Internal and external cyclidic harmonics Hans Volkmer University of - PowerPoint PPT Presentation

Internal and external cyclidic harmonics Hans Volkmer University of Wisconsin at Milwaukee

Introduction A solution u ( x , y , z ), ( x , y , z ) ∈ D , of the Laplace equation ∆ u = u xx + u yy + u zz = 0 is called harmonic in D . In 1894 Maxime Bˆ ocher showed that the Laplace equation can be solved by the method of separation of variables in 17 coordinate systems. There are 11 quadratic systems and 6 cyclidic systems. Hans Volkmer Cyclidic harmonics

References M. Bˆ ocher. Ueber die Reihenentwickelungen der Potentialtheorie . B. G. Teubner, Leipzig, 1894. H. S. Cohl and H. Volkmer. Eigenfunction expansions for a fundamental solution of Laplace’s equation on R 3 in parabolic and elliptic cylinder coordinates. Journal of Physics A: Mathematical and Theoretical , 45(35):355204, 2012. H. S. Cohl and H. Volkmer. Separation of variables in an asymmetric cyclidic coordinate system. Journal of Mathematical Physics , 54(6):063513, 2013. W. Miller, Jr. Symmetry and separation of variables . Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. Encyclopedia of Mathematics and its Applications, Vol. 4. Hans Volkmer Cyclidic harmonics

Spherical coordinates Coordinates: r > 0, 0 < θ < π , − π < φ < π . x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. Coordinate surfaces: x 2 + y 2 + z 2 = r 2 , spheres ( x 2 + y 2 ) cot 2 θ = z 2 , circular cones planes x sin φ = y cos φ. Hans Volkmer Cyclidic harmonics

Separation of variables Laplace equation ∆ u = 0 in spherical coordinates: 1 1 ( r 2 u r ) r + sin θ (sin θ u θ ) θ + sin 2 θ u φφ = 0 . Separation of variables u = u 1 ( r ) u 2 ( θ ) u 3 ( φ ): r 2 u ′′ 1 + 2 ru ′ 1 − n ( n + 1) u 1 = 0 , m 2 � � u ′′ 2 + cot θ u ′ 2 + n ( n + 1) − u 2 = 0 , sin 2 θ u ′′ 3 + m 2 u 3 = 0 , where m , n are separation parameters. Hans Volkmer Cyclidic harmonics

Internal and external spherical harmonics Internal harmonics: n (cos θ ) e im φ , G m n ( x , y , z ) = r n P m − n ≤ m ≤ n , where P m n is an associated Legendre function (Ferrer’s function). G m n is a harmonic functions in R 3 . They are polynomials in x , y , z . External harmonics: H m n ( x , y , z ) = r − n − 1 P m n (cos θ ) e im φ , − n ≤ m ≤ n . n is a harmonic functions in R 3 \ { 0 } . H m Hans Volkmer Cyclidic harmonics

Expansion of reciprocal distance Theorem. Let r , r ′ ∈ R 3 be such that � r � < � r ′ � . Then ∞ n 1 ( n − m )! � � ( n + m )! G m n ( r ) H m n ( r ′ ) . � r − r ′ � = n =0 m = − n This formula may be used to find the Green’s function for the ball. Hans Volkmer Cyclidic harmonics

Sphero-conal coordinates √ Parameter: 0 < k < 1, k ′ = 1 − k 2 . Coordinates: r > 0, 0 < s < 1 < t < k − 2 . x = krst , √ √ k y = k ′ r 1 − s t − 1 , 1 � � 1 − k 2 s 1 − k 2 t . z = k ′ r Coordinate surfaces: x 2 + y 2 + z 2 = r 2 , spheres x 2 y 2 z 2 elliptical cones s − 1 − s − k − 2 − s = 0 , x 2 y 2 z 2 elliptical cones t + t − 1 − k − 2 − t = 0 . Hans Volkmer Cyclidic harmonics

Sphero-conal coordinates Hans Volkmer Cyclidic harmonics

Separation of variables Laplace equation in sphero-conal coordinates: 1 4 ( t − s )( r 2 u r ) r + ω ( s )( ω ( s ) u s ) s + ω ( t )( ω ( t ) u t ) t = 0 , where ω ( s ) = | s (1 − s )( k − 2 − s ) | 1 / 2 . Separation of variables u = u 1 ( r ) u 2 ( s ) u 3 ( t ): r 2 u ′′ 1 + 2 ru ′ 1 − n ( n + 1) u 1 = 0 , and v = u 2 , u 3 satisfy Lam´ e’s equation v ′ + k − 2 h − n ( n + 1) s v ′′ + 1 � 1 1 1 � s + s − 1 + 4 s ( s − 1)( s − k − 2 ) v = 0 . s − k − 2 2 where n , h are separation parameters. Hans Volkmer Cyclidic harmonics

Internal and external sphero-conal harmonics Internal sphero-conal harmonics: G m n ( x , y , z ) = r n E m n ( s ) E m n ( t ) . where E m e polynomials. G m n are Lam´ n is a harmonic polynomial. External sphero-conal harmonics: H m n ( x , y , z ) = r − n − 1 E m n ( s ) E m n ( t ) . n is harmonic in R 3 \ { 0 } . H m Hans Volkmer Cyclidic harmonics

Expansion of reciprocal distance Let r , r ′ ∈ R 3 be such that � r � < � r ′ � . Then ∞ n 1 � � C m n G m n ( r ) H m n ( r ′ ) . � r − r ′ � = n =0 m = − n Hans Volkmer Cyclidic harmonics

Toroidal coordinates Coordinates: 0 < σ < ∞ , − π < ψ, φ < π . sinh σ cos φ = cosh σ − cos ψ, x sinh σ sin φ y = cosh σ − cos ψ, sin ψ = cosh σ − cos ψ. z Coordinate surfaces: (1 + x 2 + y 2 + z 2 ) 2 = 4( x 2 + y 2 ) coth 2 σ tori 1 ( z − cot ψ ) 2 + x 2 + y 2 = spherical bowls sin 2 ψ, planes x sin φ = y cos φ. Hans Volkmer Cyclidic harmonics

Toroidal coordinates Hans Volkmer Cyclidic harmonics

Separation of variables Laplace equation: � sinh σ u σ � � sinh σ u ψ � u φφ + + (cosh σ − cos ψ ) sinh σ = 0 . cosh σ − cos ψ cosh σ − cos ψ σ ψ Separation of variables: � u = cosh σ − cos ψ u 1 ( σ ) u 2 ( ψ ) u 3 ( φ ) , m 2 � � 1 n 2 − 1 � ′ − sinh σ u ′ � 4 + u 1 = 0 , sinh 2 σ 1 sinh σ u ′′ 2 + n 2 u 2 = 0 , u ′′ 3 + m 2 u 3 = 0 . where m , n are separation parameters. Hans Volkmer Cyclidic harmonics

Internal and external toroidal harmonics Internal toroidal harmonics: m , n ∈ Z 2 (cosh σ ) e in ψ e im φ , G m � cosh σ − cos ψ Q m n ( x , y , z ) = n − 1 where Q is the associated Legendre function. These are harmonic functions in R 3 except for the z -axis. External sphero-conal harmonics: H m � cosh σ − cos ψ P m 2 (cosh σ ) e in ψ e im φ . n ( x , y , z ) = n − 1 These are harmonic function in R 3 except for the unit circle z = 0, x 2 + y 2 = 1. Hans Volkmer Cyclidic harmonics

Expansion of reciprocal distance Let r , r ′ ∈ R 3 be such that σ ′ < σ . Then ∞ ∞ π ( − 1) m Γ( n − m + 1 1 1 2 ) � � 2 ) G m n ( r ) H m n ( r ′ ) . � r − r ′ � = Γ( n + m + 1 n = −∞ m = −∞ Hans Volkmer Cyclidic harmonics

Stereographic projection Let x 0 , x 1 , x 2 , x 3 be cartesian coordinates in R 4 . We consider the stereographic projection P : S 3 \ { 1 , 0 , 0 , 0) } → R 3 given by 1 P ( x 0 , x 1 , x 2 , x 3 ) = ( x 1 , x 2 , x 3 ) . 1 − x 0 The inverse map is 1 x 2 + y 2 + z 2 + 1( x 2 + y 2 + z 2 − 1 , 2 x , 2 y , 2 z ) . P − 1 ( x , y , z ) = Example: The intersection of the quadric hypersurface 2 = tanh 2 σ x 2 1 + x 2 with S 3 is mapped to the torus (1 + x 2 + y 2 + z 2 ) 2 = 4( x 2 + y 2 ) coth 2 σ. The stereographic projection maps a quadric surface to a cyclidic surface. Hans Volkmer Cyclidic harmonics

Stereographic projection and harmonic functions Theorem. Let D be an open subset of S 3 not containing (1 , 0 , 0 , 0), let E = { ( rx 0 , rx 1 , rx 2 , rx 3 ) : r > 0 , ( x 0 , x 1 , x 2 , x 3 ) ∈ D } , and let F = P ( D ) be the stereographic image of D . Let the function U : E → R be homogeneous of degree − 1 2 or − 3 2 , and let w : F → R satisfy U = w ◦ P on D . Then U is harmonic on E if and only if w ( x , y , z )( x 2 + y 2 + z 2 + 1) − 1 / 2 is harmonic on F . Hans Volkmer Cyclidic harmonics

Sphero-conal coordinates on R 4 Coordinates: r > 0, a 0 < s 1 < a 1 < s 2 < a 2 < s 3 < a 3 , � 4 i =1 ( s i − a j ) x 2 j = r 2 . � 4 j � = i =0 ( a i − a j ) Coordinate surfaces: r 2 = x 2 0 + x 2 1 + x 2 2 + x 2 3 and 4 x 2 j � = 0 for i = 1 , 2 , 3 . s i − a j j =0 Sphero-conal coordinates are orthogonal. Hans Volkmer Cyclidic harmonics

Separation of variables in sphero-conal coordinates Assume a harmonic function of the form U ( x 0 , x 1 , x 2 , x 3 ) = w 0 ( r ) w 1 ( s 1 ) w 2 ( s 2 ) w 3 ( s 3 ) . Then 0 + 4 0 + 4 λ 0 w ′′ r w ′ r 2 w 0 = 0 and, for w = w 1 , w 2 , w 3 , � 2   3 3 �  w ′′ + 1 1 � � w ′  + � λ i s 2 − i ( s − a j ) w = 0 . 2 s − a j j =0 j =0 i =0 This is a Fuchsian equation with five regular singular points a 0 , a 1 , a 2 , a 3 , ∞ with exponents 0 and 1 / 2 at a 0 , a 1 , a 2 , a 3 . The separation parameters are λ 0 , λ 1 , λ 2 . Hans Volkmer Cyclidic harmonics

Five-cyclide coordinate system on R 3 Sphero-conal coordinates s 1 , s 2 , s 3 form a coordinate system for the intersection of the hypersphere S 3 with the positive cone in R 4 . Using the stereographic projection P we project these coordinates to R 3 . We obtain an orthogonal coordinate system for the set T = { ( x , y , z ) : x , y , z > 0 , x 2 + y 2 + z 2 > 1 } . Explicitly, x 1 x 2 x 3 x = , y = , z = , 1 − x 0 1 − x 0 1 − x 0 where � 3 i =1 ( s i − a j ) x 2 j = , j = 0 , 1 , 2 , 3 . � 3 j � = i =0 ( a i − a j ) Coordinate surfaces: ( x 2 + y 2 + z 2 − 1) 2 4 x 2 4 y 2 4 z 2 + + + = 0 . s − a 0 s − a 1 s − a 2 s − a 3 Hans Volkmer Cyclidic harmonics

Coordinate surface s 2 = const Hans Volkmer Cyclidic harmonics