's equations, show by direct substitution that $$L' = L + \frac{\mathrm{d}F(q_1,\dots,q_n,t)}{\mathrm{d}t}$$ also ...
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### What is a Lagrangian of a photon? [duplicate]

In sense of classical mechanics+special relativity what is lagrangian of a photon? Lagrangian of a relativistic massive particle is as follows: $$L_{massive}= -mc\sqrt{c^2-v^2}$$ So is it a zero?
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### Why does overall action need to have an extremum?

Quoting from Landau's and Lifshitz' Mechanics : The integral ${\int\limits_{t_1}^{t_2}}L(q, \dot{q},t)\,dt$ for the entire path must have an extremum, but not necessarily a minimum. This, ...
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### Conserved quantities for the system, classical mechanics

In certain situations, particularly one-dimensional systems, it is possible to incorporate frictional effects without introducing the dissipation function. As an example, find the equations of motion ...
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### Canonical Momentum Conjugate vs. Momentum

I stumbled upon this while reading about Legendre Transforms today. So consider an n-particle system. The Lagrangian is a function of $q_i$'s and $\dot q_i$'s. If you consider the manifold $M$ where ...
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### Noether's Theorem for Hamiltonians and Lagrangians

Looking around I see one version of Noether's Theorem that creates conserved quantities from symmetries that preserve the Lagrangian (e.g. http://math.ucr.edu/home/baez/noether.html), and another ...
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### When can I apply Lagrangian mechanics?

I am trying to understand Lagrangian mechanics. I am having trouble capturing all of the nuances in one gulp. I can see the equations, but not necessarily the semantics behind such equations. I ...
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### Lagrange: when a potential force, when a generalized force?

Consider the following case of a drum unrolling a mass that is on a massless string wrapped around the drum: According to my professor, here we must consider the mass $m$ being in a gravitational ...
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### Strong interaction under $SO(3)$ isospin transformation

I'm given the following strong interaction: $$S = \int d^{4}x [\frac{1}{2} \partial_{\mu} \phi^{a} \partial^{\mu} \phi^{a} - \frac{m^2}{2} \phi^{a} \phi^{a}] ,\qquad a = 1,2,3 \text{.}$$ It is stated ...
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### How do you find potential by Lagrangian formalism?

Suppose a ball is falling towards earth and hence by Lagrange equation we can find $T$ and $V$ where $V$ is $mgh$. But we know $V$ only because we know $F = mg$. Now since Lagrange equation doesn't ...
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### Is the phrase “coupling constant” interchangable with “ strength of interactions”?

Can I use the terms coupling constant and strength of interactions, interchangeably, or are there more subtleties to the term coupling constant that I am not aware of? Coupling Constants from ...
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### Lagrangian for $\mathcal{N}=4$ SQED in 3D

What is the Lagrangian for $3D$ $\mathcal{N}=4$ supersymmetric QED, with $N_f$ hypermultiplets? In particular, which is the form of the Fayet-Iliopulos terms, and the real mass terms?
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### Help on understanding a concept in Noether's first theorem

Given a Lie group $G$, whose most general transform depends on $\rho$ parameters, under the action of which an integral $I$ is invariant, there are $\rho$ linearly independent combinations of the ...
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### Why do Lagrange multipliers work in mechanics?

I understand that it is not always simple to find generalized coordinates that satisfy the constraint equations, so we try to find an alternative (more mechanical) method that yields curves that ...
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### Trouble understanding Landau & Lifshitz writing about Lagrangians and Galilean Relativity [duplicate]

We have two inertial coordinate systems, $K'$ and $K$. $K$ is moving with infinitesimal velocity ${\epsilon}$ relative to $K'$. Using Galilean relativity we can transform this into $v'=v+{\epsilon}$. ...
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### Strain energy density (potential energy of elastic continua)

This question has to do with writing down the potential energy of an elastic body, which obeys a generalized Hooke's law [; \sigma_{ik} = \sum_{klm} \lambda_{iklm} u_{lm} ;] Where $\sigma$ is the ...