All Questions
Tagged with conservation-laws coordinate-systems
22
questions
0
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2
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82
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Generalized momentum
I am studying Hamiltonian Mechanics and I was questioning about some laws of conservation:
in an isolate system, the Lagrangian $\mathcal{L}=\mathcal{L}(q,\dot q)$ is a function of the generalized ...
- 49
1
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1
answer
60
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Unusual example of Compton Scattering (+Four-momentum approach, +nonrelativistic) [closed]
An electron of kinetic energy $k=100 keV$ (first note, doesn't this mean that its energy is much lower than $0.511 MeV$, and thus that it is a nonrelativistic electron we are dealing with?), collides ...
11
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2
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5k
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I don’t understand Noether’s theorem… there is nothing to prove?
I don’t understand Noether’s theorem… there is nothing to prove?
If I understand Noether’s theorem correctly it says: if there is coordinate where the Lagrangian is invariant, then the conjugate ...
- 2,035
1
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1
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51
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Integrals of motion for a rotational symmetric 3D Hamiltonian $H=\frac{{\bf p}^2}{2m}+V(r)$ [closed]
A particle of mass $m$ moves in three dimensions under the action of the conservative force with potential energy $V(r)$. Using the spherical coordinates $r, \theta, \phi$ find the Hamiltonian of the ...
- 11
2
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1
answer
129
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Adjustment to finite difference scheme for continuity equation in axisymmetric coordinates
I am working on a problem in which I need to numerically solve the continuity equation in an axisymmetric coordinate system (i.e. cylindrical with no $\phi$ dependence). For concreteness, I will use ...
- 23
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1
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105
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Invariance of continuity equation for Galilei transformations
I want to prove that the continuity equation for fluids, $$\dfrac {\partial \rho}{\partial t} + \nabla \cdot (\rho \textbf{u}) = 0$$ is invariant by Galilei transformations. My attempt:
Using index ...
- 115
1
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1
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128
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Separation of Hamilton's Principal Function
In Goldstein's "Classical Mechanics", at page 437, a two-dimensional anisotropic harmonic oscillator is studied by means of Hamilton-Jacobi formalism. In particular, Goldstein claims that:
...
- 402
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1
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109
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Singularities/infinities of continuity equation in polar coordinate
I encountered a bit of a difficulty in solving the continuity equation for polar coordinates. For a "fluid" or density of particles moving radially outwards with constant velocity, its flux ...
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-1
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2
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620
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Conservation theorem for cyclic coordinates in the Lagrangian
Suppose $q_1,q_2,...,q_j,..,q_n$ are the generalized coordinates of a system.
$q_j$ is not there in the Lagrangian (it is cyclic).
Then $\frac{\partial L}{\partial\dot q_j}=constant$
In Goldstein, it ...
- 293
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1
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98
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Why Goldstein's book is claiming that radius and angle doesn't contain time variable even there is $\dot{r}$ and $\dot{\theta}$?
$$L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2)-V(r)$$
$$p_\theta=\frac{\partial L}{\partial \theta}=mr^2\dot{\theta}$$
$$\dot{p}_\theta=\frac{d}{dt}(mr^2\dot{\theta})$$
Goldstein wrote that $\dot{P}_\...
- 91
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2
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117
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Where does this formula come from? [closed]
I am doing revision for my module stellar & galactic astrophysics and have come upon this formula which I cannot seem to derive. Could someone please explain where it comes from?
"For an ...
1
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0
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66
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$L_3$ conserved on geodesics?
Let's take a simple $E^3$ space with coordinates $(x,y,z)$ and metric tensor
$$ g = \mathrm{d} x \otimes \mathrm{d} x + \mathrm{d} y \otimes \mathrm{d} y + \mathrm{d} z \otimes \mathrm{d} z $$
The ...
- 274
2
votes
2
answers
520
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Invariance of Lagrangian under active coordinate transformation
This question is related to symmetry properties of the Lagrangian and conservation laws. Let us consider a one-dimensional case of a particle of mass $m$ moving along the $x$ axis such that the ...
- 101
2
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1
answer
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Noether's Theorem and conservation of momentum
So as we all know for a system that has translational symmetry Noether's Theorem states that momentum is conserved, more precisely the theorem states that the quantity:
$$\frac{\partial L}{\partial \...
- 4,577
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1
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54
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How constraining are conservation laws and continuity principles?
Suppose there are $N$ particles with masses $m_1, m_2, ..., m_n$. Consider the $3N$-dimensional classical configuration space of such particles. Consider some arbitrary physically possible trajectory $...
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