All Questions
Tagged with classical-mechanics noethers-theorem
117
questions
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Number of conservation laws
I saw a discussion about the relation of symmetries of Lagrangian and conservation laws on a textbook of analytical mechanics. A part that was counterintuitive to me was that all the discussion was ...
- 63
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Why do we study symmetries only via Noether conserved currents?
In general, we say a transformation is a symmetry of a theory if it leaves the action invariant, i.e. if
$$S \to S' = S,$$
up to, perhaps, a boundary term (b.t.). However, it is known (see e.g. this ...
- 1,154
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Conservations for time or space translational invariance, why $\delta L\vert_{\text{time trans}}\neq 0$? but $\delta L\vert_{\text{space trans}}= 0$?
To summarize my question first,
Given a classical mechanics Lagrangian,
$L=L(x(t), \dot{x}(t); t)$,
Why the conservation law for time $t$-translational invariant system, under time variance $\delta ...
- 4,336
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Noether's Theorem: There are conserved quantities corresponding to symmetries of position, orientation, and time, but why not velocity?
Noether's Theorem seems to be one of the most fundamental and beautiful results in all of physics. As I understand it, the fact that the laws of physics are the same independent of position, ...
- 2,403
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112
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How conserved quantities lead to equations of motion in Lagrangian mechanics
In a classical mechanics exercise, we were asked to derive a system of ODEs from conserved quantities. As we know from Lagrangian mechanics, Euler-Lagrange equations lead to equations of motion.
I ...
- 297
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166
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Noetherian versus Hamiltonian symmetries
There are 2 ideas of symmetry I have seen in Classical Mechanics:
Noetherian symmetry: Here they discuss infinitesimal point transformations where only position coordinates (and their derivatives ) ...
- 69
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193
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Conserved quantities and symmetries of the free 1D particle
For a classical free 1-d particle, the conserved quantities of the dynamics are:
$Q_1=p$
$Q_2=q-\frac{pt}{m}$.
The symmetry associated with $Q_1$ is translation symmetry, as I know.
What is the ...
- 1,490
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Symmetry Condition in Noether's Theorem
Suppose $q = \{q_1,\cdots, q_i\}$ is a coordinate system for Lagrangian $L(q,\dot{q},t)$. In this text by David Morin, on page 16 in chapter 6, it states that a symmetry is a transformation of the ...
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Problem with symmetry and integral of motion in Classical Mechanics [closed]
I am currently working through the problem 4.10 in Kotkin and Serbo's book "Collection of Problems in Classical Mechanics". The problem consists in showing that the quantity
\begin{equation}
...
- 39
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222
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Why does the conserved quantity generate the transformation in quantum mechanics? [duplicate]
If we have a lagrangian with a symmetry then we get a conserved quantity: $$Q=Q(p,q)$$ which is a function of the conjugate momentum and the coordinates.
If we move over to quantum mechanics then we ...
- 1,933
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Problem with Noether Theorem to prove that energy is conserved
Suppose an action $S = \int _{t_1}^{t_2} L(q(t),\dot{q}(t))$ that is invariant under an infinitesimal constant time translation $t \longrightarrow t' = t + \epsilon$, of course with $\epsilon = ...
- 1,290
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479
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Quantity conserved for the 3D spherically symmetric harmonic potential $V(r)=\alpha r^2$ [duplicate]
I know that in the case of the Kepler problem there is a quantity (other than energy, momentum,...) conserved which is the Runge-Lenz vector.
Is there also an "exotic" quantity conserved for a 2-Body ...
- 1,168
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92
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Force is to pressure as torque is to ...?
Using Noether's theorem to derive the conserved current for spacial translation symmetry, one gets that the time component is momentum and the spatial component is pressure (or stress). The continuity ...
- 101
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Conserved quantity when potential is invariant under transformation
I am very new to Noether's theorem and in our (first!) mechanics class it was proved using generators $X_i$,$X$ of a Lie group. Because I didn't really understand this proof I have trouble solving the ...
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