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

00:51:24
https://www.youtube.com/watch?v=56UzPKEnub8

Sintesi

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.

Punti di forza

  • 🎓 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.

Linea temporale

  • 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.

Mostra di più

Mappa mentale

Mind Map

Domande frequenti

  • 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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Sottotitoli
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Scorrimento automatico:
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    today let me just remind we are planning
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    to do
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    this mon
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    or we
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    [Music]
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    sotru
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    here from
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    University
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    so 2010 at University and then at
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    experience at n
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    University and University of Canada and
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    then in 2013 he joined Charmers where
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    got full profess
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    in
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    so severals and words
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    mention from European
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    and
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    as by
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    also
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    also
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    start so concerning mathematical
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    contribution one this major contribution
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    is development
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    Camp
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    IM which preserve
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    certain geometrical
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    structure anyway
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    this is
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    particular Sy for long to
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    investigate
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    long
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    Sy and here from here you see see
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    thatch is quite a lot of that among
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    different FS uh of course
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    geometry
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    [Music]
  • 00:02:52
    also okay thank you very much can you
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    hear me well yes okay
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    good thanks a lot for that very very
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    nice introduction and thanks for
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    inviting me here it's my second time in
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    Piza last time was not that long ago
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    about half a
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    year and then I also got gave a a
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    presentation
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    um about related a related topic uh so
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    there will be probably some overlap but
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    I hope not too much in fact uh yesterday
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    during
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    lunch we had a very nice lunch and then
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    Dario said okay okay so it was a nice
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    talk you gave last time but maybe you
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    can say something more about this
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    quantization business which is related
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    to the numerical methods that we're
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    using and then uh he shouldn't have said
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    that because because because then I got
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    all excited about what I think is the
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    most beautiful part of um of the oiler
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    equations namely their the rich geometry
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    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
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    saying so um we are I'm interested in
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    right today I'm interested in the 2D uh
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    Oiler
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    equations so so here you see uh a
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    formulation of the two de Oiler
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    equations more or less as Oiler wrote
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    them down in 1757 he did it in both two
  • 00:04:28
    and three dimensions
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    um so the only thing that is slightly
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    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
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    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
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    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
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    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
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    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
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    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
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    okay but it's but we don't fix the
  • 00:09:13
    stream function it's it evolves in time
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    so it's not like we're just integrating
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    one vector field the stream function is
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    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
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    you get a lot you see a lot of
  • 00:09:30
    structure and
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    uh I mean why are the 2 de equations uh
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    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
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    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
Tag
  • 2D Euler Equations
  • Symplectic Geometry
  • Hamiltonian Systems
  • Numerical Simulations
  • Fluid Dynamics
  • Quantization
  • Conservation Laws
  • Geometric Properties