Generating Functions Thomas Bisig 16.01.2006 Generating Functions - PowerPoint PPT Presentation

Generating Functions Thomas Bisig 16.01.2006

Generating Functions Idea: A certain function S moves a PDE problem and S is itself a solution of a partial differential equation (Hamilton-Jacobi PDE). Such a function S is directly connected to any simplectic map. The function S is called the Generating Function. Introduction

Generating Functions 1. Existence of Generating Functions 2. Generating Function for Symplectic Runge- Kutta Methods 3. The Hamilton-Jacobi PDE 4. Methods Based on Generating Functions Introduction

Generating Functions Initial values: p 1 , . . . , p d q 1 , . . . , q d Final values: P 1 , . . . , P d Q 1 , . . . , Q d Theorem 1 : A mapping is ϕ : ( p, q ) �→ ( P, Q ) symplectic if and only if there exists locally a function such that S ( p, q ) P T dQ − p T dq = dS P T dQ − p T dq This means that is a total differential. Existence of Generating Function

Generating Functions Change of Coordinates: ( p, q ) ( q, Q ) S ( p, q ) S ( q, Q ) Reconstruction of the transformation from S ( q, Q ) P = ∂S p = − ∂S ∂Q ( q, Q ) ∂q ( q, Q ) Any sufficiently smooth and nondegenerate function S generates a symplectic mapping. Existence of Generating Function

Generating Functions Lemma 1 : Let be a smooth ( p, q ) �→ ( P, Q ) transformation, close to identity. It is symplectic if and only if one of the following conditions hold locally: Q T dP + p T dq = d ( P T q + S 1 ) 1. ; S 1 ( P, q ) P T dQ + q T dp = d ( p T Q − S 2 ) S 2 ( p, Q ) 2. ; ( Q − q ) T d ( P + p ) − ( P − p ) T d ( Q + q ) = 2 S 3 3. ; S 3 (( P + p ) / 2 , ( Q + q ) / 2) Existence of Generating Function

Generating Functions Symplectic Euler Method S 1 Adjoint of the Symplectic Euler S 2 Method Implicit Midpoint Rule S 3 Q = q + ∂S 1 p = P + ∂S 1 ∂P ( P, q ) S 1 ∂q ( P, q ) P = p − ∂ 2 S 3 (( P + p ) / 2 , ( Q + q ) / 2) S 3 Q = q + ∂ 1 S 3 (( P + p ) / 2 , ( Q + q ) / 2) Existence of Generating Function

Generating Functions Theorem 2 : Suppose, we have a Runge-Kutta method which satisfies b i a ij + b j a ji = b i b j i.e. it is symplectic. Then the method s s � � P i = p − h a ij H q ( P j , Q j ) P = p − h b i H q ( P i , Q i ) j =1 i =1 s s � � Q i = q + h a ij H p ( P j , Q j ) Q = q + h b i H p ( P i , Q i ) j =1 i =1 can be written as s s b i a ij H q ( P i , Q i ) T H p ( P j , Q j ) � � S 1 ( P, q, h ) = h b i H ( P i , Q i ) − h 2 j =1 i,j =1 Generating Function for Symplectic Runge-Kutta Methods

Generating Functions Theorem 2 gives the explicit formula for the generating function. Lemma 1 guarantees the local existence of a generating function where the explicit formula shows that the generating function is globally defined in the sense that it is well-defined in the region where is defined. H ( p, q ) Generating Function for Symplectic Runge-Kutta Methods

Generating Functions We wish to construct a smooth generating function for a symplectic S ( q, Q, t ) transformation but the final points shall move in the flow of the Hamiltonian system P ( t ) Q ( t ) P Q has to satisfy: S ( q, Q, t ) p i ( t ) = − ∂S P i ( t ) = ∂S ( q, Q ( t ) , t ) � ( q, Q ( t ) , t ) ∂q i ∂Q i d ⇒ 0 = ∂ 2 S ∂ 2 S ( q, Q ( t ) , t ) · H � ∂q i ∂t ( q, Q ( t ) , t ) + ( P ( t ) , Q ( t )) ∂q i ∂Q j P j j =1 The Hamilton-Jacobi Partial Differential Equation

Generating Functions Using the chain rule: ( ∂ S ∂ t + H ( ∂ S , . . . , ∂ S ∂ , Q 1 , . . . , Q d )) = 0 ∂ q i ∂ Q 1 ∂ Q d Theorem 3 : If is a smooth solution of S ( q, Q, t ) ∂S ∂t + H ( ∂S , . . . , ∂S , Q 1 , . . . , Q d ) = 0 ∂Q 1 ∂Q d ∂ 2 S and if the matrix is invertible, there is a ( ) ∂q i ∂Q j map defined by which is the flow of ϕ t ( p, q ) � the Hamiltonian system. The Hamilton-Jacobi Partial Differential Equation

Generating Functions We write the Hamilton-Jacobi PDE in the coordinates used in Lemma 1: S 1 ( P, q, t ) = P T ( Q − q ) − S ( q, Q, t ) ∂S 1 ∂t ( P, q, t ) = P T ∂Q ∂t − ∂S ∂Q ( q, Q, t ) ∂Q ∂t − ∂S ∂t ( q, Q, t ) = − ∂S ∂t ( q, Q, t ) ∂ S 1 ∂ P ( P, q, t ) = Q − q + P T ∂ Q ∂ P − ∂ S ∂ Q ( q, Q, t ) ∂ Q ∂ P = Q − q The Hamilton-Jacobi Partial Differential Equation

Generating Functions Theorem 4 : If is a solution of the S 1 ( P, q, t ) partial differential equation ∂S 1 ∂t ( P, q, t ) = H ( P, q + ∂S 1 ∂P ( P, q, t )) , S 1 ( P, q, t 0 ) = 0 then the mapping is the exact ( p, q ) �→ ( P ( t ) , Q ( t )) flow of the Hamilton system. The Hamilton-Jacobi Partial Differential Equation

Generating Functions Approximate solution of the Hamilton-Jacobi equation using the ansatz: S 1 ( P, q, t ) = tG 1 ( P, q ) + t 2 G 2 ( P, q ) + t 3 G 3 ( P, q ) + . . . G 1 ( P, q ) = H ( P, q ) G 2 ( P, q ) = 1 2( ∂H ∂H ∂q )( P, q ) ∂P Problem: Higher order derivatives! Methods Based on Generating Functions

Generating Functions Try to avoid higher order derivatives (Miesbach & Pesch). We use generating functions of the following form: s � S 3 ( w, h ) = h b i H ( w + hc i J − 1 ∇ H ( w )) i =1 We only have to determine the coefficients according to the solution of the Hamilton-Jacobi equation. But: We still need second order derivatives. Methods Based on Generating Functions

Generating Functions First Order Pendulum

Generating Functions Second Order Pendulum