A tutorial on Hamiltonian mechanics - Convection: Home

Hamiltonian ODEsHamiltonian PDEs

Application of Hamiltonian mechanics

A tutorial on Hamiltonian mechanics

Alexander Bihlo

Faculty of MathematicsUniversity of Vienna

[email protected]

03.05.2011

A. Bihlo COST WG1+2 meeting 1 / 31



Outline

1 Hamiltonian ODEsCanonical Hamiltonian mechanicsNon-canonical Hamiltonian mechanicsThe inclusion of dissipation

2 Hamiltonian PDEsFrom finite- to infinite-dimensional Hamiltonian mechanicsEulerian fluid mechanics using a Hamiltonian representationMathematical issues of non-canonical fluid mechanics

3 Application of Hamiltonian mechanicsHamiltonian finite-mode modelsStatistical fluid mechanics




Motivation

Hamiltonian mechanics emerged in 1833 as another convenient formulation of classical Newtonianmechanics.

Over the next 150 years, it was gradually realized that the Hamiltonian formulation of a system ofdifferential equations has several advantages, including

a very general and appealing underlying geometric framework.

a good perspective for generalization to various disciplines.

a strong connection between symmetries and conservation laws.

a unified method for establishing nonlinear stability of equilibrium points.

a perspective for the application of statistical mechanics.

Additionally, the Hamiltonian description of a fluid mechanical system allows re-deriving severalclassical results in this field in a (more) systematic manner (e.g. Lorenz’ available potential energy,the derivation of simplified models, etc.).

This is why it is worth to have a closer look on the subject of Hamiltonian mechanics.




Motivation













Motivation













Motivation













Canonical Hamiltonian mechanicsNon-canonical Hamiltonian mechanicsThe inclusion of dissipation

Outline








Classical Hamiltonian systems

The Euler–Lagrange equations follow from the Hamiltonian principle

δSδq

!= 0,

where

S[q] =

∫ t1

t0

L(t, q, q) dt,

is the Hamiltonian action functional with L = T −U being the Lagrange function (“Kinetic energyminus potential energy”).

The variational derivative is

δSδq

:=∂L

∂q−

d

dt

∂L

∂q,

where in analogy with classical mechanics we denote

q Generalized coordinates

q Generalized velocities

p := ∂L∂q Generalized momenta

∂L∂q Generalized forces

The Hamiltonian equations are derived from the Legendre transformation of L with respect to q,

H(t, q, p) = pq − L(t, q, q),

i.e. by using p as an independent variable instead of q.







δSδq

!= 0,

where

S[q] =

∫ t1

t0

L(t, q, q) dt,



δSδq

:=∂L

∂q−

d

dt

∂L

∂q,







H(t, q, p) = pq − L(t, q, q),








δSδq

!= 0,

where

S[q] =

∫ t1

t0

L(t, q, q) dt,



δSδq

:=∂L

∂q−

d

dt

∂L

∂q,







H(t, q, p) = pq − L(t, q, q),







Taking the Legendre transformations converts the Euler–Lagrange equations into the equivalentsystem of canonical Hamiltonian equations:

dq

dt=∂H

∂p,

dp

dt= −

∂H

∂q.

In classical mechanics, H is the total energy of the system under consideration, i.e. H = T + U.

Although canonical Hamiltonian mechanics reports great success in the study of mechanical systems,it is too restrictive for our purposes. The main drawback is that canonical Hamiltonian mechanicsrequires an even-dimensional phase-space, which is spanned by tuples (q, p).

A convenient method to generalize the notion of a Hamiltonian system is by using the Poissonbracket. For a canonical system, it is given by

f , g :=∂f

∂q

∂g

∂p−∂f

∂p

∂g

∂q. (1)

The Poisson bracket can be used to describe the evolution of a function f = f (q, p) via

df

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂H

∂p−∂f

∂p

∂H

∂q= f ,H.

The bracket (1) has the following two properties (which serve as the definition of a Poisson bracket):

f , g = −g , f Anti-symmetry

f , g , h + g , h, f + h, f , g = 0 Jacobi identity

Due to anti-symmetry of the Poisson bracket, it follows that Hamiltonian systems conserve energy

dH

dt= H,H ≡ 0.







dq

dt=∂H

∂p,

dp

dt= −

∂H

∂q.




f , g :=∂f

∂q

∂g

∂p−∂f

∂p

∂g

∂q. (1)


df

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂H

∂p−∂f

∂p

∂H

∂q= f ,H.





dH

dt= H,H ≡ 0.







dq

dt=∂H

∂p,

dp

dt= −

∂H

∂q.




f , g :=∂f

∂q

∂g

∂p−∂f

∂p

∂g

∂q. (1)


df

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂H

∂p−∂f

∂p

∂H

∂q= f ,H.





dH

dt= H,H ≡ 0.







dq

dt=∂H

∂p,

dp

dt= −

∂H

∂q.




f , g :=∂f

∂q

∂g

∂p−∂f

∂p

∂g

∂q. (1)


df

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂H

∂p−∂f

∂p

∂H

∂q= f ,H.





dH

dt= H,H ≡ 0.







dq

dt=∂H

∂p,

dp

dt= −

∂H

∂q.




f , g :=∂f

∂q

∂g

∂p−∂f

∂p

∂g

∂q. (1)


df

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂H

∂p−∂f

∂p

∂H

∂q= f ,H.





dH

dt= H,H ≡ 0.







dq

dt=∂H

∂p,

dp

dt= −

∂H

∂q.




f , g :=∂f

∂q

∂g

∂p−∂f

∂p

∂g

∂q. (1)


df

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂H

∂p−∂f

∂p

∂H

∂q= f ,H.





dH

dt= H,H ≡ 0.







dq

dt=∂H

∂p,

dp

dt= −

∂H

∂q.




f , g :=∂f

∂q

∂g

∂p−∂f

∂p

∂g

∂q. (1)


df

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂H

∂p−∂f

∂p

∂H

∂q= f ,H.





dH

dt= H,H ≡ 0.





The Liouville Theorem

Definition

The phase flow is the one-parametric mapping

g t : (q(0), p(0)) 7→ (q(t), p(t)),

where q(t) and p(t) are solutions of the Hamiltonian equations.

Theorem

The phase flow preserves volume in phase space: For any region D we have

volume of g t D = volume of D.

The proof of the Liouville Theorem amounts to the statement that the change of volume V is

dV

dt=

∫D

div f dz,

wheredz

dt= f (z)

is the dynamical system under consideration.

As for a canonical Hamiltonian system z = (q, p) and f = (∂H/∂p,−∂H/∂q) it follows that

div f =∂

∂q

∂H

∂p−

∂

∂p

∂H

∂q= 0

and therefore Vt = Vt=0.






Definition


g t : (q(0), p(0)) 7→ (q(t), p(t)),


Theorem




dV

dt=

∫D

div f dz,

wheredz

dt= f (z)



div f =∂

∂q

∂H

∂p−

∂

∂p

∂H

∂q= 0







Definition


g t : (q(0), p(0)) 7→ (q(t), p(t)),


Theorem




dV

dt=

∫D

div f dz,

wheredz

dt= f (z)



div f =∂

∂q

∂H

∂p−

∂

∂p

∂H

∂q= 0







Definition


g t : (q(0), p(0)) 7→ (q(t), p(t)),


Theorem




dV

dt=

∫D

div f dz,

wheredz

dt= f (z)



div f =∂

∂q

∂H

∂p−

∂

∂p

∂H

∂q= 0







−2 −1 0 1 2−2

−1

0

1

2

q

p

Figure: Evolution of a region D in phase space M = R2 under the action of the mathematical

pendulum, q = p, p = − sin q.

The Liouville Theorem has a number of important consequences:

A basis for statistical mechanics (see later!)

Impact on the stability of Hamiltonian systems (they cannot be asymptotically stable)

Poincare recurrence theorem (any Hamiltonian system will return arbitrary close to its initialstate after a sufficiently long time)






−2 −1 0 1 2−2

−1

0

1

2

q

p

Figure: Evolution of a region D in phase space M = R2 under the action of the mathematical

pendulum, q = p, p = − sin q.

The Liouville Theorem has a number of important consequences:

A basis for statistical mechanics (see later!)

Impact on the stability of Hamiltonian systems (they cannot be asymptotically stable)

Poincare recurrence theorem (any Hamiltonian system will return arbitrary close to its initialstate after a sufficiently long time)





Non-canonical Hamiltonian mechanics

The generalization of Hamiltonian systems to phase-spaces with arbitrary dimensions can be doneusing the Poisson bracket. This has the advantage to also allow introducing infinite-dimensionalHamiltonian systems (Hamiltonian PDEs).

Definition

Let there be given a phase-space M and a function H : M → R together with a Poisson bracketon M, ·, · : M ×M → M, i.e. a bracket operation such that f , g = −g , f andf , g , h + g , h, f + h, f , g = 0 hold for all f , g , h on M.

A system is called Hamiltonian if it is possible to write its evolution via

dz

dt= z,H,

where z = (z1, . . . , zn) ∈ M.

In the definition of a Hamiltonian system we do not require that the Poisson bracket is non-degenerate. That is, for some Poisson brackets there can exist functions C : M → R such thatf ,C = 0, for all functions f .

These functions are called Casimirs and arise due to the singular nature of the given Poisson bracket.

Casimirs are in particular conserved quantities, as

dC

dt= C ,H ≡ 0.







Definition



dz

dt= z,H,

where z = (z1, . . . , zn) ∈ M.




dC

dt= C ,H ≡ 0.







Definition



dz

dt= z,H,

where z = (z1, . . . , zn) ∈ M.




dC

dt= C ,H ≡ 0.







Definition



dz

dt= z,H,

where z = (z1, . . . , zn) ∈ M.




dC

dt= C ,H ≡ 0.







Definition



dz

dt= z,H,

where z = (z1, . . . , zn) ∈ M.




dC

dt= C ,H ≡ 0.





Example: The Lorenz–1963 model

Consider the famous Lorenz–1963 model

dx

dt= σy − mσx,

dy

dt= rx − xz − my ,

dz

dt= xy − mbz. (2)

To have the chance casting this model into Hamiltonian form, we must neglect dissipation, m = 0.

The conservative Lorenz model has two conserved quantities, which are given by

H :=1

2y 2 +

1

2z2 − rz, C :=

1

2x2 − σz.

It can be verified that the conservative part of the Lorenz system is indeed Hamiltonian, providedwe use the function H as the Hamiltonian function and

f , g := σ(fx gy − gx fy )− x(fy gz − gy fz )

as Poisson bracket.

It can be verified that this bracket indeed satisfies the properties that f , g = −g , f andf , g , h + g , h, f + h, f , g = 0.

That is, we can represent (2) as

dx

dt= x,H,

dy

dt= y ,H,

dz

dt= z,H.

The Poisson bracket of the Lorenz system is singular and the Casimir is precisely the function C .

Due to the existence of two conserved quantities, the conservative Lorenz system evolves along theintersection of the C - and H-surface.







dx

dt= σy − mσx,

dy


dz

dt= xy − mbz. (2)



H :=1

2y 2 +

1

2z2 − rz, C :=

1

2x2 − σz.



as Poisson bracket.



dx

dt= x,H,

dy

dt= y ,H,

dz

dt= z,H.









dx

dt= σy − mσx,

dy


dz

dt= xy − mbz. (2)



H :=1

2y 2 +

1

2z2 − rz, C :=

1

2x2 − σz.



as Poisson bracket.



dx

dt= x,H,

dy

dt= y ,H,

dz

dt= z,H.









dx

dt= σy − mσx,

dy


dz

dt= xy − mbz. (2)



H :=1

2y 2 +

1

2z2 − rz, C :=

1

2x2 − σz.



as Poisson bracket.



dx

dt= x,H,

dy

dt= y ,H,

dz

dt= z,H.









dx

dt= σy − mσx,

dy


dz

dt= xy − mbz. (2)



H :=1

2y 2 +

1

2z2 − rz, C :=

1

2x2 − σz.



as Poisson bracket.



dx

dt= x,H,

dy

dt= y ,H,

dz

dt= z,H.









dx

dt= σy − mσx,

dy


dz

dt= xy − mbz. (2)



H :=1

2y 2 +

1

2z2 − rz, C :=

1

2x2 − σz.



as Poisson bracket.



dx

dt= x,H,

dy

dt= y ,H,

dz

dt= z,H.









dx

dt= σy − mσx,

dy


dz

dt= xy − mbz. (2)



H :=1

2y 2 +

1

2z2 − rz, C :=

1

2x2 − σz.



as Poisson bracket.



dx

dt= x,H,

dy

dt= y ,H,

dz

dt= z,H.









dx

dt= σy − mσx,

dy


dz

dt= xy − mbz. (2)



H :=1

2y 2 +

1

2z2 − rz, C :=

1

2x2 − σz.



as Poisson bracket.



dx

dt= x,H,

dy

dt= y ,H,

dz

dt= z,H.







Hamiltonian systems and dissipation

It is dissatisfactory that the occurrence of dissipation makes it impossible to cast a system intoHamiltonian form.

However, recently the notion of a metriplectic system was introduced to combine the ideas ofHamiltonian mechanics with dissipative systems.

Idea: Formulate the dissipative term using a symmetric (metric) bracket,

〈f , g〉 = 〈g , f 〉,and define the metriplectic bracket as the combination of the Poisson bracket and the metric bracket

〈〈f , g〉〉 := f , g + 〈f , g〉.


The Lorenz–1963 model has metriplectic form upon using

〈f , g〉 = σ2r−1fx gx − fy gy − bfz gz

as metric bracket and G = H − rσ−1C as “Hamiltonian”, i.e.

dx

dt= x,G + m〈x,G〉,

dy

dt= y ,G + m〈y ,G〉,

dz

dt= z,G + m〈z,G〉,

or (m = 1)

dx

dt= 〈〈x,G〉〉,

dy

dt= 〈〈y ,G〉〉,

dz

dt= 〈〈z,G〉〉,

give the dissipative Lorenz system.

Still, metriplectic systems are not Hamiltonian systems and do not necessarily possess conservedquantities!










〈〈f , g〉〉 := f , g + 〈f , g〉.





dx


dy

dt= y ,G + m〈y ,G〉,

dz


or (m = 1)

dx

dt= 〈〈x,G〉〉,

dy

dt= 〈〈y ,G〉〉,

dz

dt= 〈〈z,G〉〉,












〈〈f , g〉〉 := f , g + 〈f , g〉.





dx


dy

dt= y ,G + m〈y ,G〉,

dz


or (m = 1)

dx

dt= 〈〈x,G〉〉,

dy

dt= 〈〈y ,G〉〉,

dz

dt= 〈〈z,G〉〉,












〈〈f , g〉〉 := f , g + 〈f , g〉.





dx


dy

dt= y ,G + m〈y ,G〉,

dz


or (m = 1)

dx

dt= 〈〈x,G〉〉,

dy

dt= 〈〈y ,G〉〉,

dz

dt= 〈〈z,G〉〉,












〈〈f , g〉〉 := f , g + 〈f , g〉.





dx


dy

dt= y ,G + m〈y ,G〉,

dz


or (m = 1)

dx

dt= 〈〈x,G〉〉,

dy

dt= 〈〈y ,G〉〉,

dz

dt= 〈〈z,G〉〉,








−20 −15 −10 −5 0 5 10 15 20−50

0

50

0

5

10

15

20

25

30

35

40

45

50

Figure: The conservative (left) and the dissipative (right) Lorenz–1963 systems.





Literature

V. I. Arnold. Mathematical methods of classical mechanics. Springer, New York, 1978.

H. Goldstein. Classical Mechanics. Addison–Wesley, Reading, 1980.

P. J. Morrison. A paradigm for joint Hamiltonian and dissipative systems. Physica D 18(1–3), 410–419, 1986.

P. J. Olver. Application of Lie groups to differential equations. Springer, New York, 2000.




From finite- to infinite-dimensional Hamiltonian mechanicsEulerian fluid mechanics using a Hamiltonian representationMathematical issues of non-canonical fluid mechanics

Outline








The transition from finite- to infinite-dimensional Hamiltonian mechanics

To generalize finite-dimensional Hamiltonian systems to partial differential equations, we must:

replace the total time derivative ddt by the partial time derivative ∂

∂t .

replace the partial derivative ∂/∂z by the variational derivative δ/δu.

replace the discrete Hamiltonian function H by the Hamiltonian functional H.

replace the discrete anti-symmetric Poisson bracket by a continuous anti-symmetric Poissonbracket using a skew-adjoint differential operator D, i.e. Dad = −D where∫

f (Dadg) :=

∫g(Df ).

The variational derivative δF/δu of a functional F [u] =∫

f dx1 · · · dxn is formally

δFδu

:=∂f

∂u−

n∑i=1

∂

∂xi

∂f

∂ ∂u∂xi

+n∑

i=1

n∑j=1

∂2

∂xi∂xj

∂f

∂ ∂2u∂xi∂xj

− + · · · .

Definition

A system of evolution equations for u = u(t, x) is of Hamiltonian form if it can be represented as

∂u

∂t= D ·

δHδu

= u,H,

where D is skew-adjoint and the associated Poisson bracket

F,G :=

∫δF

δu·(

D ·δGδu

)dx

satisfies the Jacobi identity.








∂t .




f (Dadg) :=

∫g(Df ).



δFδu

:=∂f

∂u−

n∑i=1

∂

∂xi

∂f

∂ ∂u∂xi

+n∑

i=1

n∑j=1

∂2

∂xi∂xj

∂f

∂ ∂2u∂xi∂xj

− + · · · .

Definition


∂u

∂t= D ·

δHδu

= u,H,


F,G :=

∫δF

δu·(

D ·δGδu

)dx









∂t .




f (Dadg) :=

∫g(Df ).



δFδu

:=∂f

∂u−

n∑i=1

∂

∂xi

∂f

∂ ∂u∂xi

+n∑

i=1

n∑j=1

∂2

∂xi∂xj

∂f

∂ ∂2u∂xi∂xj

− + · · · .

Definition


∂u

∂t= D ·

δHδu

= u,H,


F,G :=

∫δF

δu·(

D ·δGδu

)dx






Examples of Hamiltonian evolution equations

Barotropic vorticity equation

Hamiltonian: H := − 12

∫Aζψ dxdy

Poisson bracket:

A,B :=

∫A

ζJ

(δAδζ,δBδζ

)dxdy

Casimirs: C :=∫

Af (ζ) dxdy

Evolution equation:

∂ζ

∂t= ζ,H = −J(ψ, ζ)

Conservative Saltzman convection equations

Hamiltonian: H := −∫

A

(12ψζ + RσTz

)dxdz

Poisson bracket:

A,B :=

∫A

(ζJ

(δAδζ,δBδζ

)+ (T − z)

[J

(δAδζ,δBδT

)+ J

(δAδT

,δBδζ

)])dxdz

Casimirs: C1 :=∫

Aζg(T − z)dxdz, C2 :=

∫A

h(T − z)dxdz

Evolution equations:

∂ζ

∂t= ζ,H = −J(ψ, ζ) + Rσ

∂T

∂x,

∂T

∂t= T ,H = −J(ψ,T ) +

∂ψ

∂x






Barotropic vorticity equation

Hamiltonian: H := − 12

∫Aζψ dxdy

Poisson bracket:

A,B :=

∫A

ζJ

(δAδζ,δBδζ

)dxdy

Casimirs: C :=∫

Af (ζ) dxdy

Evolution equation:

∂ζ

∂t= ζ,H = −J(ψ, ζ)

Conservative Saltzman convection equations

Hamiltonian: H := −∫

A

(12ψζ + RσTz

)dxdz

Poisson bracket:

A,B :=

∫A

(ζJ

(δAδζ,δBδζ

)+ (T − z)

[J

(δAδζ,δBδT

)+ J

(δAδT

,δBδζ

)])dxdz

Casimirs: C1 :=∫

Aζg(T − z)dxdz, C2 :=

∫A

h(T − z)dxdz


∂ζ

∂t= ζ,H = −J(ψ, ζ) + Rσ

∂T

∂x,

∂T

∂t= T ,H = −J(ψ,T ) +

∂ψ

∂x






Ideal fluid equations

Hamiltonian: H :=∫

Vρ(

12 v2 + e(ρ, s) + Φ

)dxdydz

Poisson bracket:

A,B :=

∫V

[∇(δAδρ

)·δBδv−∇

(δBδρ

)·δAδv

+∇× v

ρ·(δAδv×δBδv

)+

∇s

ρ·(δAδv

δBδs−δBδv

δAδs

)]dxdydz

Casimirs: C :=∫

Vρf (s,Π)dxdydz, Π = ρ−1(∇× v) · ∇s


∂v

∂t= v,H = −

1

2∇v2 + v × (∇× v)−∇Φ−

1

ρ∇p

∂ρ

∂t= ρ,H = −v · ∇ρ

∂s

∂t= s,H = −v · ∇s





How are these Hamiltonian formulations derived?

There are two prevailing ways to derive a Hamiltonian formulation in non-canonical coordinates:

Guessing!

Explicitly finding the non-canonical coordinate transformation.

Having some experience, using the first method it is usually quite easy to find a Poisson bracketformulation. Then, however, the proof of the validity of the Jacobi identity can be a painful exercise(and is often not done!). Alternatively, advanced mathematical concepts must be used in order toproof the Jacobi identity, which can be painful as well.

Transforming the canonical Poisson bracket using a non-canonical transformation automaticallyensures the validity of the Jacobi identity. In order to efficiently find such non-canonical transfor-mations, one usually utilizes symmetry properties of the underlying system of differential equations.







Guessing!










Guessing!








Lagrangian and Eulerian fluid mechanics from the viewpoint of HM

A transformation from a canonical to a non-canonical system is usually possible due to a symme-try that the Hamiltonian admits. This symmetry helps finding both a closed, transformed (non-canonical) Poisson bracket and an alternative representation of the Hamiltonian function(al).

In the case of ideal fluid mechanics, this symmetry is the parcel relabeling symmetry. This symmetryis responsible for the existence of a closed Eulerian description of fluid mechanics.

In the course of passing from canonical to non-canonical coordinates (Euler–Poincare reduction),the conserved quantities associated (by Noether’s theorem) with the symmetry used to carry outthe change of variables will appear as Casimirs in the non-canonical formulation of that system.

The conserved quantities corresponding to the parcel relabeling symmetry are the various forms ofpotential vorticity integrals in two-dimensional fluid mechanics. This is why they appear as Casimirsin Eulerian fluid mechanics.

Note that this transformation from Lagrangian to Eulerian coordinates is essentially analog to thetransformation enabling the representation of the free rigid body equations in terms of the body-associated reference frame (on the rotational group SO(3)).













































Literature

J. E. Marsden and T. S. Ratiu. Introduction to mechanics and symmetry. Springer, NewYork, 1999.

P. J. Morrison. Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2),467–521, 1998.

P. J. Olver. Application of Lie groups to differential equations. Springer, New York, 2000.

R. Salmon. Lectures on geophysical fluid dynamics. Oxford University Press, New York,1998.




Hamiltonian finite-mode modelsStatistical fluid mechanics

Outline








Structure-preserving finite-mode models

As almost all relevant models of geophysical fluid dynamics possess a Hamiltonian formulation, therearises the question of whether it is possible to preserve this formulation in the course of derivingreduced models (e.g. truncated spectral models).

Unfortunately, the general answer to this problem is still unknown!

For several classes of fluid dynamics equations, however, it is possible to derive finite-mode modelsthat retain some (or all) of the properties of the original system of differential equations.

V. Zeitlin found a truncation for the incompressible Euler equations which is an example for a perfectstructure-preserving truncation.
































The Zeitlin truncation

The problem

The vorticity equation has an infinite number of conserved quantities, the vorticity integrals. Whentruncating the Fourier expansion of the vorticity on the square (±N,±N) one only preserves theenstrophy and looses the Jacobi property. Is there a truncation that resolves this problem?

The evolution equations for the Fourier coefficients cm of the vorticity in the barotropic vorticityequation read

dcm

dt=∑

k

m× k

k2cm+kc−k, (3)

where m = (m1,m2), k = (k1, k2) and m×k := m1k2−m2k1. This system is a form of generalizedEuler equations

dzi

dt= almC k

imzl zk (4)

for some coordinates zi with alm being a symmetric tensor and C kim being the structure constants of

a Lie algebra (anti-symmetric in the lower indices).

Specifically, System (3) is found from System (4) by setting

alm = m−2δ(m + l), C k

im = i× mδ(k− i− m),

where δ denotes the delta-function.

That is, when approximating System (3) with a finite number of modes, we have the freedom to

separately approximate alm and C kim.






The problem



dcm

dt=∑

k

m× k

k2cm+kc−k, (3)


dzi

dt= almC k

imzl zk (4)














The problem



dcm

dt=∑

k

m× k

k2cm+kc−k, (3)


dzi

dt= almC k

imzl zk (4)














The problem



dcm

dt=∑

k

m× k

k2cm+kc−k, (3)


dzi

dt= almC k

imzl zk (4)














The naive truncation on the unit square (±N,±N) implies the approximation

alm =1

|l|2δ

0l+m, C k

im = (i× m)δki+m.

Zeitlin approximated the constants alm and C kim by

alm =1

|l|2δ

0l+m, C k

im =N

2πsin

(2π

Ni× m

)δ

ki+m|mod N .

As limx→∞

sin x/x = 1 this approximation indeed converges to the Fourier expansion of the vorticity

equation as N →∞.

The crucial advantage of Zeitlin’s approximation is that with N2 − 1 coefficients, there are N − 1Casimir invariants admitted by the resulting Euler equations. These Casimirs have the form

CM =∑

ci1· · · ciM

cos

(4π

NA(i1, . . . , iM )

),

where A(i1, . . . , iM ) = 0.5 ∗ (i2 × i1 + i3 × (i1 + i2) + · · · + iM × (i1 + · · · iM−1)), 2 ≤ M ≤ N,which in the limit N →∞ tend to the vorticity moments.

Other approximations for the metric alm can be applied and play a role when consistent truncationsof the two-dimensional incompressible Navier–Stokes equations are desired.







alm =1

|l|2δ

0l+m, C k

im = (i× m)δki+m.


alm =1

|l|2δ

0l+m, C k

im =N

2πsin

(2π

Ni× m

)δ

ki+m|mod N .

As limx→∞




CM =∑

ci1· · · ciM

cos

(4π

NA(i1, . . . , iM )

),









alm =1

|l|2δ

0l+m, C k

im = (i× m)δki+m.


alm =1

|l|2δ

0l+m, C k

im =N

2πsin

(2π

Ni× m

)δ

ki+m|mod N .

As limx→∞




CM =∑

ci1· · · ciM

cos

(4π

NA(i1, . . . , iM )

),









alm =1

|l|2δ

0l+m, C k

im = (i× m)δki+m.


alm =1

|l|2δ

0l+m, C k

im =N

2πsin

(2π

Ni× m

)δ

ki+m|mod N .

As limx→∞




CM =∑

ci1· · · ciM

cos

(4π

NA(i1, . . . , iM )

),







Limits of the Zeitlin truncation

The Zeitlin truncation can be extended to other two-dimensional models of hydrodynamics. It can,however, not be generalized to three-dimensional cases. Also, it is only applicable for non-divergentmodels.

Basically, this is due to the particular choice of the coefficients C kim, which are a special basis for the

Lie algebra of SU(N), the special unitary group.

In the continuous limit N → ∞, these coefficients C kim tend to those of the Lie algebra of area-

preserving vector fields, the dual of which is the phase space of the incompressible Euler equations.

Unfortunately, for other flow configurations (e.g. three-dimensional non-divergent flow) there is nosuch finite-dimensional Lie algebra that in the limit tends to the infinite-dimensional Lie algebra withdual corresponding to the phase space of some non-canonical Hamiltonian fluid model. Therefore,no such special construction as in the case of the Zeitlin truncation is possible.






































Another (quite) structure-preserving truncation

The conservative Saltzman convection equations have Hamiltonian form with Poisson bracket

A,B :=

∫A

(ζJ

(δAδζ,δBδζ

)+ (T − z)

[J

(δAδζ,δBδT

)+ J

(δAδT

,δBδζ

)])dxdz

We now look for a truncation of the expansion

ψ = (φnm sin anπx + ϕnm cos anπx) sin mπz, T = (ϑnm sin anπx + θnm cos anπx) sin mπz,

that preserves (some of) the Hamiltonian structure of the conservative Saltzman equations.

There is a finite-dimensional Poisson bracket for a six-component model that has the same algebraicand geometric structure as the above bracket. It is given by

F1, F2 := −π · ∇πF1 ×∇πF2 − Γ · (∇πF1 ×∇ΓF2 +∇ΓF1 ×∇πF2),

where π = (A,B,C)T and Γ = (D, E , F )T.

It can be shown that the six-component model

A =a

2bπ(1 + a2)((a2 − 3)π3BC + 2eRσE), D =

aπ

2be(eπCE − 2b2f πBF − 2b2B),

B = −a

2bπ(1 + a2)((a2 − 3)π3AC + 2eRσD), E = −

aπ

2be(eπCD − 2b2f πAF − 2b2A),

C = 0, F =abeπ2

2f(BD − AE),

is a truncation of the Saltzman equations admitting the above Poisson bracket, where φ11 = bA,ϕ11 = bB, ϕ02 = cC , ϑ11 = eD, θ11 = eE , θ02 = fF .







A,B :=

∫A

(ζJ

(δAδζ,δBδζ

)+ (T − z)

[J

(δAδζ,δBδT

)+ J

(δAδT

,δBδζ

)])dxdz








A =a

2bπ(1 + a2)((a2 − 3)π3BC + 2eRσE), D =

aπ


B = −a

2bπ(1 + a2)((a2 − 3)π3AC + 2eRσD), E = −

aπ


C = 0, F =abeπ2

2f(BD − AE),








A,B :=

∫A

(ζJ

(δAδζ,δBδζ

)+ (T − z)

[J

(δAδζ,δBδT

)+ J

(δAδT

,δBδζ

)])dxdz








A =a

2bπ(1 + a2)((a2 − 3)π3BC + 2eRσE), D =

aπ


B = −a

2bπ(1 + a2)((a2 − 3)π3AC + 2eRσD), E = −

aπ


C = 0, F =abeπ2

2f(BD − AE),








A,B :=

∫A

(ζJ

(δAδζ,δBδζ

)+ (T − z)

[J

(δAδζ,δBδT

)+ J

(δAδT

,δBδζ

)])dxdz








A =a

2bπ(1 + a2)((a2 − 3)π3BC + 2eRσE), D =

aπ


B = −a

2bπ(1 + a2)((a2 − 3)π3AC + 2eRσD), E = −

aπ


C = 0, F =abeπ2

2f(BD − AE),







The above model preserves energy

H =1

4ab2π((1 + a2)b4

π3(A2 + B2) + 2π3C 2 + 4Rb2f σF )

and Casimirs

C = −π

2ab((1 + a2)b2eπ(AD + BE) + 4f πCF + 4C) from C1 :=

∫Aζg(T − z)dxdy ,

S =Rσ

12aπ

(3e2π(D2 + E 2) + 6f 2πF 2 + 12fF

)from C2 :=

∫A

h(T − z)dxdy .

The above system is a special form of the heavy top equations, a so-called Lagrange top. As thereis the additional conserved quantity C = 0, the six-component model of the conservative Saltzmanconvection equations is completely integrable.

The obvious drawback of the extended Lorenz model is that is was derived so as its Poisson bracketkeeps the analogy with the continuous bracket of the Saltzman convection equations. There is noobvious way for generalizing this model to a Hamiltonian truncation using more than six components(again, use Zeitlin!).







H =1

4ab2π((1 + a2)b4

π3(A2 + B2) + 2π3C 2 + 4Rb2f σF )

and Casimirs

C = −π



S =Rσ

12aπ

(3e2π(D2 + E 2) + 6f 2πF 2 + 12fF

)from C2 :=

∫A

h(T − z)dxdy .









H =1

4ab2π((1 + a2)b4

π3(A2 + B2) + 2π3C 2 + 4Rb2f σF )

and Casimirs

C = −π



S =Rσ

12aπ

(3e2π(D2 + E 2) + 6f 2πF 2 + 12fF

)from C2 :=

∫A

h(T − z)dxdy .







Application to statistical mechanics

In order to apply ideas of equilibrium statistical mechanics to dynamical systems, two conditionshave to be satisfied:

The Liouville Theorem must be valid, i.e. div f = 0, must hold for the system z = f (z).

There should exist a number of conserved quantities.

Owing to the above two requirements, Hamiltonian systems are of potential interest for statisticalmechanics.

In statistical mechanics we are interested in the evolution of an initial probability measure p0(z).Using the flow map g t of the dynamical system z = f (z), we can define a function

p(z, t) = p0((g t )−1(z)),

which thanks to the validity of the Liouville equation is a probability measure for all times.

Concerning the second ingredient to statistical mechanics, consider the conserved quantities Ei (z(t)) =Ei (z0), i = 1, . . . , L. By introducing the ensemble average of Ei w.r.t. the probability measure p,we denote

Ei = 〈Ei 〉p :=

∫Rn

Ei p dz.

Again using the Liouville Theorem, it is possible to prove that the ensemble average of conservedquantities is conserved too, i.e.

〈Ei 〉p = 〈Ei 〉p0.











p(z, t) = p0((g t )−1(z)),



Ei = 〈Ei 〉p :=

∫Rn

Ei p dz.


〈Ei 〉p = 〈Ei 〉p0.











p(z, t) = p0((g t )−1(z)),



Ei = 〈Ei 〉p :=

∫Rn

Ei p dz.


〈Ei 〉p = 〈Ei 〉p0.











p(z, t) = p0((g t )−1(z)),



Ei = 〈Ei 〉p :=

∫Rn

Ei p dz.


〈Ei 〉p = 〈Ei 〉p0.











p(z, t) = p0((g t )−1(z)),



Ei = 〈Ei 〉p :=

∫Rn

Ei p dz.


〈Ei 〉p = 〈Ei 〉p0.





From Shannon entropy to the Gibbs measure

Let there be given the probability measure p on Rn. Define the following functional of p:

S[p] = −∫Rn

p(z) ln p(z) dz,

which is called the Shannon entropy.

We are now interested in finding the probability measure that has the least bias for doing furthermeasurements. This probability measure is found by invoking the maximum entropy principle.

Maximum entropy principle

Let there be given the set of constraints

C = p(z) ≥ 0,

∫Rn

p(z) dz = 1, 〈Ei 〉p = Ei , i = 1, . . . , L.

The least biased probability measure p∗ consistent with C is found by maximizing the Shannonentropy S subject to the constraints imposed by C, i.e.

S(p∗) = maxp∈CS(p).







S[p] = −∫Rn

p(z) ln p(z) dz,





C = p(z) ≥ 0,

∫Rn

p(z) dz = 1, 〈Ei 〉p = Ei , i = 1, . . . , L.









S[p] = −∫Rn

p(z) ln p(z) dz,





C = p(z) ≥ 0,

∫Rn

p(z) dz = 1, 〈Ei 〉p = Ei , i = 1, . . . , L.







The invariant Gibbs measure

In order to find the probability measure p∗, we have to employ the Lagrange multiplier method, i.e.we set up

L = S − λ0

(∫Rn

p dz − 1

)−

L∑i=1

λi (〈Ei 〉p − Ei ),

for Lagrangian multipliers λj , j = 0, . . . , L.

The probability measure p∗ is found by computing δLδp = 0. This yields

δLδp

= −(1 + ln p∗)− λ0 −L∑

i=1

λi Ei = 0.

This can be cast into the form

p∗ = C−1 exp

(−

L∑i=1

λi Ei

), C =

∫Rn

exp

(−

L∑i=1

λi Ei

)dz,

which is analog to the Gibbs measure in statistical mechanics.

It can be verified that the probability measure p∗ is invariant under the flow map g t of the dynamicalsystem z = f (z), i.e.

p∗((g t )−1(Ω)) = p∗(Ω),

where Ω ⊆ Rn. Therefore, p∗ can be used to predict the statistical behavior of an ensemble ofsolutions to the dynamical system z = f (z) at all times t.







L = S − λ0

(∫Rn

p dz − 1

)−

L∑i=1

λi (〈Ei 〉p − Ei ),



δLδp

= −(1 + ln p∗)− λ0 −L∑

i=1

λi Ei = 0.


p∗ = C−1 exp

(−

L∑i=1

λi Ei

), C =

∫Rn

exp

(−

L∑i=1

λi Ei

)dz,



p∗((g t )−1(Ω)) = p∗(Ω),








L = S − λ0

(∫Rn

p dz − 1

)−

L∑i=1

λi (〈Ei 〉p − Ei ),



δLδp

= −(1 + ln p∗)− λ0 −L∑

i=1

λi Ei = 0.


p∗ = C−1 exp

(−

L∑i=1

λi Ei

), C =

∫Rn

exp

(−

L∑i=1

λi Ei

)dz,



p∗((g t )−1(Ω)) = p∗(Ω),








L = S − λ0

(∫Rn

p dz − 1

)−

L∑i=1

λi (〈Ei 〉p − Ei ),



δLδp

= −(1 + ln p∗)− λ0 −L∑

i=1

λi Ei = 0.


p∗ = C−1 exp

(−

L∑i=1

λi Ei

), C =

∫Rn

exp

(−

L∑i=1

λi Ei

)dz,



p∗((g t )−1(Ω)) = p∗(Ω),






Literature

A. Bihlo and J. Staufer. Minimal atmospheric finite-mode models preserving symmetry andgeneralized Hamiltonian structures. Physica D 240 (7), 599–606, 2011.

A. Majda and X. Wang. Non-linear dynamics and statistical theories for basic geophysicalflows, Cambridge University Press, Cambridge, 2006.

V. Zeitlin. Finite-mode analogs of 2d ideal hydrodynamics: Coadjoint orbits and localcanonical structure. Physica D 49 (3), 353–362, 1991.

V. Zeitlin. On self-consistent finite-mode approximations in (quasi-)two-dimensionalhydrodynamics and magnetohydrodynamics. Phys. Lett. A 339 (3–5), 316–324, 2005.


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