All Questions
Tagged with classical-mechanics noethers-theorem
117
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Continuity equation and constant of motion
In wikipedia, in the page for constant of motion, it says
"In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion."
...
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answer
164
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Why is there a relationship between symmetries and conservation laws?
I am reading through my professor's notes and I am unsure as to what the intimate relationship between the symmetry property of a physical system and the conservation laws of energy, momentum, and ...
- 65
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If the lagrangian density changes by a total derivative of the lagrangian density
When we derive energy momentum tensor current by actively transforming field. We see that lagrangian ( density) changes by a total derivative of the lagrangian. If a total derivative of the function ...
- 69
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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 $...
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Conserved quantities of a simple Lagrangian
Suppose the following Lagrangian with 2 degrees of freedom:
$$L = \frac{3}{2}\dot{q}^2_{1} \ + \frac{3}{2}\dot{q}^2_{2} \ + \dot{q}_{1}\dot{q}_{2}$$
My aim is to find all the conserved quantities of ...
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196
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Symmetries and conservation laws for a falling cat
For Hamiltonian systems, symmetries and conservation laws are defined and related to each other by Noether's theorem. Symmetry means the invariance of the Hamiltonian for a transformation group, and ...
- 745
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Physical meaning of constants (momenta?) generated by Noether's theorem via an ${\rm SO}(3)$-action
Let $\Bbb R^3$ be our configuration space. Consider the Lagrangian $L\colon T\Bbb R^3 \cong \Bbb R^6 \to \Bbb R$ given by$$L(x,y,z,\dot{x},\dot{y},\dot{z}) = \frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^3) ...
- 545
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Is there any reason that Landau and Lifshitz don't discuss Noether's theorem in their mechanics book? [closed]
I'm currently working my way through the Course volume one. Unless I've completely missed it, the authors omit any discussion of Noether's theorem, instead deriving various conservation laws on a case ...
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Free Fall Conservation of Momentum
So I looked at the invariance of the Lagrangian under the Gallilei Transformations.
So for the free fall we have the Lagrangian
$$L = \frac{m}{2}\dot{z}^2 -mgz$$
Then I applied the transformation
$$x\...
2
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1
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321
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Proof of Noether's theorem: How to deal with transformations in time?
I was following the proof of Noether's theorem in Lemos - Analytical Mechanics, page 73.
He considers a full infinitesimal transformation:
$$t'=t+\epsilon X(q(t),t),$$
$$q'(t')=q(t)+\epsilon\Psi(q(t),...
- 17.8k
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Getting a Conserved Quantity from a Lagrangian [duplicate]
So I've been messing around with the implications of Noether's theorem, and though I conceptually get what it's saying, I'm having a hard time actually using it to retrieve a conserved quantity from a ...
- 167
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3
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How can we derive from $\{G,H\}=0$ that $G$ generates a transformations which leaves the form of Hamilton's equations unchanged?
In the Hamiltonian formalism, a symmetry is defined as transformation generated by a function $G$ is a symmetry if
$$\{G,H\}=0 ,$$
where $H$ denotes the Hamiltonian.
On the other hand, a symmetry is ...
- 10.1k
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Why are symmetries in phase space generated by functions that leave the Hamiltonian invariant?
Hamilton's equation reads
$$ \frac{d}{dt} F = \{ F,H\} \, .$$
In words this means that $H$ acts on $T$ via the natural phase space product (the Poisson bracket) and the result is the correct time ...
- 10.1k
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How do we define the quantity $Q$, in the conservation of energy? And what does it rely on?
Noether's theorem to me explains how a certain defined quantity (Q) is conserved (locally) in time due to the time translation symmetry, and to be more specific; if we had a ball that is placed in a ...
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Problem using Noether's theorem in time-dependent lagrangian
I have some problems calculating the conserved quantity for a lagrangian of the
form
$$
L = \frac{1}{2}m\dot{q}^2 - f(t) a q,
$$
because I found the general problem too abstract, I tried at
first ...
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