What is the main focus of the lecture?

The lecture focuses on the 2D Euler equations, symplectic geometry, Hamiltonian systems, and numerical simulations.

Who is the speaker, and where have they worked?

The speaker has worked at various universities and joined Charmers in 2013 as a full professor.

What is the Euler equation in 2D?

The 2D Euler equations describe a vector field that is time-dependent, primarily used to model fluid dynamics on a two-dimensional surface.

Why are the geometric properties of the Euler equations significant?

The geometric properties highlight the underlying structure, leading to a better understanding of the equations and the preservation of certain conservation laws.

What is a Hamiltonian system?

A Hamiltonian system is a type of dynamical system governed by Hamilton's equations, which are based on a function called the Hamiltonian.

What practical application does the speaker discuss regarding numerical simulations?

The speaker discusses using numerical simulations to understand the longtime behavior of 2D Euler equations.

How does quantization relate to Euler equations?

Quantization is used to approximate symplectic structures and provides a way to understand the Euler equations in terms of matrix approximations.

What are some challenges the lecture mentions in using traditional numerical methods?

Traditional numerical methods struggle with preserving the geometry and conservation laws inherent in the Euler equations, especially for long-time behavior.

What does the speaker propose for improving numerical methods?

The speaker suggests preserving symplectic structures in numerical simulations to better understand qualitative long-term behavior of Euler equations.

How does the speaker describe the complexity of analyzing Euler equations?

The speaker describes it as complex, noting that true mixing in a smooth sense cannot happen due to the transportation of level sets.

Colloquium - Klas Modin - Statistical mechanics, 2-D fluids, and structure preserving numerics

00:51:24

https://www.youtube.com/watch?v=56UzPKEnub8

Ringkasan

TLDRThe lecture explores the intricacies of the 2D Euler equations, linking their geometric properties and behavior to symplectic geometry and Hamiltonian systems. The speaker, an experienced professor, shares insights into these mathematical frameworks, highlighting their significance in understanding fluid dynamics on two-dimensional surfaces. The lecture also addresses the challenges associated with numerical simulations, which often fail to preserve essential geometric structures and long-term behavior. The speaker discusses innovative approaches, such as quantization, to create more effective numerical models. This perspective enables a deeper understanding of the underlying conservation laws and dynamics intrinsic to the Euler equations. Overall, the talk underscores the beauty and complexity of these equations, detailing mathematical contributions and the development of symplectic numerical methods that potentially offer more accurate approximations and solutions.

Takeaways

🎓 The speaker has a rich academic background, having worked at multiple universities.
🔍 The 2D Euler equations are central to the discussion, relevant in fluid dynamics modeling.
🥇 Symplectic geometry plays a significant role in understanding these equations.
💡 Geometric conservation laws are crucial for the Euler equations' structure.
🖥️ Quantization offers innovative numerical approaches to approximate these equations.
🎯 Numerical simulations need to preserve the equations' inherent geometry for accuracy.
🔄 Traditional numerical methods may not capture long-term behavior effectively.
📊 Matrix methods are proposed to enhance understanding and simulation accuracy.
🌊 Vorticity and vector fields are key concepts in the Euler equations analysis.
↔️ Hamiltonian systems provide a framework for exploring dynamical equations.

Garis waktu

00:00:00 - 00:05:00
The speaker begins by summarizing the academic and professional journey of a colleague named 'Sotru' who has received recognition for his contributions, particularly in preserving geometrical structures in mathematical developments. The speaker is also preparing to discuss their interest in the 2D Euler equations and their geometrical implications.
00:05:00 - 00:10:00
The speaker acknowledges the introduction and mentions being in Pisa for the second time. They recall having a related discussion about the Euler equations and their geometric beauty, specifically focusing on the 2D Euler equations' formulation on two-dimensional surfaces and manifold structures.
00:10:00 - 00:15:00
Euler's perspective on pressure and how it relates to vector fields being divergence-free is discussed. The speaker explains how the Euler equations connect with Newton's equations and highlights the Helmholtz decomposition in achieving incompressibility within fluids.
00:15:00 - 00:20:00
The speaker explains the Euler equations' geometric aspects, having researched related equations in shape analysis. They emphasize the vorticity function, derived using curl, to understand the equations better. The description includes how the vorticity is transported by velocity fields related to Hamiltonian functions, a stream function's geometric interpretation, and the 2D Euler equations' unique structure.
00:20:00 - 00:25:00
The speaker discusses the long-term behavior of 2D Euler equations, acknowledging their global solutions, unlike 3D Euler equations. Statistical mechanics and their application to 2D Euler equations are mentioned, along with historical references to using these equations to understand phenomena like planetary storms.
00:25:00 - 00:30:00
The speaker describes three approaches to studying 2D Euler equations: statistical mechanics by Onsager, numerical simulations, and rigorous PDE analysis. They highlight challenges in proving longtime behavior with PDE analysis and introduce numerical solutions that might approximate dynamics closely.
00:30:00 - 00:35:00
The speaker highlights how, unlike traditional numerics, structure-preserving numerical methods are crucial for capturing qualitative longtime behavior. While traditional methods focus on tracking single trajectories, newer approaches incorporate conservation laws and phase flow properties for more accurate analysis.
00:35:00 - 00:40:00
To understand the geometrical properties, the speaker introduces the concept of using Lie groups and invariant Hamiltonian systems. This includes introducing the configuration space as a Lie group and discussing how the dynamics on this space illustrate certain geometrical structures, which relate to 2D Euler equations.
00:40:00 - 00:45:00
The speaker explains how vorticity and the concept of point vortices arise from the symplectic structure of fluids, Newtonian mechanics, and Hamiltonian formulations. They discuss how the 2D Euler equations reflect dynamics on coadjoint orbits, with a focus on Casimir functions and how they maintain certain properties during flow.
00:45:00 - 00:51:24
Discussing quantization of Euler equations, the speaker outlines Vladimir Zeitlin's approach from 1991, which involves approximating the symplectic structure via finite-dimensional operators. They explain how this framework allows for statistical mechanics interpretations and potentially helps in discretized numerical applications.

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Pertanyaan yang Sering Diajukan

What is the main focus of the lecture?
The lecture focuses on the 2D Euler equations, symplectic geometry, Hamiltonian systems, and numerical simulations.
Who is the speaker, and where have they worked?
The speaker has worked at various universities and joined Charmers in 2013 as a full professor.
What is the Euler equation in 2D?
The 2D Euler equations describe a vector field that is time-dependent, primarily used to model fluid dynamics on a two-dimensional surface.
Why are the geometric properties of the Euler equations significant?
The geometric properties highlight the underlying structure, leading to a better understanding of the equations and the preservation of certain conservation laws.
What is a Hamiltonian system?
A Hamiltonian system is a type of dynamical system governed by Hamilton's equations, which are based on a function called the Hamiltonian.
What practical application does the speaker discuss regarding numerical simulations?
The speaker discusses using numerical simulations to understand the longtime behavior of 2D Euler equations.
How does quantization relate to Euler equations?
Quantization is used to approximate symplectic structures and provides a way to understand the Euler equations in terms of matrix approximations.
What are some challenges the lecture mentions in using traditional numerical methods?
Traditional numerical methods struggle with preserving the geometry and conservation laws inherent in the Euler equations, especially for long-time behavior.
What does the speaker propose for improving numerical methods?
The speaker suggests preserving symplectic structures in numerical simulations to better understand qualitative long-term behavior of Euler equations.
How does the speaker describe the complexity of analyzing Euler equations?
The speaker describes it as complex, noting that true mixing in a smooth sense cannot happen due to the transportation of level sets.

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Teks

Gulir Otomatis:

00:00:16
today let me just remind we are planning
00:00:19
to do
00:00:20
this mon
00:00:22
or we
00:00:29
[Music]
00:00:47
sotru
00:00:50
here from
00:00:53
University
00:00:54
so 2010 at University and then at
00:01:00
experience at n
00:01:02
University and University of Canada and
00:01:06
then in 2013 he joined Charmers where
00:01:10
got full profess
00:01:14
in
00:01:17
so severals and words
00:01:22
mention from European
00:01:25
and
00:01:28
as by
00:01:32
also
00:01:34
also
00:01:39
start so concerning mathematical
00:01:41
contribution one this major contribution
00:01:44
is development
00:01:49
Camp
00:01:52
IM which preserve
00:01:56
certain geometrical
00:01:58
structure anyway
00:02:06
this is
00:02:09
particular Sy for long to
00:02:13
investigate
00:02:15
long
00:02:19
Sy and here from here you see see
00:02:23
thatch is quite a lot of that among
00:02:27
different FS uh of course
00:02:32
geometry
00:02:43
[Music]
00:02:52
also okay thank you very much can you
00:02:54
hear me well yes okay
00:02:57
good thanks a lot for that very very
00:03:00
nice introduction and thanks for
00:03:01
inviting me here it's my second time in
00:03:04
Piza last time was not that long ago
00:03:06
about half a
00:03:08
year and then I also got gave a a
00:03:12
presentation
00:03:13
um about related a related topic uh so
00:03:18
there will be probably some overlap but
00:03:20
I hope not too much in fact uh yesterday
00:03:24
during
00:03:25
lunch we had a very nice lunch and then
00:03:28
Dario said okay okay so it was a nice
00:03:31
talk you gave last time but maybe you
00:03:33
can say something more about this
00:03:36
quantization business which is related
00:03:39
to the numerical methods that we're
00:03:42
using and then uh he shouldn't have said
00:03:44
that because because because then I got
00:03:48
all excited about what I think is the
00:03:50
most beautiful part of um of the oiler
00:03:53
equations namely their the rich geometry
00:03:57
okay and um so I will tell you a little
00:04:00
bit about this and then try to connect
00:04:03
it uh to to various Fields as uh as
00:04:06
Mario was
00:04:08
saying so um we are I'm interested in
00:04:12
right today I'm interested in the 2D uh
00:04:15
Oiler
00:04:15
equations so so here you see uh a
00:04:19
formulation of the two de Oiler
00:04:21
equations more or less as Oiler wrote
00:04:24
them down in 1757 he did it in both two
00:04:28
and three dimensions
00:04:30
um so the only thing that is slightly
00:04:33
different here is that we we do it
00:04:35
generically on on any two-dimensional
00:04:37
surface I call that surface M so that
00:04:40
has a manifold
00:04:41
structure and also with a remanion
00:04:43
structure on
00:04:45
it um so so these are the equations and
00:04:48
and the equations kind of describe a a
00:04:51
vector field that is time dependent and
00:04:55
this Vector field describes how the
00:04:56
fluid is moving the velocity field of
00:04:59
the
00:05:00
fluid okay so probably most of you have
00:05:04
seen this before there is also this
00:05:06
mysterious it's actually not so
00:05:08
mysterious this pressure here pressure
00:05:10
is a function the physicists have their
00:05:13
point of view but actually if you read
00:05:16
the original paper by by Oiler you see
00:05:18
that his view on this is exactly the
00:05:20
same view that many mathematicians have
00:05:24
namely that this comes from a projection
00:05:26
this is due to the fact that when you
00:05:28
look at these Oilers equations and they
00:05:30
they Oiler derive them from Newton's
00:05:33
equations so if you if you look at this
00:05:36
term here that is not going to be
00:05:38
Divergence free and we want the vector
00:05:40
field to be Divergence free because we
00:05:42
assume that our fluids is
00:05:43
incompressible okay so you want the the
00:05:46
the the the vector field to remain
00:05:48
Divergence free but this one qu of
00:05:50
points outside the space of Divergence
00:05:54
re Vector Fields so you need to
00:05:55
compensate for that and what Oiler
00:05:58
realized that the optim way to
00:06:00
compensate namely kind of the orthogonal
00:06:02
to to this subset of vector fields we
00:06:05
given by the
00:06:06
gradients so today we just refer to the
00:06:09
helm hols decomposition but Oiler
00:06:12
figured this out long before Helm hols
00:06:14
was even born it's quite
00:06:16
impressive okay so so so that's where
00:06:19
the how we think of the pressure and
00:06:20
these are the equations so I said I you
00:06:24
know these Oiler equations they have a
00:06:26
lot of kind of geometry attached to them
00:06:29
and I will try to say something about
00:06:31
this geometry in fact that for many many
00:06:34
years understanding that geometry was uh
00:06:38
my field of research um that's where the
00:06:43
first postto I did in New Zealand was
00:06:45
about understanding that not for fluid
00:06:47
equations but for kind of related
00:06:49
equations that shows up in shape
00:06:51
analysis and various Fields so I was not
00:06:53
so interested in fluids but then um I
00:06:57
had a student
00:07:00
milu Vivani uh and we we found some
00:07:04
interesting numerical problem related to
00:07:06
the two de Oiler equations and then I I
00:07:09
gradually realized that the gem among
00:07:12
all this class of geometric equations
00:07:15
that I'm going to express a little bit
00:07:16
is actually this two dimensional one
00:07:18
because it's the one that has most of
00:07:22
the structure in
00:07:24
fact uh so so let me kind of describe
00:07:28
that a little bit probably if you have
00:07:30
studied the two the oiler equations you
00:07:32
know that when you want to understand
00:07:34
this equation what you typically do you
00:07:36
don't think of it in terms of the vector
00:07:38
field instead you construct what is
00:07:40
called a verticity function you can do
00:07:42
this in three dimensions as well only in
00:07:44
three dimensions it's not a function
00:07:46
okay but here the verticity is a
00:07:49
function and kind of you know if if you
00:07:53
don't want to explain this in terms of
00:07:55
geometry what you're doing is you just
00:07:56
take your vector field and you take the
00:07:58
curl of that Vector field that is your
00:08:00
vorticity function and then they try to
00:08:02
express everything just in terms of this
00:08:04
this function and then you're led to to
00:08:07
new set of equations which I've write
00:08:09
like this so the this here is a Pon
00:08:13
bracket so this stresses actually the
00:08:16
the underlying geometry that kind of
00:08:18
pops out here uh but what this means
00:08:21
really is that the the verticity
00:08:24
function is
00:08:26
transported by the the velocity f
00:08:30
and furthermore the velocity field since
00:08:32
it is Divergence free that velocity
00:08:35
field corresponds to a hamiltonian
00:08:37
function on this two-dimensional
00:08:39
manifold that hamiltonian function we
00:08:41
call the stream function okay so here is
00:08:44
the geometric interpretation you you
00:08:46
take you take uh the level sets of this
00:08:49
stream function which is related to the
00:08:52
to the vorticity via the the laas
00:08:54
operator Quon
00:08:56
equation and then uh you look at the
00:08:59
gradient of the stream function and you
00:09:01
just rotate it it's called The skew
00:09:02
gradient and this in fact corresponds to
00:09:05
the the hamiltonian vector field for the
00:09:08
stream function as a hiltonium function
00:09:11
okay but it's but we don't fix the
00:09:13
stream function it's it evolves in time
00:09:16
so it's not like we're just integrating
00:09:18
one vector field the stream function is
00:09:20
part of the
00:09:21
Dynamics so so then you get this
00:09:23
vorticity formulation of the oil
00:09:25
equations and just by looking at this
00:09:28
you get a lot you see a lot of
00:09:30
structure and
00:09:33
uh I mean why are the 2 de equations uh
00:09:38
interesting well for several reasons
00:09:41
mathematically I suppose it's
00:09:43
interesting because they
00:09:45
are let's say easy enough that we can
00:09:49
prove that there are Global Solutions
00:09:51
Global in time solutions to these
00:09:54
equations
00:09:56
okay so so that's nice this some
00:09:59
analysis that shows us that solution
00:10:01
exists for all time you know that we we
00:10:04
don't know if this true and it's
00:10:05
probably not true for the 3D
00:10:07
equations you know that's a different
00:10:09
story but for 2D we know that it's true
00:10:13
okay yovich did this already in the 60s
00:10:15
I think
00:10:18
so uh so so once you have that you can
00:10:22
ask okay so what is going on then if if
00:10:25
you look at kind of is there some sort
00:10:27
of asymptotic behavior what happens when
00:10:30
you run when you you run these equations
00:10:33
for very long is there some kind of
00:10:36
asymptotic behavior that you can can
00:10:38
look at and initially you would say uh
00:10:41
probably not because
00:10:45
uh you know why should there be it's a
00:10:47
hamiltonian system hamiltonian systems
00:10:50
are not like gradient flows that kind of
00:10:52
converge to something so we don't expect
00:10:55
them to converge to something right in
00:10:58
fact they will not converge to something
00:11:00
I will talk more about this but still
00:11:02
there is something you can say about the
00:11:03
longtime
00:11:04
Behavior okay and this story to try to
00:11:09
understand the longtime Behavior well in
00:11:12
some sense it started already by the
00:11:14
time that people understood that there
00:11:16
are storms right because if you take as
00:11:19
a simple approximation of the atmosphere
00:11:21
just the two de Oiler equations then
00:11:23
eventually you see these storms forming
00:11:25
and it seems to be that they are quite
00:11:27
stable not completely stable but quite
00:11:30
stable if you look at other planets like
00:11:32
Jupiter for example then you see very
00:11:35
big storms that are very stable like the
00:11:38
Red Spot on Jupiter is actually a big
00:11:40
storm so you can ask what makes these
00:11:43
kind of
00:11:44
formations stable why why do they
00:11:50
appear and so that's a very natural
00:11:54
question and there are different
00:11:56
approaches for studying this problem
00:11:58
studying the long time behavior of these
00:12:00
two de oil equations and what I would
00:12:02
say one of the gems in the field is to
00:12:06
make progress on this problem okay so I
00:12:10
I I've listed here the the three
00:12:12
approaches that I consider there might
00:12:13
be others but I consider these like the
00:12:15
the main approaches and this
00:12:18
goes uh you know this covers both
00:12:21
physics and
00:12:23
Mathematics so the first approach I I
00:12:25
mean maybe I should not list it at this
00:12:28
first but this one approach which the
00:12:29
physicist like very much is to apply
00:12:33
statistical mechanics to the problem so
00:12:36
the first one to do this to apply
00:12:38
statistical mechanics to the two de
00:12:39
Oiler equations was l uner in 1949 he
00:12:43
wrote one paper about this which became
00:12:46
a legendary paper in fluid dynamics only
00:12:48
one paper uh but it has led to a a big
00:12:52
field of both mathematics and physics
00:12:54
that that paper Al but what is
00:12:57
statistical mechanics when you think
00:12:59
about it well it's something you take
00:13:01
some dynamical system and you try to
00:13:05
understand what I mean Bas it's like
00:13:09
probability applied to uh to dynamical
00:13:13
systems hamiltonian
00:13:15
systems so and you try to understand
00:13:18
where in phas base is it most common to
00:13:21
be you know where where does most States
00:13:24
wants to be in face space so this is the
00:13:26
underlying idea if you like of
00:13:28
statistical mechanics that the boltzman
00:13:31
uh was developing for gas Dynamics but
00:13:34
unog's brilliant idea was to apply this
00:13:37
to the fluids I will tell you more about
00:13:38
exactly
00:13:39
how um but the interesting thing here is
00:13:42
that what do you use from the actual
00:13:44
Dynamics the dynamical system that you
00:13:46
start with which in this case is
00:13:48
infinite dimensional what do you
00:13:50
actually use so you use that um there is
00:13:54
a
00:13:55
leoville structure to the equations
00:13:57
namely that pH BAS volume is
00:14:00
preserved so for hamiltonian systems
00:14:02
that's automatic because it follows from
00:14:04
the fact that the flow is simplec so you
00:14:07
always have this Lille property and
00:14:09
that's very
00:14:10
important H the other thing you use from
00:14:12
the Dynamics is that there are
00:14:13
conservation laws typically at least
00:14:16
energy could be others as
00:14:19
well but the rest of the Dynamics you
00:14:21
kind of throw away and then you just say
00:14:24
uh assuming that I know these kind of
00:14:26
macroscopic variables and I put them as
00:14:29
constraint and then I make the
00:14:30
assumption that all the states
00:14:34
microscopic States corresponding to this
00:14:37
uh microscopic state have the same
00:14:40
probability more or less this is the
00:14:42
intuition behind statistical mechanics
00:14:45
okay so you throw away a lot of the
00:14:47
Dynamics and just see okay what if I
00:14:50
make these assumptions and by the way
00:14:52
this is connected to the assumption that
00:14:54
the flow is erotic it's very much
00:14:56
connected to this so you make this
00:14:58
assumption and then normally you need to
00:15:01
make some sort of discretization as well
00:15:03
at least if this system is initially
00:15:05
infinite dimensional
00:15:08
because you know otherwise uh it doesn't
00:15:12
statistical mechanics essentially
00:15:13
doesn't make sense you need to to
00:15:15
truncate it somehow make it finite
00:15:17
dimensional so there's some
00:15:18
discretization involved also here but
00:15:20
not discretization to kind of of the
00:15:23
Dynamics but rather of the face bace
00:15:26
somehow
00:15:30
um so and then this gives you
00:15:32
predictions about what will happen or
00:15:34
what is most common so and then and then
00:15:37
the saying is if you're in a place in
00:15:39
face base which is not so common and
00:15:41
probably most of the time it will kind
00:15:43
of move towards some other part of face
00:15:45
base which is more
00:15:47
common okay that's a neat idea and then
00:15:51
you work surprisingly well actually so
00:15:55
the other approach is to to carry out
00:15:57
numerical simulations
00:15:59
so this is uh another approach where you
00:16:04
the it comes in two flavors I will say
00:16:06
something about this but normally what
00:16:08
you do you start with the equations and
00:16:10
then you discretize them in space and in
00:16:13
time you end up with some new Dynamics
00:16:17
which you now can actually Implement as
00:16:19
an algorithm in the computer you solve
00:16:21
for this and then you hope that whatever
00:16:24
discretizations you used will
00:16:26
approximate the flow good enough so that
00:16:30
somehow it corresponds to the the the
00:16:33
true dynamics that you're interested in
00:16:35
and then if you're if you're a good uh
00:16:37
numerical analyst you will not only make
00:16:40
the discretization you will also prove
00:16:43
that as I made the discretization better
00:16:45
and better this will actually converge
00:16:47
to the true solution of the of the
00:16:49
equations this is the traditional
00:16:51
numerical approach to to PDS let's
00:16:55
say and then of course uh the other
00:16:58
branch
00:16:59
uh which is probably you know the most
00:17:02
mathematical one and for some people
00:17:04
it's the only one is that you you apply
00:17:07
as much analysis in PD geometric
00:17:10
analysis you can to these equations to
00:17:12
try to understand the solutions okay so
00:17:15
this is uh here you you you really look
00:17:18
at the exact Dynamics you you try to
00:17:21
prove rigorous results but it's usually
00:17:24
very hard okay we can Pro we can prove
00:17:27
things like Global existence but to say
00:17:29
something about the longtime Behavior
00:17:32
rigorously with PD analysis is a very
00:17:35
very hard
00:17:36
problem okay so normally what you can
00:17:39
say the the state-ofthe-art today for
00:17:41
this kind of approach is that you start
00:17:44
with some uh near steady state solution
00:17:48
and then you maybe say that it will
00:17:50
remain near steady state for for that's
00:17:53
the kind of predictions you can get with
00:17:54
this this is like perturbations of
00:17:57
things that we we fully understand
00:18:01
so so in a way I I put numerics here in
00:18:04
the middle because I think of it as
00:18:07
something in
00:18:09
between this and this right so let me
00:18:12
explain why
00:18:16
so in numerics you take some concern to
00:18:20
the Dynamics your approximate the
00:18:23
Dynamics okay so you you do care about
00:18:25
the the Dynamics but if you do the
00:18:27
numerics carefully not everyone is doing
00:18:30
that but if you do it carefully and
00:18:31
you're you're interested in longtime
00:18:32
Behavior you should also care about
00:18:35
these other things that that that you do
00:18:39
that are important in statistical
00:18:42
hydrodynamics for example that the phase
00:18:44
flow is is preserves deliv property is
00:18:48
volume preserving and also that you have
00:18:51
conservation laws okay so you should try
00:18:53
to replicate those in your
00:18:56
discretizations and this is kind of
00:18:59
what separates what I what what I will
00:19:02
call Matrix hydrodynamics applied to
00:19:05
this uh equation from traditional
00:19:07
numerics where you traditional numerics
00:19:09
you essentially try to follow a single
00:19:11
trajectory as well as you can and
00:19:15
sometimes that's what you want you want
00:19:16
to follow a single trajectory as well as
00:19:19
you can to uh understand what is
00:19:22
happening for that single trajectory but
00:19:24
that's not really the problem we're
00:19:25
interested in here we're interested in
00:19:27
the qualitative long time Behavior okay
00:19:30
and for the oiler equations if you
00:19:32
follow a single trajectory even if you
00:19:34
do that extremely well eventually they
00:19:37
will diverge from the true solution
00:19:39
because uh because it's a kind of a
00:19:42
chaotic system it's very very sensitive
00:19:44
for perturbations of your initial
00:19:47
data so that's why I claim that
00:19:50
traditional numerics is good for
00:19:52
shorttime trajectory tracking but it's
00:19:55
not good for long time qualitative
00:19:57
Behavior then you really need to involve
00:20:00
some of the structure that we know is
00:20:02
important from statistical mechanics
00:20:04
namely all the conservation laws and
00:20:05
their phase flow
00:20:07
property okay so that was uh kind of a
00:20:10
long
00:20:11
introduction um so now I said that it's
00:20:14
important to preserve all this geometry
00:20:16
let me say something about more about
00:20:19
this geometry what it what it actually
00:20:21
comes from and I will do it first in a
00:20:24
kind of abstract way and U if you
00:20:27
haven't seen this before it might be uh
00:20:31
a bit much but relax because I will
00:20:34
connect it to the oiler equations the
00:20:37
two de Oiler equations so that you see
00:20:39
what's what's going on in fact I'm
00:20:41
essentially only interested in two the
00:20:43
oiler equations on the two
00:20:45
sphere
00:20:47
okay so here is the kind of abstract
00:20:50
framework you start with some
00:20:51
configuration space which is not like
00:20:54
particles in space but rather consists
00:20:57
of a Le group so so there's a group
00:20:59
structure to your configuration and then
00:21:02
you look at hamiltonian system for that
00:21:04
configuration space that means you have
00:21:05
some hamiltonian which involves not just
00:21:08
the configuration variable which is you
00:21:10
can think of as for example a rotation
00:21:12
Matrix if it describes a rigid body but
00:21:15
also some Associated momentum variable
00:21:18
geometrically this means we look at
00:21:19
points the face base consists of the
00:21:21
cotangent ble of this uh of this leag
00:21:24
group and then to get hamiltonian
00:21:27
Dynamics what you need you need a
00:21:30
hamiltonian uh so that's a function of
00:21:32
the configuration variable q and the
00:21:35
corresponding momentum variable which is
00:21:37
kind of the momenta sitting above the
00:21:38
point think of it as as
00:21:40
[Music]
00:21:42
a an configuration and a corresponding
00:21:45
uh momentum variable or velocity
00:21:48
variable associated with that but but we
00:21:51
put some constraints on our D Dynamics
00:21:54
so we say that in fact because we have
00:21:56
this group structure we say that our
00:21:57
hamiltonian is invariant under the
00:22:00
action of the group
00:22:01
itself okay so what does this mean think
00:22:05
of a rigid body rigid body is described
00:22:09
by I take this pan by by an a matrix an
00:22:14
orthogonal Matrix with determinant one
00:22:17
that describes its orientation right so
00:22:20
that's my all my
00:22:21
configurations and then I want my
00:22:23
hamiltonian describing this to be
00:22:25
invariant under how I kind of rotate the
00:22:28
object so this has a natural Symmetry
00:22:30
and a lot of problems in physics
00:22:32
actually have the Symmetry and the cool
00:22:35
thing that in fact the fluid also has
00:22:37
this symmetry you think about it like
00:22:39
this I'll come to that but the fact that
00:22:41
you have this symmetry means that in the
00:22:44
end you can reduce the Dynamics from
00:22:46
this cotangent bundle in quent out by
00:22:49
the Symmetry and what you end up with
00:22:51
and this is if you like this is uh like
00:22:55
Le
00:22:56
algebra things but think of it as you
00:22:59
have some symmetry you reduce out that
00:23:01
Symmetry and you end up if your
00:23:03
configuration space is a Le group you
00:23:05
actually end up in a vector space that
00:23:07
Vector space is just the Dual of the Le
00:23:10
algebra of of your your your Le group
00:23:15
okay I want to kind of go quickly here
00:23:18
because the point is once you have such
00:23:21
a system you can abstractly write it as
00:23:23
some equation just on the Duel of the
00:23:26
Lee algebra okay this is called the Le s
00:23:28
system those were the kind of systems
00:23:30
that I was I was studying long before I
00:23:33
got interested in fluids but not in
00:23:36
particular for for the fluid
00:23:39
okay but such systems also have a
00:23:42
hamiltonian structure that they kind of
00:23:45
inherit from the original Honan
00:23:47
structure it's just that it's not a
00:23:49
canonical Honan structure it's not the Q
00:23:51
and P
00:23:52
variable uh it's it's it's what it's
00:23:55
called a pon pon structure
00:23:59
okay but in fact you can still recover
00:24:02
the Q and P variables somehow because
00:24:06
every such Pon structure Pon manifold uh
00:24:10
is kind of foliated
00:24:13
into sub manifolds in the best case they
00:24:15
are sub manifolds and these are called
00:24:18
the simplec leaves and each such sub
00:24:21
manifold is a simplec manifold and then
00:24:23
you know the daru theorem that on each
00:24:25
simplec manifold you can put canonical
00:24:27
coordinates so locally you can in fact
00:24:30
reduce this system even more by
00:24:32
restricting to one of these simplec
00:24:34
leaves because once you start on one you
00:24:36
will never leave it okay and this by the
00:24:39
way for the fluids I'm skipping apart
00:24:41
has to do with the fact that there are
00:24:42
infinitely many conservation laws for
00:24:44
two the oiler equations so the Casimir
00:24:46
functions it just means that you remain
00:24:48
on these coant orbits
00:24:50
okay and formally the coadjoint orbits
00:24:53
are given you take the the action of the
00:24:57
group on its algebra and then that
00:25:00
induces an action on the Dual of the Le
00:25:02
algebra that's called a coint action we
00:25:04
write add star this operator here okay
00:25:08
so the Dynamics will remain on this
00:25:10
coadjoint orbits that's one of the main
00:25:12
messages from from this Vos Dynamics so
00:25:15
let's apply this now to the two de Oiler
00:25:17
equations what is the
00:25:19
group okay the group has to be infinite
00:25:22
dimensional because we know it's a flow
00:25:24
on some infinite dimensional space the
00:25:26
beautiful thing and this is due to oural
00:25:28
is that the group is the space of
00:25:31
simplecom morphisms or area preserving
00:25:34
maps of the
00:25:36
manifolds
00:25:38
okay uh
00:25:40
so that's the group write it like this
00:25:43
diff mu mu is the volume or the area
00:25:45
form so in this case it's the simplec
00:25:48
structure on these two dimensional
00:25:50
manifold just just think of it as the
00:25:51
area
00:25:52
form so the Le algebra of a Le group is
00:25:56
the tangent space at the identity this
00:25:58
comes out as what Divergence free Vector
00:26:01
fields which in the two-dimensional case
00:26:03
corresponds to simplec Vector Fields so
00:26:06
these are really the the simplec vector
00:26:08
fields and I put a little star here
00:26:11
because this is not always true you have
00:26:12
to assume that they're uh there some
00:26:15
topological properties of your manifold
00:26:17
the first cohomology has to be trivial
00:26:19
if that's true which is true on the
00:26:20
sphere then every simplec manifold every
00:26:23
simplec Vector Fields can be written in
00:26:25
terms of a generator or a hamiltonian
00:26:27
function that's the one we call the
00:26:29
stream function in fact so this space of
00:26:33
hamiltonian vector Fields is
00:26:35
parameterized by stream function and the
00:26:37
stream function is unique up to some
00:26:40
constant so the natural kind of space of
00:26:44
simple of hamiltonan vector Fields is
00:26:47
just the smooth functions modul the
00:26:52
constants okay so I said we wanted to
00:26:54
work with the Dual so now we need to
00:26:56
take the Dual of this Beast here here of
00:26:58
course the Dual of some space of smooth
00:27:02
functions is a nasty
00:27:05
object it's a you know you can put if
00:27:08
you want to put topologies on this basis
00:27:11
the the if m is compact the natural
00:27:13
thing is to put the fresh a topology
00:27:15
here and then the Dual of a fresh a
00:27:18
space which is not the Bono space not
00:27:19
even a fresh a space so this is a nasty
00:27:22
thing okay so what you do is you say
00:27:25
okay let us restrict restrict ourself a
00:27:29
little bit just to what is called the
00:27:31
smooth part of the jewel so you consider
00:27:34
elements in the jewel so those are are
00:27:38
one forms like this which are determined
00:27:41
by some smooth function okay and in
00:27:44
order for this to actually descend to
00:27:46
the the quotient here you you you have
00:27:48
to require that this function integrates
00:27:50
to zero so it has mean value
00:27:53
zero this is the how you kind of
00:27:55
normally do this but I wanted to stress
00:27:57
it because it's in fact important to
00:28:00
realize that sometimes you need to move
00:28:03
to this full
00:28:04
Jewel why is it okay to restrict to the
00:28:07
smooth duel well if you remember the
00:28:10
Dynamics remains on the coadjoint orbits
00:28:13
and the point is if you start on some
00:28:16
smooth with on a smooth um verticity it
00:28:21
will remain smooth if so if you start in
00:28:23
the smooth duel it will remain smooth
00:28:25
because you're acting on it with diffuse
00:28:27
so the actual of of a function the
00:28:29
action by the way is just by composition
00:28:31
of the inverse so this is going to
00:28:34
preserve the smooth structure of the
00:28:35
function okay so if you start with
00:28:38
something smooth then and apply a nice
00:28:40
diffo you remain smooth okay so in that
00:28:43
sense it's okay to work with just a
00:28:44
smooth
00:28:46
duel uh and now you can ask okay so what
00:28:51
is Theon equation to get the leason
00:28:53
equation we have to also specify a
00:28:55
hamiltonian function and the hamiltonian
00:28:57
function we specify is just this one so
00:29:01
it's quadratic corresponding to the H
00:29:04
minus1 Norm on the space of
00:29:07
functions okay so if you write it in
00:29:11
terms of the string function is this
00:29:12
thing here and then you write down the L
00:29:14
Leon system and out pops exactly the two
00:29:18
the oiler equations in vorticity form so
00:29:20
that's the beauty of these equations it
00:29:22
can be interpreted as some hiltonium
00:29:25
flow on the cotangent bundle of a space
00:29:27
of dorph
00:29:29
a simplec diff morphisms and in fact
00:29:33
this hamiltonian description describes
00:29:35
geodesics on the space of diffuse this
00:29:38
is Arnold's big Discovery from
00:29:40
1966 I don't say so much here but but
00:29:43
the hamiltonian side is interesting
00:29:45
enough let's say you can also have a
00:29:48
Rania side of this so so so this is how
00:29:51
kind of the geometry pops out see where
00:29:53
I am in
00:29:55
time yeah uh
00:29:58
um and I said something that you know
00:30:01
there are some a lot of conserved
00:30:04
properties here and essentially where
00:30:06
does the from the geometric perspective
00:30:08
where do these properties come from well
00:30:11
if you take a function now in this large
00:30:13
pH base of all vorticity
00:30:16
functions um and you know you're going
00:30:19
to remain on the coant orbit so if you
00:30:21
take any
00:30:22
function on or any
00:30:25
functional uh on the space of verticity
00:30:31
functions that is constant on the coant
00:30:33
orbits that's going to be preserved by
00:30:36
the
00:30:37
flow and these functions are the ones
00:30:39
called Casmir Casmir
00:30:41
functions and and here is how you
00:30:43
construct them uh you just take some
00:30:46
function from R to R
00:30:49
okay and then you you just compose this
00:30:52
function with with your uh your uh your
00:30:55
smooth verticity field and you integrate
00:30:57
with respect to your simplec structure
00:31:00
your area form okay and the proof that
00:31:03
this is preserved well you see easily by
00:31:05
a change of variable that this is
00:31:07
preserved on the coent orbit and because
00:31:10
we remain on the coent orbit it has to
00:31:12
be preserved okay so so so these are
00:31:15
infinitely many custom unit one for each
00:31:18
function
00:31:20
f normally you take F to be polinomial
00:31:22
monomial
00:31:24
even um
00:31:26
okay so here is connecting back a bit to
00:31:30
this full story of the full Jewel which
00:31:32
was a nasty object so what happens if
00:31:35
you start on a coint orbit which is not
00:31:38
a smooth coint
00:31:40
orbit so maybe I didn't say that but the
00:31:46
interpretation of this coadjoint orbit
00:31:48
is that you you kind of look at the
00:31:50
level sets of your vorticity function
00:31:54
and and and the action is just moving
00:31:56
these level sets so the only thing that
00:31:58
you're allowed to do in the Dynamics is
00:32:00
to move around the level sets of your
00:32:02
verticity function that's kind of what
00:32:04
you're allowed to do that's puts a lot
00:32:06
of constraints on what you can do right
00:32:08
and in fact it it kind of explains why
00:32:11
fluids are so complicated because in
00:32:13
fluids you sometimes see like vorticity
00:32:17
Blobs of the same sign that kind of
00:32:20
undergo some
00:32:22
mixing but true mixing in the smooth
00:32:26
sense cannot happen because you're just
00:32:29
transporting level sets so the only
00:32:31
thing that happens is that this becomes
00:32:32
more and more and more intricate and in
00:32:35
fact it's so intricate that in the end
00:32:38
if you just take some completion in a
00:32:40
slightly rougher topology you cannot see
00:32:44
what has happened okay so this is why
00:32:46
say at least from my perspective the
00:32:48
analysis of this equation is is far from
00:32:51
trivial okay
00:32:54
um so so you're just transporting levels
00:32:56
so what happens if you start with some
00:32:58
verticity that has support just on some
00:33:01
single on some points you know so you
00:33:04
start with a Sit situation where the
00:33:06
verticity is zero everywhere except on a
00:33:09
few points these points here okay let's
00:33:13
say four of
00:33:15
them so so then you can construct the
00:33:18
corresponding
00:33:20
uh element in the Dual right this is an
00:33:23
element in the Dual I just take some
00:33:25
some strengths here gamma and I and and
00:33:28
I construct this thing here it's it's
00:33:31
linear okay so it's an element in the
00:33:33
Dual um and in fact formally you can
00:33:36
write this in terms of direct Delta so
00:33:39
kind of what happens here is that you
00:33:41
you put you put vorticity it's not mass
00:33:44
now we put verticity just in a few
00:33:48
points
00:33:49
okay and then the point is in the smooth
00:33:53
setting if I started with a smooth
00:33:54
vorticity field then it remain Smooth by
00:33:56
the co induction so what happens if I
00:33:59
apply the coant action to to this
00:34:01
strange configuration well it's easy to
00:34:04
see it's going to uh remain of the same
00:34:07
form I just moved around these Delta
00:34:10
pulses okay with my diff
00:34:13
morphis so in fact it also means that I
00:34:15
have these finite dimensional coagent
00:34:17
orbits because I start with finitely
00:34:19
many let's say n of these direct deltas
00:34:23
and I'm just moving them
00:34:25
around okay and it's clear that the ACT
00:34:28
on these is transitive by the diffuse
00:34:30
you can move the points to any where you
00:34:33
like except maybe I so I should take
00:34:35
away the diagonal here sorry that's a
00:34:37
mistake because you cannot take two
00:34:40
points
00:34:41
and put them on top of each other
00:34:43
because then you you're no longer a diff
00:34:45
morphis okay so that cannot happen so I
00:34:47
should take out the diagonal here but if
00:34:49
I do that it's a kind of transitive
00:34:50
action on that on that manifold and that
00:34:54
manifold is now finite dimensional so I
00:34:55
have finite dimensional coordinate
00:34:57
orbits and then I can describe the
00:34:59
Dynamics on these finite dimensional
00:35:00
spaces and what comes
00:35:03
out is point vertices okay so the
00:35:07
correct geometric interpretation of the
00:35:10
point Vortex solutions to the oiler
00:35:12
equations which were found by the way
00:35:15
long
00:35:16
before uh this geometry was discovered
00:35:19
so and it's kind of natural as soon as
00:35:21
you realize that verticity is being
00:35:23
transported it's natural to say what
00:35:25
happens if I transport just a few sing
00:35:27
points but this is kind of the geometric
00:35:30
description that these Point vertices
00:35:32
just correspond to finite dimensional
00:35:34
coint
00:35:35
orbits okay of the fluids so now what
00:35:40
was unog's idea his idea was okay we
00:35:44
have these two deiler equations I want
00:35:46
to do statistical mechanics to try to
00:35:48
predict what happens see if I can see
00:35:49
these storms forming okay uh but how do
00:35:54
I do this on an infinite dimensional
00:35:56
face space doesn't make makes sense it
00:35:58
needs to be finite dimensional in fact
00:36:00
it kind of needs to be
00:36:02
compact so his idea was let's take Point
00:36:07
vortices instead and restrict them to
00:36:09
some manifold which is
00:36:12
compact uh so he he used I think the the
00:36:15
flat two tourus the dou periodic
00:36:18
square and he put N Point veses there
00:36:21
then he gets a finite dimensional
00:36:23
hamiltonian
00:36:25
system H and then he said this is a
00:36:28
realm where I can apply statistical
00:36:31
mechanics to this finite dimensional
00:36:33
hamiltonian system and he did that and
00:36:38
out came This brilliant
00:36:40
interpretation at uh I think I talked
00:36:43
about this last time I was here I I
00:36:45
don't want to spend too much time on it
00:36:46
but one of oner's great ideas was that
00:36:50
or what he realized is that even if you
00:36:52
don't restrict to a specific energy
00:36:55
level um the the
00:36:58
energy
00:37:00
uh uh the the pH base volume is is is
00:37:04
finite let's say one okay H and because
00:37:08
of that you see that the the this
00:37:11
thermodynamical temperature or the
00:37:13
statistical temperature can be both
00:37:15
positive and negative and when it is
00:37:19
negative which cannot happen for a gas
00:37:21
but it can happen for for this system
00:37:23
when it's negative then you kind of
00:37:25
expect that uh uh clusters of Point
00:37:30
vortices with the same sign will come
00:37:34
together because that's what the
00:37:36
statistical mechanics then
00:37:38
predicts so that was his insight and
00:37:40
that kind of explains why you see this
00:37:42
merging of storms right that that
00:37:45
behavior happening he also realized uner
00:37:48
himself that that this uh model has it
00:37:53
its
00:37:53
weaknesses it's based on point vortices
00:37:56
typically he said
00:37:58
say
00:38:00
uh distributions of vorticity which
00:38:03
occurs occur in the actual flow of
00:38:05
normal liquids are continuous of course
00:38:08
they very far from continuous when
00:38:10
you're approximating by uh by direct
00:38:13
deltas and then he says and then he
00:38:16
realized and here he's saying without
00:38:18
saying it explicitly for me what he's
00:38:20
saying is in the continuous case the
00:38:22
coadjoint orbits look very different
00:38:24
from in the point Vortex case he saying
00:38:27
that uh so that convective processes can
00:38:31
build V vortices only in the sense of
00:38:33
bringing together volume elements of
00:38:35
great initial vorticity what he's saying
00:38:37
is we're just transporting around the
00:38:39
level sets of vity so not everything can
00:38:42
be
00:38:43
reached okay and for me this is just
00:38:46
means soer kind of realized
00:38:48
that constraining yourself to staying on
00:38:52
these coadjoint orbits is is important
00:38:55
and the coint orbits in the point Vortex
00:38:56
model are very very different from the
00:38:58
continuous or smooth quadrant
00:39:02
orbits so okay so then you can ask this
00:39:05
is a nice idea of one Sager to take a
00:39:07
fluid approximate it by Point vortices
00:39:11
and apply statistical mechanics you can
00:39:12
also use this to discretize the equation
00:39:14
think of this as a numerical method
00:39:16
because in the end you have a finite
00:39:17
dimensional homoni system you can apply
00:39:19
some numerical integration scheme to
00:39:21
this and see what
00:39:23
happens but still there is this problem
00:39:26
that unar himself lifted so the question
00:39:29
is is there another model another finite
00:39:32
dimensional model of the oiler equations
00:39:35
where you have all this nice structure
00:39:38
which kind of fits into this
00:39:42
framework but still have the something
00:39:46
corresponding to smooth coant
00:39:48
orbits okay or or smooth verticity
00:39:52
fields and the answer is
00:39:55
yes if you relax a little bit the what
00:39:58
you mean so you we're not now not no
00:40:01
longer looking for Exact Solutions we're
00:40:03
modifying structures a little bit we're
00:40:06
approximating the structures and this is
00:40:08
an idea that goes back to Vladimir zlin
00:40:11
from 1991 and his idea was to take this
00:40:15
infinite dimensional Bon system and he
00:40:18
just looked at it well I imagine that he
00:40:21
just looked at it looked at the
00:40:22
verticity formulations formulation of
00:40:25
this system and he said what what do I
00:40:27
need I need the pon brackets and I need
00:40:31
a
00:40:31
lashan how can I approximate a Pon
00:40:34
bracket and a lashan that's what
00:40:37
quantization is doing for you okay the
00:40:40
field that was uh invented by Paul Dr
00:40:44
that was his idea you know you you you
00:40:47
you take a Pon algebra of smooth
00:40:51
functions and you replace it with
00:40:53
operators in such a
00:40:55
way that the the Pome bracket is
00:40:59
approximated by The Matrix
00:41:01
commutation and and it's not there's no
00:41:04
exact equality here essentially cannot
00:41:06
get exact quantization it it it cannot
00:41:09
happen Okay so you always get an
00:41:10
approximation here and and then you have
00:41:13
this H bar also which kind of has to do
00:41:16
with when this approximation breaks down
00:41:19
completely if you
00:41:21
like and so so in our case or what what
00:41:25
zlin did here was he was thinking about
00:41:29
taking a verticity field applying to
00:41:32
quantization to replace this Vector
00:41:35
field by a matrix because he also knew
00:41:39
something about quantization Theory and
00:41:41
he knew that if your underlying manifold
00:41:43
m is compact there is some chance that
00:41:46
these operators can be finite
00:41:47
dimensional so they can be
00:41:49
matrices okay and this going to be
00:41:51
because it's quantum mechanics going to
00:41:53
be a skew herian Matrix actually in
00:41:56
quantum mechanics you work with herian
00:41:57
matrices but it's just a matter of
00:41:59
multiplying with I everywhere so from
00:42:01
the Le algebraic perspective it's more
00:42:04
natural to think about SK Herm missan
00:42:07
Matrix so this is uh the approach and
00:42:09
this H bar in this case corresponds to
00:42:11
the size of the Matrix which numerically
00:42:14
makes perfect sense you have a very
00:42:15
large Matrix meaning many degrees of
00:42:18
freedom you can resolve more meaning you
00:42:20
have a smaller H bar in fact H bar
00:42:22
scales like one over n where n is the
00:42:24
size of the Matrix we're looking at n *
00:42:26
n Matrix skewer Mission
00:42:29
matrices okay so this is this is the
00:42:32
idea and the only problem is how do you
00:42:34
actually do this because we want to in
00:42:36
the end get something which we can work
00:42:38
with and Implement in the computer and
00:42:40
so on so you need explicit schemes for
00:42:42
how to quantize things and there are
00:42:45
such schemes um and the ones that sain
00:42:48
used were developed by Jen Hopper just a
00:42:50
few years
00:42:52
earlier um and he was doing it on on on
00:42:55
various manifolds so in particular this
00:42:57
flat Taurus and also on S2 I think zlin
00:43:00
initially did this just on the
00:43:03
Taurus um but I will be more interested
00:43:06
in doing this on the sphere in fact and
00:43:08
this is I this exactly how this works is
00:43:11
interesting uh and you know every
00:43:15
mathematician at some point you have to
00:43:16
study some representation Theory uh and
00:43:19
I probably did as a student but I didn't
00:43:22
quite get the points uh but now I
00:43:25
realize why representation Theory is
00:43:27
very very important uh because you see
00:43:30
in this quantization how it kind of pops
00:43:32
out and it's the it's a it's a very
00:43:35
natural way to describe this whole
00:43:36
process so I'm not going to go through
00:43:38
that because it will take too long but
00:43:40
essentially this is how it works if you
00:43:42
think about you know algorithmically how
00:43:45
you construct these matrices you start
00:43:47
with some function you expand it in
00:43:49
spherical
00:43:50
harmonics and then what the quantization
00:43:53
gives you is some approximation of these
00:43:55
spherical harmonics and B matrices and
00:43:58
the way you you you construct this is it
00:44:01
comes directly out from from
00:44:03
representation Theory so you start with
00:44:05
some symmetry which is in this case is
00:44:06
the the S SO3 Symmetry and from and then
00:44:09
you look at all the irreducible
00:44:11
representations and that you know we
00:44:13
know there are connections to spherical
00:44:15
harmonics and then this is how you
00:44:17
construct these matrices and out from
00:44:20
this you also get a
00:44:22
laian which is called the H HOA laian
00:44:26
which then approximates the the the llin
00:44:29
on the sphere okay so here is the the
00:44:32
idea of these Zin equations you take
00:44:35
your verticity state which is you know
00:44:39
the variable in your equation you
00:44:41
quantize it you get a corresponding
00:44:44
verticity
00:44:45
Matrix and then what else do you need
00:44:48
well you need a laian and if you
00:44:49
quantize that and H Yao tells us that we
00:44:52
get this H Yao laian which again you can
00:44:57
explicitly write down uh the formula
00:45:00
for and uh and then once we have this we
00:45:04
know what to do because this is how the
00:45:05
equations look like and now we know that
00:45:08
the pon bracket is just replaced by the
00:45:10
commutator so then we get the Matrix
00:45:11
flow just like this okay and then we can
00:45:15
once we solve this we can go back and
00:45:18
ask uh you know uh and interpret this as
00:45:22
a solution or some approximation to the
00:45:25
oiler equations what is the benefit of
00:45:27
this
00:45:27
approach well first of all what's the
00:45:30
bad thing with this approach if you if
00:45:32
you're a numerical analyst typical
00:45:35
questions you ask are like how well is
00:45:39
this uh discretization converging when
00:45:43
the number of degrees of freedom
00:45:44
increases okay and the other question
00:45:47
you ask is Convergence how fast does
00:45:51
Solutions converge assuming that they
00:45:52
you know if they do converge at
00:45:55
all uh so so this as an as a kind of
00:45:59
numerical scheme to approximate smooth
00:46:02
functions this has very bad order the
00:46:06
order is like one half because the error
00:46:10
goes down the discretization error and
00:46:12
this comes from quation Theory goes down
00:46:15
like one over n but the number of
00:46:17
degrees of freedoms is n
00:46:20
SAR so it's like an one half order
00:46:24
spatial discretization okay so that
00:46:26
that's kind of bad for a method right I
00:46:28
mean if you come say ha look I have a
00:46:31
numerical method the order the conver
00:46:33
the convergence order is one half it's
00:46:35
not so impressive okay so that you have
00:46:37
to live with because there is an
00:46:40
something that you gain from this also
00:46:41
which is that you preserve all this
00:46:43
richly plusone structure it's all there
00:46:46
so you have the analog you have coint
00:46:48
orbits you have all this casmere
00:46:51
functions or approximations of the
00:46:53
Casmir functions just contained in this
00:46:55
framework which is beautiful because
00:46:58
that means that we have the things or at
00:47:01
least some approximation of the
00:47:02
structure that we needed uh in order to
00:47:05
for statistical mechanics to apply so
00:47:08
somehow if you use this as a numerical
00:47:09
method you're you're in between you know
00:47:13
statistical mechanics and and the true
00:47:16
Dynamics somehow approach that's at
00:47:18
least how I think of it and um and once
00:47:21
you you have these things sorry let me
00:47:24
just check the time so once you have
00:47:27
these things you you can play a game the
00:47:29
game is very fun you take whatever you
00:47:32
want on the fluid side and you try to
00:47:34
look at it on the Matrix side okay and
00:47:38
then you get this dictionary and you can
00:47:39
go the other way around you can say Okay
00:47:41
Le theory is a very rich mathematical
00:47:44
Theory maybe it has some tools that I
00:47:46
can transfer back to the fluids so this
00:47:49
is the game that uh I've been playing
00:47:52
for several years together with milu
00:47:55
Vivani who did his about this and uh one
00:47:59
of the
00:48:00
key results that Milo had in his thesis
00:48:03
was that he in addition to using
00:48:06
this uh this spatial discretization on
00:48:09
the sphere he was able to find also a
00:48:12
way to discretize this equation in time
00:48:15
so that you preserve all the leoson
00:48:17
structure Which is far from
00:48:19
non-trivial so so so that was a very so
00:48:21
we now have everything we need to
00:48:23
preserve all the structure in the
00:48:25
equations in on the computer except for
00:48:28
round of Errors we still have round of
00:48:31
Errors so you play this game you have
00:48:34
verticity which corresponds to a matrix
00:48:36
you have the cmir functions which
00:48:38
corresponds to the integral corresponds
00:48:40
to taking the trace so you have trace of
00:48:42
of you know Matrix functions uh and then
00:48:45
you have hamiltonian which just looks
00:48:47
like this the values of the Omega this
00:48:50
is a key point in quantum mechanics
00:48:53
right the values of your function
00:48:55
corresponds to the igen values of
00:48:56
operator or I times the I values if it's
00:48:59
a skew if it's a skew Herm missan
00:49:02
Matrix uh and you we have this kind of
00:49:05
interesting interpretation that the
00:49:07
level sets of our proticity somehow
00:49:10
corresponds to the igen vectors of the
00:49:13
function okay so so you see it's natural
00:49:16
to each smooth function you associate a
00:49:19
level set level sets and values on that
00:49:22
level sets corresponding to the value of
00:49:23
the function in the same sense on The
00:49:25
Matrix you have I
00:49:27
values and to each igen value you
00:49:29
associate one or in the generic case
00:49:31
maybe just one igen
00:49:34
vector okay so the we also get a lot of
00:49:38
results from quantization Theory because
00:49:40
this is a big field of mathematical
00:49:42
physics so in particular we know that
00:49:45
and you can even prove convergence here
00:49:47
when n goes to Infinity the L Infinity
00:49:50
Norm corresponds to the spectral norm
00:49:53
and for example the L2 Norm corresponds
00:49:56
to The Matrix forus Norm things like
00:49:59
that uh here is something that mil and I
00:50:01
was working with if you take like
00:50:03
averages along the level set of the
00:50:06
stream function so you take your
00:50:07
verticity and then you take averages
00:50:09
along the level sets of your stream
00:50:12
function I mean you just think about
00:50:15
that in the smooth setting it's kind of
00:50:18
complicated but that's something that
00:50:20
you want to do normally because it has
00:50:22
to do with some smoothing and you know
00:50:24
conver anyway so so on The Matrix side
00:50:27
this is a very natural thing to do
00:50:29
because you just project onto the
00:50:31
stabilizer of your stream Matrix okay so
00:50:35
yeah so now I I told you kind of what I
00:50:38
why I think these two de Oiler equations
00:50:41
are so beautiful I hope that I have one
00:50:43
or two minutes more I will just show you
00:50:45
some simulations as well uh for for
00:50:48
these
00:50:49
things and I mean so once Milo had
00:50:52
developed this integrator we I mean we
00:50:54
thought okay this is a nice publication
00:50:56
in a good numerical Journal uh but we
00:50:59
want more because we want to see what
00:51:01
actually happens in the Dynamics so we
00:51:03
just say okay let's start we do it on
00:51:05
the sphere because that's what we did
00:51:06
and let's start with some random
00:51:08
randomly chosen but very smooth
00:51:10
verticity field and run it through this
00:51:13
numerical integrator that preserves all
00:51:14
the structure and see what happens so we
00:51:16
did this for for for many cases and and
00:51:20
this is a typical thing that happens so
00:51:22
you see we actually see