1 Hamiltonian formulation: water waves Lecture 3 Wm.. Hamilton V.E. Zakharov2 Hamiltonian formulation: water waves This lecture: A. apid review of Ham...

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Wm. R. Hamilton

V.E. Zakharov

Hamiltonian formulation: water waves This lecture: A. Rapid review of Hamiltonian machinery (see also extra notes) B. Hamiltonian formulation of water waves - Zakharov, 1967, 1968 [Lagrangian formulation - Luke, 1967] C. Some consequences of Hamiltonian structure

A. Review of Hamiltonian systems 1. Example: nonlinear oscillator

˙"˙ + # 2" + $" 3 = 0, !

!

d# " 2 > 0, #˙ = . dt

A. Review of Hamiltonian systems 1. Example: nonlinear oscillator

˙"˙ + # 2" + $" 3 = 0, a)

Find an energy integral

"˙ # (eq' n) $

! !

d# " 2 > 0, #˙ = . dt

!

1 ˙ 2 # 2! 2 $ 4 E = (" ) + (" ) + (" ) = const. 2 2 4

A. Review of Hamiltonian systems 1. Example: nonlinear oscillator

˙"˙ + # 2" + $" 3 = 0, a)

Find an energy integral

"˙ # (eq' n) $

! b) !

d# " 2 > 0, #˙ = . dt

1 ˙ 2 # 2! 2 $ 4 E = (" ) + (" ) + (" ) = const. 2 2 4

Write eq’n as a first-order system Define q = " (t), p = "˙ (t) !

equivalent:

!

q˙ = p, p˙ = "# 2q " $q 3

A. Review of Hamiltonian systems 1. Example: nonlinear oscillator

˙"˙ + # 2" + $" 3 = 0, a)

Find an energy integral

"˙ # (eq' n) $

! b) !

d# " 2 > 0, #˙ = . dt

1 ˙ 2 # 2! 2 $ 4 E = (" ) + (" ) + (" ) = const. 2 2 4

Write eq’n as a first-order system Define q = " (t), p = "˙ (t)

p

!

equivalent:

!

q˙ = p, p˙ = "# 2q " $q 3

q

! !

Review of Hamiltonian systems 2. Definition: A system of 2N first-order ODEs is Hamiltonian if there exist N pairs of coordinates on the phase space,

{pj(t), qj(t)},

j = 1,2,…,N,

and a real-valued Hamiltonian function,

H(p(t), q(t), t), such that the original equations are equivalent to dq j "H q˙ j = = , dt "p j

!

"H p˙ j = # , "q j

j = 1,2,...,N.

Review of Hamiltonian systems 2. Definition: A system of 2N first-order ODEs is Hamiltonian if there exist N pairs of coordinates on the phase space,

{pj(t), qj(t)},

j = 1,2,…,N,

and a real-valued Hamiltonian function,

H(p(t), q(t), t), such that the original equations are equivalent to dq j "H q˙ j = = , dt "p j

"H p˙ j = # , "q j

In example, N = 1, choose ! !

j = 1,2,...,N.

1 2 "2 2 # 4 H= E = p + q + q . 2 2 4

Review of Hamiltonian systems 2. Definition: A system of 2N first-order ODEs is Hamiltonian if there exist N pairs of coordinates on the phase space,

{pj(t), qj(t)},

j = 1,2,…,N,

and a real-valued Hamiltonian function,

H(p(t), q(t), t), such that the original equations are equivalent to dq j "H q˙ j = = , dt "p j

"H p˙ j = # , "q j

In example, N = 1, choose !

"H q˙ = p = "p

✔ !

j = 1,2,...,N.

1 2 "2 2 # 4 H= E = p + q + q . 2 2 4

Review of Hamiltonian systems 2. Definition: A system of 2N first-order ODEs is Hamiltonian if there exist N pairs of coordinates on the phase space,

{pj(t), qj(t)},

j = 1,2,…,N,

and a real-valued Hamiltonian function,

H(p(t), q(t), t), such that the original equations are equivalent to dq j "H q˙ j = = , dt "p j

"H p˙ j = # , "q j

In example, N = 1, choose !

"H q˙ = p = "p

✔

j = 1,2,...,N.

1 2 "2 2 # 4 H= E = p + q + q . 2 2 4 p˙ = "# 2q " $q 3 = "

!

%H %q

✔

Review of Hamiltonian systems 3. Comments a) Not every system of 2N first-order ODEs is Hamiltonian. b) An essential property of a Hamiltonian system: the flow preserves volume in phase space. (The volume of a “ball” of initial data is preserved.) c) H is often the physical energy, but not necessarily. d) H is often a constant of the motion, but not necessarily.

Review of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. •

Start with M first-order ODEs dx j ! = v j ( x,t), dt

!

j = 1,2,..., M.

Review of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. •

Start with M first-order ODEs dx j ! = v j ( x,t), dt

j = 1,2,..., M.

• Imagine a “fluid” that fills the M-dimensional phase space. ! {x1(t), x2(t), …, xM(t)} are the coordinates of a fluid particle, {v1(t), v2(t), …, vM(t)} are the components of fluid velocity.

Review of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. (See p. 69 of Arnold’s “Classical Mechanics” for a real proof)

•

Start with M first-order ODEs dx j ! = v j ( x,t), dt

• •

j = 1,2,..., M.

Imagine a “fluid” that fills the M-dimensional phase space. {x1(t), !x2(t), …, xM(t)} are the coordinates of a fluid particle, {v1(t), v2(t), …, vM(t)} are the components of fluid velocity. The fluid is “incompressible”, so volume is preserved if ! M $v j "#v =% =0 j=1 $x j

Review of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. • Claim: Any Hamiltonian system of ODEs with a smooth Hamiltonian is volume-preserving, because " # v! = 0

!

Review of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. • Claim: Any Hamiltonian system of ODEs with a smooth Hamiltonian is volume-preserving, because " # v! = 0 •

Proof: dq j

"H dp j "H = , =# , dt "p j dt "q j

! !

j = 1,2,...,N.

Review of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. • Claim: Any Hamiltonian system of ODEs with a smooth Hamiltonian is volume-preserving, because " # v! = 0 •

Proof: dq j

"H dp j "H = , =# , dt "p j dt "q j

! !

j = 1,2,...,N.

N ! N $ dq j $ dp j "#v =% ( )+% ( ) dt dt j=1 $q j j=1 $p j N

N $ $H $ $H =% ( ) +% (& ) $q j j=1 $q j $p j j=1 $p j = 0.

Review of Hamiltonian systems 5. Hamiltonian PDEs Example: nonlinear wave equation, periodic b.c. 2

2

2

2

"t # = c " x # $ % # $ &#

!

3

Review of Hamiltonian systems 5. Hamiltonian PDEs Example: nonlinear wave equation, periodic b.c. 2

2

2

2

"t # = c " x # $ % # $ &# a)

3

Find energy integral 2 2 1 c % & E = " [ (# t$ ) 2 + (#x$ ) 2 + $ 2 + $ 4 ]dx, 2 2 2 4

! !

!

dE = 0. dt

Review of Hamiltonian systems 5. Hamiltonian PDEs Example: nonlinear wave equation, periodic b.c. 2

2

2

2

"t # = c " x # $ % # $ &# a)

3

Find energy integral 2 2 1 c % & E = " [ (# t$ ) 2 + (#x$ ) 2 + $ 2 + $ 4 ]dx, 2 2 2 4

! !

b)

Choose “conjugate variables”

p(x,t) = "t# (x,t), q(x,t) = # (x,t). !

“index” (j) becomes continuous (x)

!

dE = 0. dt

Review of Hamiltonian systems 5. Hamiltonian PDEs Example: nonlinear wave equation, periodic b.c. 2

2

2

2

"t # = c " x # $ % # $ &# a)

3

Find energy integral 2 2 1 c % & E = " [ (# t$ ) 2 + (#x$ ) 2 + $ 2 + $ 4 ]dx, 2 2 2 4

!

b)

dE = 0. dt

Choose “conjugate variables”

p(x,t) = "t# (x,t), q(x,t) = # (x,t).

!

!

“index” (j) becomes continuous (x)

c)

!

Guess:

1 2 c2 $2 2 % 4 2 H( p,q,t) = " [ p + (# x q) + q + q ]dx 2 2 2 4

Review of Hamiltonian systems Q: What happens to

!

(

"H ) "p j

in the PDE setting?

Review of Hamiltonian systems Q: What happens to

(

"H ) "p j

in the PDE setting?

Define variational derivative: Start with H( p,q,t) ! = " [...]dx !

Review of Hamiltonian systems "H ) "p j

Q: What happens to in the PDE setting? Define variational derivative: = " [...]dx Start with H( p,q,t) ! (

H( p + "p,q,t) # H( p,q,t) =

2 [(**) " p + O(( " p) ]dx $

! defines variational derivative: this

!

!

"H "p

Review of Hamiltonian systems "H ) "p j

Q: What happens to in the PDE setting? Define variational derivative: = " [...]dx Start with H( p,q,t) ! (

H( p + "p,q,t) # H( p,q,t) =

2 [(**) " p + O(( " p) ]dx $

! defines variational derivative: this

!

H( p,q + "q,t) # H( p,q,t) =

$ [(

! !

"H "p

"H )"q + O(("q) 2 ]dx "q

Review of Hamiltonian systems "H ) "p j

Q: What happens to in the PDE setting? Define variational derivative: = " [...]dx Start with H( p,q,t) ! (

H( p + "p,q,t) # H( p,q,t) =

2 [(**) " p + O(( " p) ]dx $

! defines variational derivative: this

!

H( p,q + "q,t) # H( p,q,t) =

Example:

"H )"q + O(("q) 2 ]dx "q

1 2 c2 $2 2 % 4 2 H( p,q,t) = " [ p + ! (# x q) + q + q ]dx 2 2 2 4

! Q: What is "H ? Why? "p !

$ [(

"H "p

Review of Hamiltonian systems Q: What happens to Continue example: !

!

(

"H ) "q j

in the PDE setting?

1 2 c2 $2 2 % 4 2 H( p,q,t) = " [ p + (# x q) + q + q ]dx 2 2 2 4

Review of Hamiltonian systems Q: What happens to Continue example:

(

"H ) "q j

in the PDE setting?

1 2 c2 $2 2 % 4 2 H( p,q,t) = " [ p + (# x q) + q + q ]dx 2 2 2 4

!

H( p,q + "q,t) # H( p,q,t) = 2 2 3 2 [c ( $ q)( $ " q) + % q " q + & q " q + O(( " q) )]dx ' !x x new twist: integrate by parts, with δq = 0 on boundaries

!

!

Review of Hamiltonian systems Q: What happens to Continue example:

(

"H ) "q j

in the PDE setting?

1 2 c2 $2 2 % 4 2 H( p,q,t) = " [ p + (# x q) + q + q ]dx 2 2 2 4

!

H( p,q + "q,t) # H( p,q,t) = 2 2 3 2 [c ( $ q)( $ " q) + % q " q + & q " q + O(( " q) )]dx ' !x x new twist: integrate by parts, with δq = 0 on boundaries

!

H( p,q + "q,t) # H( p,q,t) =

2 2 2 3 2 [(#c $ q + % q + & q ) " q + O(( " q) )]dx ' x

"H "q

!

Review of Hamiltonian systems Q: What happens to Continue example:

(

"H ) "q j

in the PDE setting?

1 2 c2 $2 2 % 4 2 H( p,q,t) = " [ p + (# x q) + q + q ]dx 2 2 2 4

!

H( p,q + "q,t) # H( p,q,t) = 2 2 3 2 [c ( $ q)( $ " q) + % q " q + & q " q + O(( " q) )]dx ' !x x new twist: integrate by parts, with δq = 0 on boundaries

!

H( p,q + "q,t) # H( p,q,t) =

2 2 2 3 2 [(#c $ q + % q + & q ) " q + O(( " q) )]dx ' x

"H "q End of lightning tour of Hamiltonian systems

B. Inviscid water waves Recall:

" t# + $% & $# = " z%, "t % +

on z = η(x,y,t)

1 ' $# | $% |2 +g# = $ & { }, 2 2 ( 1+ | $# |

!

! !

on z = η(x,y,t)

" 2# = 0

-h(x,y) < z < η(x,y,t)

"n # = 0

on z = -h(x,y)

B. Inviscid water waves Recall:

" t# + $% & $# = " z%, "t % +

on z = η(x,y,t)

1 ' $# | $% |2 +g# = $ & { }, 2 2 ( 1+ | $# |

!

" 2# = 0

-h(x,y) < z < η(x,y,t)

"n # = 0

on z = -h(x,y)

Q: Where does t-evolution occur? ! A. (Zakharov): on z = η(x,y,t) !

on z = η(x,y,t)

B. Inviscid water waves Recall:

" t# + $% & $# = " z%, "t % +

on z = η(x,y,t)

1 ' $# | $% |2 +g# = $ & { }, 2 2 ( 1+ | $# |

!

on z = η(x,y,t)

" 2# = 0

-h(x,y) < z < η(x,y,t)

"n # = 0

on z = -h(x,y)

Q: Where does t-evolution occur? ! A. (Zakharov): on z = η(x,y,t) !

Propose conjugate variables:

"(x, y,t), # (x, y,t) = $ (x, y,z,t) |z= "

Water waves as Hamiltonian system "(x, y,t), # (x, y,t) = $ (x, y,z,t) |z= " z ="

Plausibility check: periodic b.c.

" 2# = 0 !

!

Suppose at some fixed t, {η(x,y,t), ψ = φ(x,y,t)|z=η } are given. !

z = "h

!

Water waves as Hamiltonian system "(x, y,t), # (x, y,t) = $ (x, y,z,t) |z= " z ="

Plausibility check: periodic b.c.

" 2# = 0 !

!

Suppose at some fixed t, {η(x,y,t), ψ = φ(x,y,t)|z=η } are given. ! Then φ(x,y,z,t) is determined uniquely in domain. [We need a procedure to find φ(x,y,z,t) from {η, ψ}].

z = "h

!

Result: At any fixed time, {η, ψ} determine the entire solution.

Water waves as Hamiltonian system Proposed conjugate variables:

"(x, y,t), # (x, y,t) = $ (x, y,z,t) |z= " Q: What is H(η,ψ) ? A: Physical energy (from

!

1 & 1 2 ' 2 H = "" [ "%h | #$ | dz + g& + ( 1+ | #& |2 %1)]dxdy 2 2 ( R

kinetic energy !

HW #1):

potential energy

Water waves as Hamiltonian system Claim (Zakharov, 1968): Let R be a fixed region in x-y plane. Let h(x,y) be continuous and differentiable on R. Define 1 " 1 ( H(",# ) = $$ [ $'h | %& |2 dz + g" 2 + ( 1+ | %" |2 '1)]dxdy 2 2 ) R

We need to show that !

$H $H " t# = , " t% = & $% $# are equivalent to the two boundary conditions on z = η(x,y,t). !

Water waves as Hamiltonian system Step 1: Rewrite 2 eq’ns on z = η in terms of {η,ψ} and normal velocity on z = η. • Define F(x,y,z,t) = z - η(x,y,t) , so F = 0 on z = η.

Water waves as Hamiltonian system Step 1: Rewrite 2 eq’ns on z = η in terms of {η,ψ} and normal velocity on z = η. • Define F(x,y,z,t) = z - η(x,y,t) , so F = 0 on z = η. • unit normal vector on z = η: n! =

!

{#$x%,#$ y%,1} "F = . 2 | "F | 1+ | "% |

Water waves as Hamiltonian system Step 1: Rewrite 2 eq’ns on z = η in terms of {η,ψ} and normal velocity on z = η. • Define F(x,y,z,t) = z - η(x,y,t) , so F = 0 on z = η. • unit normal vector on z = η: n! =

{#$x%,#$ y%,1} "F = . 2 | "F | 1+ | "% |

• Normal component of velocity on z = η: "n # = $# % nˆ =

!

!

&$# % $' + " z# 1+ | $' |2

Water waves as Hamiltonian system Step 1: Rewrite 2 eq’ns on z = η in terms of {η,ψ} and normal velocity on z = η. • Define F(x,y,z,t) = z - η(x,y,t) , so F = 0 on z = η. • unit normal vector on z = η: n! =

{#$x%,#$ y%,1} "F = . 2 | "F | 1+ | "% |

• Normal component of velocity on z = η: !

"n # = $# % nˆ =

&$# % $' + " z#

• Eq’n #1 on z = η: "t# + $%!& $# = "z%

"

1+ | $' |2

"t# = 1+ | $# |2 "n % .

Water waves as Hamiltonian system Step 2: Rewrite 2nd eq’n on z = η: •

!

1 ' $% 2 "t # + | $# | +g% & $ ) { }=0 2 2 ( 1+ | $% |

Water waves as Hamiltonian system Step 2: Rewrite 2nd eq’n on z = η: •

1 ' $% 2 "t # + | $# | +g% & $ ) { }=0 2 2 ( 1+ | $% |

• But !

"t# = " t $ |z= % +" z$ |z= % "t%

! !

" (x, y,t) = # (x, y,z,t) |z= $ (x,y,t ) (chain rule)

Water waves as Hamiltonian system Step 2: Rewrite 2nd eq’n on z = η: •

1 ' $% 2 "t # + | $# | +g% & $ ) { }=0 2 2 ( 1+ | $% |

• But !

" (x, y,t) = # (x, y,z,t) |z= $ (x,y,t ) "t# = " t $ |z= % +" z$ |z= % "t%

! ! !

(chain rule)

= "t # |z= $ +"z# |z= $ {"z# % &# ' &$} |z= $

Water waves as Hamiltonian system Step 2: Rewrite 2nd eq’n on z = η: •

1 ' $% 2 "t # + | $# | +g% & $ ) { }=0 2 2 ( 1+ | $% |

• But !

" (x, y,t) = # (x, y,z,t) |z= $ (x,y,t ) "t# = " t $ |z= % +" z$ |z= % "t%

!

(chain rule)

= "t # |z= $ +"z# |z= $ {"z# % &# ' &$} |z= $

!• Eq’n #2 on z = η: ! 1 2 2

) &( "t# + [(" x $ ) + ("y $ ) % (" z$ ) 2 ] + (" z$ )&$ ' &( + g( % & ' { }=0 2 2 * 1+ | &( |

Water waves as Hamiltonian system The test: 1 & 1 ' H = "" [ "%h | #$ |2 dz + g& 2 + ( 1+ | #& |2 %1)]dxdy 2 2 ( R Hkin

!

Q:

$H " t# =

%$%H "t# = $ %&

?

(check this)

!

Hpot

?

(see Zakharov’s paper)

!

Water waves as Hamiltonian system The test: 1 & 1 ' H = "" [ "%h | #$ |2 dz + g& 2 + ( 1+ | #& |2 %1)]dxdy 2 2 ( R Hkin

!

Q:

$H " t# =

%$2) !

%H "t# = $ %&

?

(check this)

! 1)

Hpot

?

(see Zakharov’s paper)

"H pot =0 "#

! (easy)

"H kin "#

(not so easy)

Water waves as Hamiltonian system The test (continued)

S

nˆ Recall divergence theorem: Let S be a piecewise smooth, closed,!oriented, 2-D ! surface with outward normal nˆ . Let F be a continuously differentiable vector field defined!on S and its interior, V. ! ! Then [ F " nˆ ]ds = [# " F ]dv

$$ S

!

$$$

!

V

!

Water waves as Hamiltonian system The test (continued)

S

nˆ Recall divergence theorem: Let S be a piecewise smooth, closed,!oriented, 2-D ! surface with outward normal nˆ . Let F be a continuously differentiable vector field defined!on S and its interior, V. ! ! Then [ F " nˆ ]ds = [# " F ]dv

$$ S

$$$ V

!

! 1 ! • Choose F = "#", where " 2# = 0. 2 ! 1 ! " # F = [| "$ |2 +$" 2$ ] 2 ! !

!

Water waves as Hamiltonian system The test (continued)

S

nˆ Recall divergence theorem: Let S be a piecewise smooth, closed,!oriented, 2-D ! surface with outward normal nˆ . Let F be a continuously differentiable vector field defined!on S and its interior, V. ! ! Then [ F " nˆ ]ds = [# " F ]dv

$$

$$$

S

V

!

! 1 ! • Choose F = "#", where " 2# = 0. 2 ! 1 ! " # F = [| "$ |2 +$" 2$ ] 2 !

!

H kin

1 = 2

%

2

!

&& [ &$h | "# | dz]dxdy = R

1 2

&& [#' # ]ds n

S

Water waves as Hamiltonian system z ="

The test (continued) H kin

1)

1 = 2

%

&& [ &$h | "# |2dz]dxdy = R

On z = -h,

!

1 2

&& [#' # ]ds n

S

! z = "h

"n # = 0 !

!

Water waves as Hamiltonian system z ="

The test (continued) H kin

!

1 = 2

%

&& [ &$h | "# |2dz]dxdy = R

1 2

&& [#' # ]ds n

S

! z = "h

1)

On z = -h,

2)

φ periodic in (x,y) on vertical sides,

"n # = 0

$$! ["# " ]ds = 0 n

!

!

H kin

1 1 = %% ["#n " ]ds = 2 z= $ 2

2 [ &# " | ] 1+ | ' $ | dxdy %% n z= $ R !

Water waves as Hamiltonian system z ="

The test (continued) H kin

!

1 = 2

%

&& [ &$h | "# |2dz]dxdy = R

1 2

&& [#' # ]ds n

S

! z = "h

1)

On z = -h,

2)

φ periodic in (x,y) on vertical sides,

"n # = 0

$$! ["# " ]ds = 0 n

!

!

H kin

1 1 = %% ["#n " ]ds = 2 z= $ 2

2 [ &# " | ] 1+ | ' $ | dxdy %% n z= $ R !

Last step: Relate "n # |z= $ to ψ

Water waves as Hamiltonian system z ="

Last step: Relate "n # |z= $ to ψ Dirchlet-to-Neumann map:

" 2# = 0

!

!

z = "h

!

!

Water waves as Hamiltonian system z ="

Last step: Relate "n # |z= $ to ψ Dirchlet-to-Neumann map: There is G(x,y;µ,ν), symmetric Green’s f’n "n # (x, y,z,t) |z=! $=

'' [% (µ,& ,t)G(x, y;µ,& )]ds

" 2# = 0

!

!

free surface

=

2 [ " ( µ , # ,t)G(x, y; µ , # )] 1+ | % & | dµd!# $$ R

!

!

z = "h

Water waves as Hamiltonian system z ="

Last step: Relate "n # |z= $ to ψ Dirchlet-to-Neumann map: There is G(x,y;µ,ν), symmetric Green’s f’n "n # (x, y,z,t) |z=! $=

'' [% (µ,& ,t)G(x, y;µ,& )]ds

" 2# = 0

!

!

z = "h

free surface

=

2 [ " ( µ , # ,t)G(x, y; µ , # )] 1+ | % & | dµd!# $$ R

!

!

Substitute into Hkin: H kin

! 1 = 2

&& dxdy R

1+ | "# |2

&& dµd$ R

1+ | "# |2% (x, y,t)% (µ,$ ,t)G(x, y;µ,$ )

Water waves as Hamiltonian system z ="

Last step: Relate "n # |z= $ to ψ Dirchlet-to-Neumann map: There is G(x,y;µ,ν), symmetric Green’s f’n "n # (x, y,z,t) |z=! $=

'' [% (µ,& ,t)G(x, y;µ,& )]ds

" 2# = 0

!

!

z = "h

free surface

=

2 [ " ( µ , # ,t)G(x, y; µ , # )] 1+ | % & | dµd!# $$ R

!

!

Substitute into Hkin: H kin

! 1 = 2

&& dxdy R

1+ | "# |2

&& dµd$ R

Finally! Vary ψ, hold η fixed.

1+ | "# |2% (x, y,t)% (µ,$ ,t)G(x, y;µ,$ )

Water waves as Hamiltonian system H kin =

1 2

&& dxdy

1+ | "# |2

R

Vary ψ, hold η fixed

&& dµd$ R

1+ | "# |2% (x, y,t)% (µ,$ ,t)G(x, y;µ,$ )

Water waves as Hamiltonian system H kin =

1 2

&& dxdy

1+ | "# |2

R

&& dµd$

1+ | "# |2% (x, y,t)% (µ,$ ,t)G(x, y;µ,$ )

R

Vary ψ, hold η fixed "H kin

1 = 2

%% dxdy R

... %% dµd# ...["$ (x, y)$ (µ,# ) + $ (x, y)"$ (µ,# )]G(...) R

Water waves as Hamiltonian system H kin =

1 2

&& dxdy

1+ | "# |2

R

&& dµd$

1+ | "# |2% (x, y,t)% (µ,$ ,t)G(x, y;µ,$ )

R

Vary ψ, hold η fixed "H kin

1 = 2

%% dxdy R

... %% dµd# ...["$ (x, y)$ (µ,# ) + $ (x, y)"$ (µ,# )]G(...) R

But G is symmetric "H kin =

%% dxdy R

"H kin =

'' dxdy R

! !

... %% dµd# ...["$ (x, y)$ (µ,# )]G(...) R

1+ | #$ |2 "% (x, y)&n ( |z= $

Water waves as Hamiltonian system H kin =

1 2

&& dxdy

1+ | "# |2

R

&& dµd$

1+ | "# |2% (x, y,t)% (µ,$ ,t)G(x, y;µ,$ )

R

Vary ψ, hold η fixed "H kin

1 = 2

%% dxdy R

... %% dµd# ...["$ (x, y)$ (µ,# ) + $ (x, y)"$ (µ,# )]G(...) R

But G is symmetric "H kin =

%% dxdy R

"H kin =

'' dxdy

... %% dµd# ...["$ (x, y)$ (µ,# )]G(...) R

1+ | #$ |2 "% (x, y)&n ( |z= $

R

! !

"H = 1+ | $% |2 &n ' |z= % = &t% "#

✔

Water waves as Hamiltonian system Conclusion: Zakharov is correct! • • •

The equations of inviscid,irrotational water waves are Hamiltonian. Conjugate variables are {η, ψ}. The Hamiltonian is the physical energy.

C. So what? Q: What does Hamiltonian structure buy? A: Volume-preserving flow • Asymptotic stability is impossible neutral stability is only choice • “attractors” and “repellers” are impossible • Symplectic integrators: numerical integrators that preserve volume in phase space • For water waves, (η,ψ) are good variables • Complete integrability

C. So what? Q: What is complete integrability? 1. Need to define Poisson bracket for correct statement. 2. If a system of 2N first-order ODEs is Hamiltonian, and if one finds N (not 2N) constants of the motion, in involution relative to the Poisson bracket, then the motion is confined to an Ndimensional submanifold of 2N dim. phase space. • If this manifold is compact, it is a torus. • The N action variables are constants of the motion. • N angle variables are coordinates on the torus. • All of soliton theory fits into this framework.

Next lecture: The (completely integrable) Korteweg-de Vries equation as an approximate model of waves of moderate amplitude in shallow water.

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