Hamiltonian formulation: water waves Lecture 3

Transcription

1 Hamiltonian formulation: water waves Lecture 3 Wm.. Hamilton V.E. Zakharov

2 Hamiltonian formulation: water waves This lecture: A. apid 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

3 A. eview of Hamiltonian systems 1. Example: nonlinear oscillator " + # 2 " + $" 3 = 0, " 2 > 0, # = d# dt.

4 A. eview of Hamiltonian systems 1. Example: nonlinear oscillator " + # 2 " + $" 3 = 0, a) Find an energy integral " 2 > 0, # = d# dt. " # (eq'n) $ E = 1 2 ( " ) 2 + # 2 2 (")2 + $ 4 (")4 = const.

5 A. eview of Hamiltonian systems 1. Example: nonlinear oscillator " + # 2 " + $" 3 = 0, a) Find an energy integral " 2 > 0, # = d# dt. " # (eq'n) $ E = 1 2 ( " ) 2 + # 2 2 (")2 + $ 4 (")4 = const. b) Write eq n as a first-order system Define equivalent: q = "(t), q = p, p = "# 2 q "$q 3 p = " (t)

6 A. eview of Hamiltonian systems 1. Example: nonlinear oscillator " + # 2 " + $" 3 = 0, a) Find an energy integral " 2 > 0, # = d# dt. " # (eq'n) $ E = 1 2 ( " ) 2 + # 2 2 (")2 + $ 4 (")4 = const. b) Write eq n as a first-order system Define q = "(t), p = " (t) p equivalent: q = p, p = "# 2 q "$q 3 q

7 eview 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, {p j (t), q j (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 q j = dq j dt = "H "p j, p j = # "H "q j, j =1,2,...,N.

8 eview 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, {p j (t), q j (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 q j = dq j dt = "H, p j = # "H, j =1,2,...,N. "p j "q j In example, N = 1, choose H= E = 1 2 p2 + " 2 2 q2 + # 4 q4.

9 eview 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, {p j (t), q j (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 q j = dq j dt = "H, p j = # "H, j =1,2,...,N. "p j "q j In example, N = 1, choose q = p = "H "p H= E = 1 2 p2 + " 2 2 q2 + # 4 q4.

10 eview 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, {p j (t), q j (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 q j = dq j dt = "H, p j = # "H, j =1,2,...,N. "p j "q j In example, N = 1, choose q = p = "H "p H= E = 1 2 p2 + " 2 p = "# 2 q "$q 3 = " %H %q 2 q2 + # 4 q4.

11 eview 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.

12 eview of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. Start with M first-order ODEs dx j dt = v j (! x,t), j =1,2,..., M.

13 eview of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. Start with M first-order ODEs dx j dt = v j (! x,t), j =1,2,..., M. Imagine a fluid that fills the M-dimensional phase space. {x 1 (t), x 2 (t),, x M (t)} are the coordinates of a fluid particle, {v 1 (t), v 2 (t),, v M (t)} are the components of fluid velocity.

14 eview 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 dt = v j (! x,t), j =1,2,..., M. Imagine a fluid that fills the M-dimensional phase space. {x 1 (t), x 2 (t),, x M (t)} are the coordinates of a fluid particle, {v 1 (t), v 2 (t),, v M (t)} are the components of fluid velocity. The fluid is incompressible, so volume is preserved if " #! v = M $v % j = 0 j=1 $x j

15 eview 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

16 eview of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. Claim: Any Hamiltonian system of ODEs with a smooth Hamiltonian is volume-preserving, because Proof: dq j dt " # v! = 0 = "H "p j, dp j dt = # "H "q j, j =1,2,...,N.

17 eview of Hamiltonian systems 4. Plausibility argument for volume-preserving flows. Claim: Any Hamiltonian system of ODEs with a smooth Hamiltonian is volume-preserving, because Proof: dq j dt " # v! = 0 = "H "p j, " #! v = = N % N % j=1 dp j dt = # "H "q j, j =1,2,...,N. $ $q j ( dq j $ ( $H ) + j=1 $q j $p j N dt ) + $ % ( dp j $p j dt ) = 0. N % j=1 $ (& $H ) j=1 $p j $q j

18 eview of Hamiltonian systems 5. Hamiltonian PDEs Example: nonlinear wave equation, periodic b.c. " t 2 # = c 2 " x 2 # $% 2 # $&# 3

19 eview of Hamiltonian systems 5. Hamiltonian PDEs Example: nonlinear wave equation, periodic b.c. " t 2 # = c 2 " x 2 # $% 2 # $&# 3 a) Find energy integral E = " [ 1 (# t $) 2 + c (# x$) 2 + % 2 2 $ 2 + & 4 $ 4 ]dx, de dt = 0.

20 eview of Hamiltonian systems 5. Hamiltonian PDEs Example: nonlinear wave equation, periodic b.c. " t 2 # = c 2 " x 2 # $% 2 # $&# 3 a) Find energy integral E = " [ 1 (# t $) 2 + c (# x$) 2 + % 2 2 $ 2 + & 4 $ 4 ]dx, b) Choose conjugate variables p(x,t) = " t #(x,t), q(x,t) = #(x,t). de dt = 0. index (j) becomes continuous (x)

21 eview of Hamiltonian systems 5. Hamiltonian PDEs Example: nonlinear wave equation, periodic b.c. " t 2 # = c 2 " x 2 # $% 2 # $&# 3 a) Find energy integral E = " [ 1 (# t $) 2 + c (# x$) 2 + % 2 2 $ 2 + & 4 $ 4 ]dx, b) Choose conjugate variables p(x,t) = " t #(x,t), q(x,t) = #(x,t). de dt = 0. index (j) becomes continuous (x) c) Guess: H( p,q,t) = " [ 1 p 2 + c (# xq) 2 + $ 2 2 q2 + % 4 q4 ]dx

22 eview of Hamiltonian systems Q: What happens to in the PDE setting? ( "H "p j )

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

24 eview of Hamiltonian systems Q: What happens to in the PDE setting? Define variational derivative: Start with ( "H ) "p j H( p,q,t) = " [...] dx H( p + "p,q,t) # H(p,q,t) = $ [(**)"p + O(("p) 2 ]dx "H "p this defines variational derivative:

25 eview of Hamiltonian systems Q: What happens to in the PDE setting? Define variational derivative: Start with ( "H ) "p j H( p,q,t) = " [...] dx H( p + "p,q,t) # H(p,q,t) = $ [(**)"p + O(("p) 2 ]dx "H "p this defines variational derivative: H( p,q + "q,t) # H(p,q,t) = $ [( "H )"q + O(("q)2 ]dx "q

26 eview of Hamiltonian systems Q: What happens to in the PDE setting? Define variational derivative: Start with this defines variational derivative: Example: Q: What is? Why? ( "H ) "p j H( p,q,t) = " [...] dx H( p + "p,q,t) # H(p,q,t) = $ [(**)"p + O(("p) 2 ]dx "H "p H( p,q + "q,t) # H(p,q,t) = $ [( "H )"q + O(("q)2 ]dx "q H( p,q,t) = " [ 1 p 2 + c (# xq) 2 + $ 2 2 q2 + % 4 q4 ]dx "H "p

27 eview of Hamiltonian systems Q: What happens to in the PDE setting? ( "H ) "q j Continue example: H( p,q,t) = " [ 1 p 2 + c (# x q)2 + $ 2 2 q2 + % 4 q4 ]dx

28 eview of Hamiltonian systems Q: What happens to in the PDE setting? Continue example: ' ( "H ) "q j H( p,q,t) = " [ 1 p 2 + c (# x q)2 + $ 2 2 q2 + % 4 q4 ]dx H( p,q + "q,t) # H(p,q,t) = [c 2 ($ x q)($ x "q) + % 2 q"q + &q 3 "q + O(("q) 2 )]dx new twist: integrate by parts, with δq = 0 on boundaries

29 eview of Hamiltonian systems Q: What happens to in the PDE setting? Continue example: ' ( "H ) "q j H( p,q,t) = " [ 1 p 2 + c (# x q)2 + $ 2 2 q2 + % 4 q4 ]dx H( p,q + "q,t) # H(p,q,t) = [c 2 ($ x q)($ x "q) + % 2 q"q + &q 3 "q + O(("q) 2 )]dx new twist: integrate by parts, with δq = 0 on boundaries H( p,q + "q,t) # H(p,q,t) = ' [(#c 2 $ 2 x q + % 2 q + &q 3 )"q "H "q + O(("q) 2 )]dx

30 eview of Hamiltonian systems Q: What happens to in the PDE setting? Continue example: ' ( "H ) "q j H( p,q,t) = " [ 1 p 2 + c (# x q)2 + $ 2 2 q2 + % 4 q4 ]dx H( p,q + "q,t) # H(p,q,t) = [c 2 ($ x q)($ x "q) + % 2 q"q + &q 3 "q + O(("q) 2 )]dx new twist: integrate by parts, with δq = 0 on boundaries H( p,q + "q,t) # H(p,q,t) = ' [(#c 2 $ 2 x q + % 2 q + &q 3 )"q "H "q End of lightning tour of Hamiltonian systems + O(("q) 2 )]dx

31 B. Inviscid water waves ecall: " t # + $% & $# = " z %, " t % $% 2 +g# = ' ( $ &{ $# 1+ $# 2 }, on z = η(x,y,t) on z = η(x,y,t) " 2 # = 0 " n # = 0 -h(x,y) < z < η(x,y,t) on z = -h(x,y)

32 B. Inviscid water waves ecall: " t # + $% & $# = " z %, " t % $% 2 +g# = ' ( $ &{ $# 1+ $# 2 }, on z = η(x,y,t) on z = η(x,y,t) " 2 # = 0 " n # = 0 Q: Where does t-evolution occur? -h(x,y) < z < η(x,y,t) on z = -h(x,y) A. (Zakharov): on z = η(x,y,t)

33 B. Inviscid water waves ecall: " t # + $% & $# = " z %, " t % $% 2 +g# = ' ( $ &{ $# 1+ $# 2 }, on z = η(x,y,t) on z = η(x,y,t) " 2 # = 0 " n # = 0 Q: Where does t-evolution occur? -h(x,y) < z < η(x,y,t) on z = -h(x,y) A. (Zakharov): on z = η(x,y,t) Propose conjugate variables: "(x, y,t), #(x, y,t) = $(x,y,z,t) z="

34 Water waves as Hamiltonian system "(x, y,t), #(x, y,t) = $(x, y,z,t) z=" Plausibility check: z = " 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

35 Water waves as Hamiltonian system "(x, y,t), #(x, y,t) = $(x, y,z,t) z=" Plausibility check: z = " 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. z = "h [We need a procedure to find φ(x,y,z,t) from {η, ψ}]. esult: At any fixed time, {η, ψ} determine the entire solution.

36 Water waves as Hamiltonian system Proposed conjugate variables: "(x, y,t), Q: What is H(η,ψ)? A: Physical energy (from HW #1): H = [ 1 2 & "" " #$ 2 dz + %h #(x, y,t) = $(x, y,z,t) z=" 1 2 g&2 + ' ( ( 1+ #& 2 %1)]dxdy kinetic energy potential energy

37 Water waves as Hamiltonian system Claim (Zakharov, 1968): Let be a fixed region in x-y plane. Let h(x,y) be continuous and differentiable on. Define H(",#) = [ 1 2 " $$ $ %& 2 dz + 'h We need to show that " t # = $H $%, " t% = & $H $# 1 2 g"2 + ( ) ( 1+ %" 2 '1)]dxdy are equivalent to the two boundary conditions on z = η(x,y,t).

38 Water waves as Hamiltonian system Step 1: ewrite 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.

39 Water waves as Hamiltonian system Step 1: ewrite 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 = "F "F = {#$ x%,#$ y %,1} 1+ "% 2.

40 Water waves as Hamiltonian system Step 1: ewrite 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 = "F "F = {#$ x%,#$ y %,1} 1+ "% 2. Normal component of velocity on z = η: " n # = $# % ˆ n = &$# % $' + " z# 1+ $' 2

41 Water waves as Hamiltonian system Step 1: ewrite 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: Normal component of velocity on z = η: Eq n #1 on z = η:! n = "F "F = {#$ x%,#$ y %,1} 1+ "% 2. " n # = $# % ˆ n = &$# % $' + " z# 1+ $' 2 " t # + $% & $# = " z % " " t # = 1+ $# 2 " n %.

42 Water waves as Hamiltonian system Step 2: ewrite 2 nd eq n on z = η: " t # $# 2 +g% & ' ( $ ) { $% 1+ $% 2 } = 0

43 Water waves as Hamiltonian system Step 2: ewrite 2 nd eq n on z = η: " t # $# 2 +g% & ' ( $ ) { $% 1+ $% } = 0 2 But "(x, y,t) = #(x,y,z,t) z=$(x,y,t ) " t # = " t $ z=% +" z $ z=% " t % (chain rule)

44 Water waves as Hamiltonian system Step 2: ewrite 2 nd eq n on z = η: " t # $# 2 +g% & ' ( $ ) { $% 1+ $% } = 0 2 But "(x, y,t) = #(x,y,z,t) z=$(x,y,t ) " t # = " t $ z=% +" z $ z=% " t % (chain rule) = " t # z=$ +" z # z=$ {" z # % &# ' &$} z=$

45 Water waves as Hamiltonian system Step 2: ewrite 2 nd eq n on z = η: " t # $# 2 +g% & ' ( $ ) { $% 1+ $% } = 0 2 But "(x, y,t) = #(x,y,z,t) z=$(x,y,t ) " t # = " t $ z=% +" z $ z=% " t % (chain rule) = " t # z=$ +" z # z=$ {" z # % &# ' &$} z=$ Eq n #2 on z = η: " t # [(" x$) 2 + (" y $) 2 % (" z $) 2 ] + (" z $)&$ ' &( + g( % ) * & '{ &( 1+ &( 2 } = 0

46 Water waves as Hamiltonian system The test: H = [ 1 2 & "" " #$ 2 dz + %h 1 2 g&2 + ' ( ( 1+ #& 2 %1)]dxdy H kin H pot " t # = $H $% " t # = $ %H %& Q:?? (check this) (see Zakharov s paper)

47 Water waves as Hamiltonian system The test: H = [ 1 2 & "" " #$ 2 dz + %h 1 2 g&2 + ' ( ( 1+ #& 2 %1)]dxdy H kin H pot " t # = $H $% " t # = $ %H %& Q:?? (check this) (see Zakharov s paper) "H pot 1) (easy) "# = 0 "H kin 2) (not so easy) "#

48 Water waves as Hamiltonian system The test (continued) ecall divergence theorem: Let S be a piecewise smooth, closed,! oriented, 2-D surface with outward normal n ˆ. Let F be a continuously differentiable vector field defined on S and its interior, V. Then $$ S [! F " ˆ $$$ n ]ds = [# " F! ]dv V ˆ n S

49 Water waves as Hamiltonian system The test (continued) ecall divergence theorem: Let S be a piecewise smooth, closed,! oriented, 2-D surface with outward normal n ˆ. Let F be a continuously differentiable vector field defined on S and its interior, V. Then Choose F = 1, where 2 "#" $$ S [! F " ˆ! $$$ n ]ds = [# " F! ]dv " #! F = 1 2 [ "$ 2 +$" 2 $] V " 2 # = 0. ˆ n S

50 Water waves as Hamiltonian system The test (continued) ecall divergence theorem: Let S be a piecewise smooth, closed,! oriented, 2-D surface with outward normal n ˆ. Let F be a continuously differentiable vector field defined on S and its interior, V. Then Choose F = 1, where 2 "#" $$ S [! F " ˆ! $$$ n ]ds = [# " F! ]dv " #! F = 1 2 [ "$ 2 +$" 2 $] V " 2 # = 0. ˆ n S H kin = 1 2 % && [ & "# 2 dz]dxdy = 1 $h 2 && S [#' n #]ds

51 Water waves as Hamiltonian system The test (continued) H kin = 1 2 && [ & "# 2 dz]dxdy = 1 $h 2 1) On z = -h, " n # = 0 % && S [#' n #]ds z = " z = "h

52 Water waves as Hamiltonian system The test (continued) H kin = 1 2 && [ & "# 2 dz]dxdy = 1 $h 2 1) On z = -h, " n # = 0 % && S [#' n #]ds z = " z = "h 2) φ periodic in (x,y) on vertical sides, $$ ["# n "]ds = 0 H kin = 1 2 %% ["# n "] ds = 1 2 z=$ %% [&# n " z=$ ] 1+ '$ 2 dxdy

53 Water waves as Hamiltonian system The test (continued) H kin = 1 2 && [ & "# 2 dz]dxdy = 1 $h 2 1) On z = -h, " n # = 0 % && S [#' n #]ds z = " z = "h 2) φ periodic in (x,y) on vertical sides, $$ ["# n "]ds = 0 H kin = 1 2 %% ["# n "] ds = 1 2 z=$ %% [&# n " z=$ ] 1+ '$ 2 dxdy Last step: elate " n # z=$ to ψ

54 Water waves as Hamiltonian system Last step: elate " n # z=$ to ψ Dirchlet-to-Neumann map: " 2 # = 0 z = " z = "h

55 Water waves as Hamiltonian system Last step: elate " 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 free surface " 2 # = 0 z = " z = "h = $$ ["(µ,#,t)g(x,y; µ,#)] 1+ %& 2 dµd#

56 Water waves as Hamiltonian system Last step: elate " 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 free surface " 2 # = 0 z = " z = "h = $$ ["(µ,#,t)g(x,y; µ,#)] 1+ %& 2 dµd# Substitute into H kin: H kin = 1 2 && dxdy 1+ "# 2 && dµd$ 1+ "# 2 %(x, y,t)%(µ,$,t)g(x, y;µ,$)

57 Water waves as Hamiltonian system Last step: elate " 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 free surface " 2 # = 0 z = " z = "h = $$ ["(µ,#,t)g(x,y; µ,#)] 1+ %& 2 dµd# Substitute into H kin: H kin = 1 2 && dxdy 1+ "# 2 && dµd$ 1+ "# 2 %(x, y,t)%(µ,$,t)g(x, y;µ,$) Finally! Vary ψ, hold η fixed.

58 Water waves as Hamiltonian system H kin = 1 2 && dxdy 1+ "# 2 Vary ψ, hold η fixed && dµd$ 1+ "# 2 %(x, y,t)%(µ,$,t)g(x, y;µ,$)

59 Water waves as Hamiltonian system H kin = 1 2 && dxdy 1+ "# 2 Vary ψ, hold η fixed && dµd$ 1+ "# 2 %(x, y,t)%(µ,$,t)g(x, y;µ,$) "H kin = 1 2 %% %% dxdy... dµd#...["$(x,y)$(µ,#) +$(x, y)"$(µ,#)]g(...)

60 Water waves as Hamiltonian system H kin = 1 2 && dxdy 1+ "# 2 Vary ψ, hold η fixed && dµd$ 1+ "# 2 %(x, y,t)%(µ,$,t)g(x, y;µ,$) "H kin = 1 2 But G is symmetric %% %% %% dxdy... dµd#...["$(x,y)$(µ,#) +$(x, y)"$(µ,#)]g(...) %% "H kin = dxdy... dµd#...["$(x,y)$(µ,#)]g(...) '' "H kin = dxdy 1+ #$ 2 "%(x,y)& n ( z=$

61 Water waves as Hamiltonian system H kin = 1 2 && dxdy 1+ "# 2 Vary ψ, hold η fixed && dµd$ 1+ "# 2 %(x, y,t)%(µ,$,t)g(x, y;µ,$) "H kin = 1 2 But G is symmetric %% %% %% dxdy... dµd#...["$(x,y)$(µ,#) +$(x, y)"$(µ,#)]g(...) "H kin = dxdy... dµd#...["$(x,y)$(µ,#)]g(...) '' %% "H kin = dxdy 1+ #$ 2 "%(x,y)& n ( z=$ "H "# = 1+ $% 2 & n ' z=% = & t %

62 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.

63 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

64 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 N- dimensional 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.

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