Hamiltonian formulation: water waves Lecture 3
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.
