Shape Dynamics of Point Vortices Tomoki Ohsawa April Fools Day, - PowerPoint PPT Presentation

Shape Dynamics of Point Vortices Tomoki Ohsawa April Fools’ Day, 2019 Tomoki Ohsawa (UT–Dallas) Georgia Tech 1 / 46

Dynamics of Hurricanes? Source: U. Washington News & NOAA Tomoki Ohsawa (UT–Dallas) Georgia Tech 2 / 46

Dynamics of Hurricanes? Source: U. Washington News & NOAA Tomoki Ohsawa (UT–Dallas) Georgia Tech 2 / 46

� � � � � � � � � � Point Vortex on R 2 Point vortex with circulation Γ at x 0 = ( x 0 , y 0 ) � Vorticity ξ ( x ) = curl u ( x ) = Γ δ ( x − x 0 ) Γ > 0 x 0 Tomoki Ohsawa (UT–Dallas) Georgia Tech 3 / 46

Point Vortex on R 2 Point vortex with circulation Γ at x 0 = ( x 0 , y 0 ) � Vorticity ξ ( x ) = curl u ( x ) = Γ δ ( x − x 0 ) ⇓ Γ Velocity field u ( x ) = 2 π r 2 ( − ( y − y 0 ) , x − x 0 ) � � Γ > 0 � x 0 - � - � - � - � � � � Tomoki Ohsawa (UT–Dallas) Georgia Tech 3 / 46

Dynamics of N Point Vortices on R 2 Γ 3 > 0 Γ 1 > 0 x 3 x 1 Γ 2 < 0 x 2 Each point vortex j located at x j ∈ R 2 is convected by the net velocity of the other vortices: � x j ( t ) = ˙ u k ( x j ( t )) , 1 ≤ k ≤ N k � = j Tomoki Ohsawa (UT–Dallas) Georgia Tech 4 / 46

Dynamics of N Point Vortices on R 2 which gives: x j = − 1 y j − y k y j = 1 x j − x k � � ˙ Γ k � x j − x k � 2 , ˙ Γ k � x j − x k � 2 , 2 π 2 π 1 ≤ k ≤ N 1 ≤ k ≤ N k � = j k � = j or, by setting q j := x j + i y j ∈ C , i q j − q k � q j = ˙ Γ k | q j − q k | 2 2 π 1 ≤ k ≤ N k � = j for j ∈ { 1 , . . . , N } . Tomoki Ohsawa (UT–Dallas) Georgia Tech 5 / 46

What is Hamiltonian System? Classical Hamiltonian system q = position, p = m ˙ q = momentum p 2 Hamiltonian H ( q , p ) = + V ( q ) = Total Energy 2 m � �� � ���� Potential Energy Kinetic Energy The Hamiltonian system � � � ∂ H � ∂ H q = ∂ H p = − ∂ H 0 I [ ˙ q ˙ p ] = ⇐ ⇒ ˙ ∂ p , ˙ − I 0 ∂ q ∂ p ∂ q gives the equations of motion (Newton’s Second Law): md 2 q dt 2 = −∇ V ( q ) . Tomoki Ohsawa (UT–Dallas) Georgia Tech 6 / 46

Symplectic Geometry and Hamiltonian Systems Symplectic manifold (“Phase space”) P = { ( q , p ) } equipped with symplectic form Ω = d q ∧ d p Vector field q ∂ p ∂ X H = ˙ ∂ q + ˙ ∂ p Contraction i X H Ω = Ω( X H , · ) = − ˙ p d q + ˙ q d p Exterior differential d H = ∂ H ∂ q d q + ∂ H ∂ p d p Hamilton’s Eq. q = ∂ H p = − ∂ H ⇒ ˙ i X H Ω = d H = ∂ p , ˙ ∂ q Tomoki Ohsawa (UT–Dallas) Georgia Tech 7 / 46

Symplectic Geometry and Hamiltonian Systems More generally... Definition (Hamiltonian System on Symplectic Manifold) T z 2 P Manifold P P T z 3 P T z 1 z 2 Symplectic form Ω (closed z 3 z 1 non-degenerate two-form) Hamiltonian H : P → R P Tomoki Ohsawa (UT–Dallas) Georgia Tech 8 / 46

Symplectic Geometry and Hamiltonian Systems More generally... Definition (Hamiltonian System on Symplectic Manifold) 2 P T Manifold P z P T T X H ( z 2 ) P z 1 z 3 X H ( z 1 ) z 2 Symplectic form Ω (closed X H ( z 3 ) z 3 z 1 non-degenerate two-form) Hamiltonian H : P → R P Define a Hamiltonian system by i X H Ω = d H = ⇒ Determines the vector field (dynamics) X H on P . Tomoki Ohsawa (UT–Dallas) Georgia Tech 8 / 46

Symplectic Geometry and Hamiltonian Systems More generally... Definition (Hamiltonian System on Symplectic Manifold) 2 P T Manifold P z P T T X H ( z 2 ) P z 1 z 3 X H ( z 1 ) z 2 Symplectic form Ω (closed X H ( z 3 ) z 3 z 1 non-degenerate two-form) Hamiltonian H : P → R P Define a Hamiltonian system by i X H Ω = d H = ⇒ Determines the vector field (dynamics) X H on P . Tomoki Ohsawa (UT–Dallas) Georgia Tech 8 / 46

Hamiltonian Dynamics of N Point Vortices on R 2 Symplectic form on C N = { ( q 1 , . . . , q N ) } : N Ω j := − 1 � 2 Im( d q j ∧ d q ∗ Ω = Γ j Ω j with j ) = d x j ∧ d y j j =1 Hamiltonian H : C N → R H ( q 1 , . . . , q N ) = − 1 � Γ j Γ k ln | q j − q k | 2 . 4 π 1 ≤ j < k ≤ N Then the Hamiltonian system i X H Ω = d H gives the equations for point vortices: For j ∈ { 1 , . . . , N } . i q j − q k � q j = ˙ Γ k | q j − q k | 2 2 π 1 ≤ k ≤ N k � = j Tomoki Ohsawa (UT–Dallas) Georgia Tech 9 / 46

Dynamics of 3 Point Vortices on R 2 Tomoki Ohsawa (UT–Dallas) Georgia Tech 10 / 46

Dynamics of 3 Point Vortices on R 2 Tomoki Ohsawa (UT–Dallas) Georgia Tech 11 / 46

Dynamics of 4 Point Vortices on R 2 Tomoki Ohsawa (UT–Dallas) Georgia Tech 12 / 46

Shape Dynamics of Vortices on R 2 Distance between vortices: Γ 1 ℓ jk := | q j − q k | 12 Γ 2 Shape dynamics or Equations of relative motion: A 123 31 � � d ij = 2 1 − 1 � dt ℓ 2 Γ k A ijk , 23 ℓ 2 ℓ 2 π jk ki 1 ≤ l ≤ N Γ 3 i � = j � = k where A ijk = signed area of triangle defined by vortices ( i , j , k ) Tomoki Ohsawa (UT–Dallas) Georgia Tech 13 / 46

Shape Dynamics of Vortices on R 2 Distance between vortices: Γ 1 ℓ jk := | q j − q k | 12 Γ 2 Shape dynamics or Equations of relative motion: A 123 31 � � d ij = 2 1 − 1 � dt ℓ 2 Γ k A ijk , 23 ℓ 2 ℓ 2 π jk ki 1 ≤ l ≤ N Γ 3 i � = j � = k where A ijk = signed area of triangle defined by vortices ( i , j , k ) Goal Is shape dynamics also Hamiltonian? Geometric structure behind it? Tomoki Ohsawa (UT–Dallas) Georgia Tech 13 / 46

Background: Lie Groups and Lie Algebras G : = T g e G G = Lie group e “continuous transformations” g := T e G = Lie algebra of G “infinitesimal transformations” g ∗ = dual of Lie algebra g Tomoki Ohsawa (UT–Dallas) Georgia Tech 14 / 46

Background: Lie Groups and Lie Algebras G : = T g e G G = Lie group e “continuous transformations” g := T e G = Lie algebra of G “infinitesimal transformations” g ∗ = dual of Lie algebra g Example: 3D rotation group SO(3) Rotations: � � R ∈ R 3 × 3 | R T R = I , det R = 1 G = SO(3) = Angular velocities: � � ξ ∈ R 3 × 3 | ξ T = − ξ ∼ = R 3 g = so (3) = T I SO(3) = Tomoki Ohsawa (UT–Dallas) Georgia Tech 14 / 46

SE(2) -Action on the Plane R 2 Lie group SE(2) := SO(2) ⋉ R 2 �� R θ � cos θ � � � a − sin θ , θ ∈ [0 , 2 π ) , a ∈ R 2 = | R θ = 0 1 sin θ cos θ = All rotations of R 2 and translations of R 2 combined SE(2)-action on R 2 : x Tomoki Ohsawa (UT–Dallas) Georgia Tech 15 / 46

SE(2) -Action on the Plane R 2 Lie group SE(2) := SO(2) ⋉ R 2 �� R θ � cos θ � � � a − sin θ , θ ∈ [0 , 2 π ) , a ∈ R 2 = | R θ = 0 1 sin θ cos θ = All rotations of R 2 and translations of R 2 combined SE(2)-action on R 2 : x R θ x θ Tomoki Ohsawa (UT–Dallas) Georgia Tech 15 / 46

SE(2) -Action on the Plane R 2 Lie group SE(2) := SO(2) ⋉ R 2 �� R θ � cos θ � � � a − sin θ , θ ∈ [0 , 2 π ) , a ∈ R 2 = | R θ = 0 1 sin θ cos θ = All rotations of R 2 and translations of R 2 combined SE(2)-action on R 2 : R θ x + a a x R θ x θ Tomoki Ohsawa (UT–Dallas) Georgia Tech 15 / 46

SE(2) -Symmetry of N Point Vortices The Hamiltonian H is invariant under the SE(2)-action. ⇒ The system of N point vortices is invariant under the SE(2)-action. = 2 3 1 3 2 a 2 1 3 1 θ These configurations are essentially the same because the shape of the vortices is the same. Tomoki Ohsawa (UT–Dallas) Georgia Tech 16 / 46

SE(2) -Symmetry of N Point Vortices The Hamiltonian H is invariant under the SE(2)-action. ⇒ The system of N point vortices is invariant under the SE(2)-action. = 2 3 1 3 2 a 2 1 3 1 θ These configurations are essentially the same because the shape of the vortices is the same. ⇒ After “dividing” it by SE(2), all that matters is the shape of point = vortices. Tomoki Ohsawa (UT–Dallas) Georgia Tech 16 / 46

Background: Noether’s Theorem Suppose G Lie group and g its Lie algebra G acts on a symplectic manifold P in a canonical manner Then one may define a corresponding map J : P → g ∗ called a momentum map . Theorem (Noether) If a Hamiltonian system on P has a G-symmetry, then J is a conserved quantity of the Hamiltonian system. Tomoki Ohsawa (UT–Dallas) Georgia Tech 17 / 46

Simple Example: A Free Particle on the Plane Configuration space R 2 = { x = ( x 1 , x 2 ) } Symplectic manifold T ∗ R 2 = { ( x , p ) = ( x 1 , x 2 , p 1 , p 2 ) } H = 1 Hamiltonian H : T ∗ R 2 → R ; 2 m ( p 2 1 + p 2 2 ) Tomoki Ohsawa (UT–Dallas) Georgia Tech 18 / 46

Simple Example: A Free Particle on the Plane Configuration space R 2 = { x = ( x 1 , x 2 ) } Symplectic manifold T ∗ R 2 = { ( x , p ) = ( x 1 , x 2 , p 1 , p 2 ) } H = 1 Hamiltonian H : T ∗ R 2 → R ; 2 m ( p 2 1 + p 2 2 ) Example Symmetry under translations by G = R 2 : ( x 1 , x 2 ) �→ ( x 1 + a , x 2 + b ). Momentum map J : T ∗ R 2 → R 2 ; J ( x , p ) = ( p 1 , p 2 ) = linear momentum Tomoki Ohsawa (UT–Dallas) Georgia Tech 18 / 46