Linked Questions
12 questions linked to/from Detailed conditions for symmetries of Lagrangian
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Why are differential equations for fields in physics of order two?
What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations?
If someone on the street would flat out ask me ...
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answers
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Noether Theorem and Energy conservation in classical mechanics
I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
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Do an action and its Euler-Lagrange equations have the same symmetries?
Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations.
Can ...
- 9,691
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votes
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Invariance of Lagrangian in Noether's theorem
Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$.
However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ (...
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Does a four-divergence extra term in a Lagrangian density matter to the field equations?
Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation:
${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^\...
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When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?
The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time ...
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Invariance of action $\Rightarrow$ covariance of field equations?
Invariance of action $\Rightarrow$ covariance of field equations? Is this statement true?
I have only seen examples of this, like the invariance of Electromagnetic action under Lorentz ...
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Off-shell and on-shell assumptions within the derivation of Noether's theorem
If we consider a transformation of a field $\Phi \rightarrow \Phi + \alpha \frac{\partial \Phi}{\partial \alpha}$ which is not a symmetry of a lagrangian then one can show that the Noether current is ...
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Lagrangian $L' = L + \frac{df}{dt}$ gives the same equations of motion
It is well known that when a Lagrangian $L$ is incremented by the total time derivative of a function $f$ that does not depend on the time derivatives of the generalized coordinates, the same ...
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Why is the symmetry variation $\delta_s q$ different from the ordinary variation $\delta q$?
I was reading about symmetry of action when I came before the symmetry variation in Particles and Quantum Fields by H. Kleinert; there he wrote:
Symmetry variations must not be confused with ...
user36790
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Clarification on Noether’s Theorem
Consider a particle which occupy $(t_1,q_1)$ and $(t_2,q_2)$ where $t$ denotes time and $q$ denotes spatial coordinate. The dynamics of the particle is determined by extremising $$S[q]=\int^{t_2}_{t_1}...
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Conflict of domain and endpoints in Noether's theorem and energy conservation
In the derivation of energy conservation, there is the transformation $q(t)\rightarrow q'(t)=q(t+\epsilon)$, whose end points are kind of fuzzy. The original path $q(t)$ is only defined from $t_1$ to $...
