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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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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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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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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How does one express a Lagrangian and Action in the language of forms?

In Lipschitzs Classical Mechanics a Lagrangian is defined as: $L(q,q',t)$ for some trajectory $q(t)$ of a particle And the action is defined as: $S:=\int^a_b L(q,q',t) dt$ How does one ...
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Rigorous version of field Lagrangian

In Classical Mechanics the configuration of a system can be characterized by some point $s\in \mathbb{R}^n$ for some $n$. In particular, if it's a system of $k$ particles then $n = 3k$ and if there ...
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Boundary term in Einstein-Hilbert action

Why is the boundary term in the Einstein-Hilbert action, the Gibbons-Hawking-York term, generally "missing" in General Relativity courses, IMPORTANT from the variational viewpoint, geometrical setting ...
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The BRST construction for YM with or without auxiliary field

I'm learning BRST symmetry for Yang-Mills theory and I see that there are two ways of writing BRST differential. In some books (for example Ryder's and Ramond's textbooks) BRST differential acts as \...
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Is it possible to project a problem of mechanics in a lower dimensionality?

I had the intuition that, in classical mechanics, when the trajectory of a body is known, then analysis of its motion can be done in the linear space of that trajectory, if all forces are projected on ...
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Where does the mass term come from in the Proca Lagrangian?

There are many good books describing how to construct the Lagrangian for an electromagnetic field in a medium. $$\mathcal{L}~=~-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}-\frac{1}{c}j^{\nu}A_{\nu}$$ ...
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Is the Lagrangian density in field theory real?

As the Lagrangian in classical mechanics corresponds to energy, it must be real. But is that the case in quantum field theory? I mean, it should still correspond to some sort of energy, but what about ...
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Lagrangian of two particles connected with a spring, free to rotate

Two particles of different masses $m_1$ and $m_2$ are connected by a massless spring of spring constant $k$ and equilibrium length $d$. The system rests on a frictionless table and may both oscillate ...
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Lagrangian mechanics vs Hamiltonian mechanics [duplicate]

First of all, what are the differences between these two: Lagrangian mechanics and Hamiltonian mechanics? And secondly, do I need to learn both in order to study quantum mechanics and quantum field ...
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How do you know if a coordinate is cyclic if its generalized velocity is not present in the Lagrangian?

Goldstein's Classical Mechanics says that a cyclic coordinate is one that doesn't appear in the Lagrangian of the system, even though its generalized velocity may appear in it (emphasis mine). For ...
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Noether current for the Yang-Mills-Higgs Lagrangian

I am trying to calculate the Noether current, more specifically, the energy density of the Yang-Mills-Higgs Lagrangian. Please refer to the equations in the Harvey lectures on Magnetic Monopoles, ...
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I am self-studying Goldstein's book "Classical Mechanics", and I need some help understanding the part where Goldstein discusses using Hamilton's principle to solve systems with holonomic constraints (...
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Higher order covariant Lagrangian

I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at Klein-...
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Why is the Lagrangian quadratic in $\dot{q}$? [duplicate]

My teacher said we only consider Lagrangians which are quadratic in $\dot{q}$, and we don't take other Lagrangians. I couldn't understand why. Can anyone please explain this?
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speed of sound and the potential energy of an ideal gas; Goldstein derivation

I am looking the derivation of the speed of sound in Goldstein's Classical Mechanics (sec. 11-3, pp. 356-358, 1st ed). In order to write down the Lagrangian, he needs the kinetic and potential ...
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Derivation of the Lagrangian method using discretized time axis

I'm watching this video lecture by Leonard Susskind of Stanford: http://www.youtube.com/watch?v=3apIZCpmdls After some preliminaries, at 34 minutes he jumps into a discretization of the time axis ...
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Is the form of the Lagrangian relevant before the renormalization procedure?

In the renormalization procedure, is writing things like $$\varphi=\sqrt{Z_{\varphi}}\ \varphi_R\ ,\ \ m_0^2=Z_m\ m_R^2\ ,\ \ g_0=Z_g \mu^{\epsilon}\ g_R$$ and Z_i=1+\sum_{\nu=1}^\infty C_i^{(\...
What kind of fields can couple naturally to a $p$-form gauge fields in a Lagrangian?
Ordinary $U(1)$ gauge fields can naturally couple to classical fields such as spin-$1/2$ fields via the Dirac Lagrangian, or to complex spin-$0$ fields via the obvious covariant derivative coupling, ...