All Questions
Tagged with lagrangian-formalism special-relativity
216 questions
3
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2
answers
129
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Why can you not take $u \cdot u = c^2$ in the relativistic free massive particle Lagrangian?
In my classical electrodynamics class, we use the Lagrangian of the relativistic free massive particle as $$L = - mc\sqrt{\dot{r}\cdot\dot{r}}.$$ Where $\dot{r}^\mu = u^\mu = \frac{dr^\mu}{d \tau}$; $...
- 1,217
1
vote
1
answer
112
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Why is integral of relativistic action $-\alpha \int_{a}^{b} \, \mathrm ds$ minimised with respect to $\mathrm ds$?
While reading some aspects concerning the conclusion that bodies follow geodesics of spacetime, I ran into relativistic action on p.24 in chapter 2 $\S8$ of the 2nd volume of Landau & Lifshitz:
$$...
- 1,999
0
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2
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61
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Lorentz force law, Action of relativistic particle in e.m. field with interaction between field and particle
This question is about the derivation of the Lorentz force law and may be answered quickly. Nonetheless, I will give some context. In lecture we defined the overall Action of a particle + field (in ...
- 13
1
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1
answer
61
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Particle-hole symmetry and relativistic theory
Why must a particle-hole symmetric theory always be relativistic? Let's suppose we have the following Lagrangian density:
$$L = K_1 \Psi^*\frac{\partial\Psi}{\partial\tau} + K_2 \left|\frac{\partial\...
- 427
2
votes
1
answer
197
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Hamiltonian for massive relativistic point particle in light-cone gauge
Given light-cone coordinates, and some worldline parameterised by some arbitrary parameter $\tau$.
$$x^{+} = \frac{1}{\sqrt{2}}(x^0+x^1),$$
$$x^{-} = \frac{1}{\sqrt{2}}(x^0-x^1).\tag{2.50}$$
And the ...
- 6,946
1
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0
answers
59
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Homogeneous Lagrangian in covariant Lagrangian formulations
I have a question in Chapter $7.10$ of Goldstein et. al.'s classical mechanics (3rd edition). Specifically, on page 320 about the discussion of covariant Lagrangian formulations.
First, they introduce ...
- 1,505
0
votes
0
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36
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For an $O(N)$ symmetric function, is the 4-derivative simply equal to the derivative w.r.t the $N$-dimensional norm?
I am working on an $O(4)$-symmetric instanton which has the Lagrangian: $$L = \frac{M^6}{4E^2T^2} \left[ \frac{1}{2}(\partial_\mu\Phi)^2 - \frac{1}{2} \Phi^2 + \frac{1}{2}\Phi^3 - \frac{\alpha}{8} \...
- 357
0
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2
answers
98
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Lorentz scalar Lagrangian in curved spacetime
This question might be very simple but I guess I'm missing something. We know that Lagrangian has to be a Lorentz scalar. I can see why that should be the case when dealing with inertial frames of ...
- 397
0
votes
1
answer
68
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Lorentz transformation on field theory using coordinate expressions not representations
This is probably a stupid question but, I want to show that a Lagrangian written in field theory is Lorentz invariant WITHOUT using the Lorentz transformation representation / generators. I know we ...
- 1
0
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0
answers
48
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Does the Lagrangians in different coordinate systems differ by a total derivative when converted to the same coordinate in special relativity?
According to this post, it seems like we should use the $$\mathbb{L}~=~L \mathrm{d}t~=~ L \dot{t}\mathrm{d}\lambda \tag1$$ rather than $$L(v'^2) = L(v^2)+ \frac{df}{dt}\tag2$$ when working with ...
- 413
0
votes
1
answer
122
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Is relativistic action too restrictive?
When I was studying special relativity, I've learned that the relativistic action for a free particle is defined as
$$
S = \lambda \int_{\tau_0}^{\tau_1} d\tau
$$
Where $\lambda$ is a constant that is ...
- 4,200
3
votes
2
answers
144
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Relativistic Lagrangian for a System of Massive Particles
The standard Lagrangian $L$ written in local coordinates for a free, relativistic particle of mass $m > 0$ is given by
$$L(q, \dot{q}) = -m \sqrt{-g_{\mu\nu}(q) \dot{q}^\mu \dot{q}^\nu}\tag{1}$$
...
- 360
0
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0
answers
47
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Deducing the (special) relativistic Lagrangian only using Lorentz transform [duplicate]
I feel like there must be a way to deduce the (special) relativstic Lagrangian using only the Lorentz transform (without other knowledge of special relativity). So far, all the derivations I have seen ...
- 382
2
votes
2
answers
179
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When is the Lagrangian a Lorentz scalar?
The Lagrangian $\mathcal{L}$ can be defined as the Legendre transform (when it exists) of the Hamiltonian $\mathcal{H}$, a non-Lorentz scalar quantity (as $\mathcal{H} =T^{00}$). My questions are,
...
- 2,246
0
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38
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Turning a Lagrangian contains superscript and subscript indices into energy
I'm recently reading the book "Solitons and Instantons" written by R. RAJARAMAN. However, for lacking of ability, I couldn't figure out how to derivate the static solution for energy with ...
- 1