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Andrius Kulikauskas

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Book.PreservationOfSymplecticArea istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2019 sausio 20 d., 12:15 atliko AndriusKulikauskas -
Pakeista 34 eilutė iš:
* How is symplectic area related to the Lie algebra/group {$C_n$}?
į:
* How is symplectic area related to the constraints on symplectic matrices? and to the Lie algebra/group {$C_n$}?
2019 sausio 20 d., 12:14 atliko AndriusKulikauskas -
Pridėta 34 eilutė:
* How is symplectic area related to the Lie algebra/group {$C_n$}?
2019 sausio 20 d., 12:11 atliko AndriusKulikauskas -
Pakeistos 42-43 eilutės iš
* [[https://en.wikipedia.org/wiki/Action_(physics) | Action]]
į:
* [[https://en.wikipedia.org/wiki/Action_(physics) | Action]]
* [[https://en.wikipedia.org/wiki/Calculus_of_variations | Calculus of variations
]]
2019 sausio 20 d., 12:11 atliko AndriusKulikauskas -
Pakeista 33 eilutė iš:
* Understand action in physics.
į:
* Understand action in physics. Action is energy x time = momentum x distance. In particular, physics is based on the minimization of action.
2019 sausio 20 d., 12:09 atliko AndriusKulikauskas -
Pakeistos 29-30 eilutės iš
į:
'''Further investigations'''

* Describe the change in the shape of the region of possibilities as the pendulum moves. Show that the area is preserved.
* Read further about Heisenberg's uncertainty principle.
* Understand action in physics.
Pakeistos 41-42 eilutės iš
* [[https://en.wikipedia.org/wiki/Stokes%27_theorem | Stokes' theorem]]
į:
* [[https://en.wikipedia.org/wiki/Stokes%27_theorem | Stokes' theorem]]
* [[https://en.wikipedia.org/wiki/Action_(physics) | Action
]]
2019 sausio 20 d., 11:57 atliko AndriusKulikauskas -
Pakeistos 22-26 eilutės iš
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}. If we have a rectangle given by {$p_{max}$} and {$p_{min}$} and {$q_{max}$} and {$q_{min}$} then the integral of the action - and the area - is given by {$(p_{max} - p_{min})(q_{max} - q_{min})=\Delta p \Delta q$}.
į:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}. If we have a rectangle given by {$p_{max}$} and {$p_{min}$} and {$q_{max}$} and {$q_{min}$} then the integral of the action - and the area - is given by {$(p_{max} - p_{min})(q_{max} - q_{min})=\Delta p \Delta q$} if we integrate clockwise.
Pridėtos 26-27 eilutės:

If we integrate clockwise, then position and momentum increase together. If we integrate counterclockwise, then one increases as the other decreases.
2019 sausio 20 d., 11:45 atliko AndriusKulikauskas -
Pakeista 22 eilutė iš:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}. If we have a rectangle given by {$p_{max}$} and {$p_{min}$} and {$q_{max}$} and {$q_{min}$} then the integral of the action - and the area - is given by {$(p_{max} - p_{min})(q_{max} - q_{min})$}.
į:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}. If we have a rectangle given by {$p_{max}$} and {$p_{min}$} and {$q_{max}$} and {$q_{min}$} then the integral of the action - and the area - is given by {$(p_{max} - p_{min})(q_{max} - q_{min})=\Delta p \Delta q$}.
2019 sausio 20 d., 11:44 atliko AndriusKulikauskas -
Pakeista 22 eilutė iš:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}. If we have a rectangle given by {$p_{max}$} and {$p_{min}$} and {$q_{max}$} and {$q_{min}$} then the integral of the action - and the area - is given by {$({p_{max} - p_{min})({q_{max} - q_{min})$}.
į:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}. If we have a rectangle given by {$p_{max}$} and {$p_{min}$} and {$q_{max}$} and {$q_{min}$} then the integral of the action - and the area - is given by {$(p_{max} - p_{min})(q_{max} - q_{min})$}.
2019 sausio 20 d., 11:43 atliko AndriusKulikauskas -
Pakeista 22 eilutė iš:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}.
į:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}. If we have a rectangle given by {$p_{max}$} and {$p_{min}$} and {$q_{max}$} and {$q_{min}$} then the integral of the action - and the area - is given by {$({p_{max} - p_{min})({q_{max} - q_{min})$}.
2019 sausio 20 d., 11:39 atliko AndriusKulikauskas -
Pakeistos 23-26 eilutės iš
\\
į:

2019 sausio 20 d., 11:24 atliko AndriusKulikauskas -
Pakeistos 22-23 eilutės iš
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p dq$} around the boundary {$\partial S$}.
į:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p \, dq$} around the boundary {$\partial S$}.
\\
2019 sausio 20 d., 11:22 atliko AndriusKulikauskas -
Pakeista 22 eilutė iš:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p dq$} around the boundary {$\partial S$}.
į:
The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p dq$} around the boundary {$\partial S$}.
2019 sausio 20 d., 11:22 atliko AndriusKulikauskas -
Pridėtos 19-22 eilutės:

'''Interpreting the area'''

The symplectic form {$dp \wedge dq$} measures the area. By Stokes' theorem, the integral of this area form equals the integral of the action {$p dq$} around the boundary {$\partial S$}.
2019 sausio 20 d., 11:18 atliko AndriusKulikauskas -
Pakeista 18 eilutė iš:
Define a region in phase space which describes the first extreme, where the velocity is close to zero but the position can vary substantially, corresponding to different amplitudes. Then the area - the variety of possibilies - will be preserved as time evolves. In particular, when the pendulum hangs down vertically - let us call this position zero - then the momentum varies substantially.
į:
Define a region in phase space which describes the first extreme, where the velocity is close to zero but the position can vary substantially, corresponding to different amplitudes. Then the area - the variety of possibilies - will be preserved as time evolves. In particular, when the pendulum hangs down vertically - let us call this position zero - then the momentum varies substantially. Note that in this case of a pendulum, the frequencies are the same, the possibilities happen to be in sync, and so the area exhibits periodic behavior and traces a closed loop.
2019 sausio 20 d., 11:15 atliko AndriusKulikauskas -
Pridėtos 20-23 eilutės:
'''Orientation of the area'''

Pakeistos 28-29 eilutės iš
* [[https://en.wikipedia.org/wiki/Old_quantum_theory | Bohr–Sommerfeld quantization condition]]
į:
* [[https://en.wikipedia.org/wiki/Old_quantum_theory | Bohr–Sommerfeld quantization condition]]
* [[https://en.wikipedia.org/wiki/Stokes%27_theorem | Stokes' theorem]]
2019 sausio 20 d., 11:14 atliko AndriusKulikauskas -
Pridėtos 19-24 eilutės:

'''Readings'''

* [[https://math.barnard.edu/sites/default/files/ewmcambrevjn23.pdf | What is symplectic geometry?]] Lisa McDuff
* [[https://en.wikipedia.org/wiki/Uncertainty_principle | Heisenberg's uncertainty principle]]
* [[https://en.wikipedia.org/wiki/Old_quantum_theory | Bohr–Sommerfeld quantization condition]]
2019 sausio 20 d., 10:56 atliko AndriusKulikauskas -
Pridėta 18 eilutė:
Define a region in phase space which describes the first extreme, where the velocity is close to zero but the position can vary substantially, corresponding to different amplitudes. Then the area - the variety of possibilies - will be preserved as time evolves. In particular, when the pendulum hangs down vertically - let us call this position zero - then the momentum varies substantially.
2019 sausio 20 d., 10:48 atliko AndriusKulikauskas -
Pakeistos 12-17 eilutės iš
Consider a simple harmonic oscillator, such as a pendulum. It may be thought of a single particle with a position and a momentum. Note that the frequency does not depend on the amplitude. We will consider the phase space given by the possibilities for the position and the momentum.
į:
Consider a simple harmonic oscillator, such as a pendulum. It may be thought of a single particle with a position and a momentum. We will consider the phase space given by the possibilities for the position and the momentum.

The symplectic area is a region of this phase space. We can consider a variation of the possibilities for position and likewise for
momentum.

Consider, in particular, two extreme cases. In one extreme, kinectic energy is near its minimum (near zero) but potential energy is near its maximum. This is when the pendulum has swung completely to one side. In the other extreme, kinectic energy is near its maximum but potential energy is near its minimum. This is when the pendulum is hanging down vertically and swinging with greatest velocity.
2019 sausio 20 d., 10:42 atliko AndriusKulikauskas -
Pridėtos 10-12 eilutės:
'''Consider the simplest illustration'''

Consider a simple harmonic oscillator, such as a pendulum. It may be thought of a single particle with a position and a momentum. Note that the frequency does not depend on the amplitude. We will consider the phase space given by the possibilities for the position and the momentum.
2019 sausio 20 d., 10:29 atliko AndriusKulikauskas -
Pakeistos 1-7 eilutės iš





Investigation
: Interpret the preservation of symplectic area
į:
>>bgcolor=#E9F5FC<<
---------------------
See
: [[Math Notebook]]


'''Interpret
the preservation of symplectic area.'''
----------------------
>><<
2019 sausio 20 d., 10:28 atliko AndriusKulikauskas -
Pridėtos 1-7 eilutės:






Investigation: Interpret the preservation of symplectic area

PreservationOfSymplecticArea


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Puslapis paskutinį kartą pakeistas 2019 sausio 20 d., 12:15
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