Computing necessary integrability conditions for planar parametrized homogeneous potentials Thierry Combot University of Burgundy, Dijon France In collaboration with Alin Bostan, INRIA Mohab Safey El Din, UPMC INRIA CNRS IUF Thierry Combot University of Burgundy, Dijon France Integrability conditions
Motion of a point mass under a force field: potential V , equations q 1 = − ∂ q 2 = − ∂ ¨ V ( q ) ¨ V ( q ) ∂ q 1 ∂ q 2 Liouville Arnold Theorem (1980): Integrability ⇔ quasi-periodic motion ⇔ exactly solvable system Integrability analysis: Darboux search for integrable systems (1901), Hietarinta list of integrable systems (1987). Thierry Combot University of Burgundy, Dijon France Integrability conditions
Problem V ∈ Q ( a )( q 1 , q 2 ) homogeneous in q 1 , q 2 with parameters a . Find all parameters values a such that this potential is integrable. Integrability ⇔ Regular motion ⇒ exceptional phenomenon! Our goal Algorithm Input: V ∈ Q ( a )( q 1 , q 2 ) homogeneous in q 1 , q 2 with parameters a . Output: Non-trivial necessary conditions on a for integrability of V . Thierry Combot University of Burgundy, Dijon France Integrability conditions
What is already done? Ziglin (1983): Theoretical conditions for integrability Morales-Ramis (1997): Explicit conditions for integrability Maciejewski-Przybylska (2000): Universal relations allowing to test these conditions in the parametrized case What remains to be done? handle automatically singular parameter’s values handle multiplicity of roots/poles and degenerated asymptotics precise when and what happens when the universal relation does not hold Thierry Combot University of Burgundy, Dijon France Integrability conditions
Our contributions Rewriting Morales-Ramis integrability conditions and Maciejewski-Przybylska relation under a simpler form Ability to deal with singular specializations of the parameters Complete automation of the computation of integrability conditions through Groebner bases Explicit application and discovery of new candidates for integrability Thierry Combot University of Burgundy, Dijon France Integrability conditions
Integrability conditions (Morales-Ramis): the eigenvalues of ∇ 2 V ( c ) at Darboux points, i.e. solutions of ∂ V ∂ V ( c 1 , c 2 ) = kc 1 ( c 1 , c 2 ) = kc 2 ∂ q 1 ∂ q 2 should belong to a (completely explicit) discrete set E k Universal relation : there exists a relation R between eigenvalues independent of parameters (Maciejewski-Przybylska) General strategy Compute the relation R . Solve R in E k , get finitely many solutions. For each solution, rewrite the conditions on eigenvalues as polynomial conditions on a . Thierry Combot University of Burgundy, Dijon France Integrability conditions
Step 1: Polar coordinates V = ( a 1 + a 2 + a 3 ) q 3 1 + i (3 a 1 − a 3 + a 2 ) q 2 1 q 2 + ( − 3 a 1 + a 2 + a 3 ) q 1 q 2 2 − i ( a 1 − a 2 + a 3 ) q 3 2 Write V in polar coordinates F ( z ) = a 1 z 3 + a 2 z + a 3 / z V = r 3 F ( e i θ ) For c = ( r cos θ, r sin θ ) a Darboux point, we have F ′ ( e i θ ) = 0 , F ( e i θ ) � = 0 6 , 3 − ( e i θ ) 2 F ′′ ( e i θ ) � � Sp ( ∇ 2 V ( c )) = = { 6 , λ } F ( e i θ ) Thierry Combot University of Burgundy, Dijon France Integrability conditions
Step 2: Computation of the relation The relation R reads λ − 3 = 1 1 − 1 � k 0 k ∞ λ eigenvalue where z → 0 a 0 z k 0 F ( z ) z →∞ a 0 z k ∞ F ( z ) ∼ ∼ 3 important parameters: number of eigenvalues p , asymptotic constants k 0 , k ∞ Parity of F allows simplifications: p is even, eigenvalues λ come by pairs Thierry Combot University of Burgundy, Dijon France Integrability conditions
Depending on specialization of a 1 , a 2 , a 3 , 8 possibilities p = 4 , k ∞ = 3 , k 0 = − 1 p = 2 , k ∞ = 1 , k 0 = − 1 p = 2 , k ∞ = 3 , k 0 = − 1 p = 2 , k ∞ = 3 , k 0 = 1 p = 0 , k ∞ = 1 , k 0 = 1 p = 0 , k ∞ = 1 , k 0 = 1 p = 0 , k ∞ = 3 , k 0 = 3 F = 0 Thierry Combot University of Burgundy, Dijon France Integrability conditions
Step 3: Solving the relation If p = 0, no integrability conditions. λ 1 − 3 = − 2 or − 4 2 3 or 2 If p = 2: 3 One solution in E 3 : λ 1 = 6 2 λ 2 − 3 = − 4 2 If p = 4: λ 1 − 3 + 3 To solve this equation in E 3 : bounding min( λ 1 , λ 2 ) min( λ 1 , λ 2 ) ≤ 3 ⇒ min( λ 1 , λ 2 ) ∈ { 0 , 1 } Recursively solve the equation with one unknown less ⇒ ( λ 1 , λ 2 ) = (0 , 0) Thierry Combot University of Burgundy, Dijon France Integrability conditions
Step 4: Conditions on a Rewriting these conditions as polynomial conditions on a If p = 2 , k ∞ = 3 , k 0 = 1, then a 3 = 0 , a 1 � = 0 , a 2 � = 0 (multiplicity & asymptotics) 3 − z 2 F ′′ ( z ) = 6 for F ′ ( z ) = 0 , F ( z ) � = 0 , z � = 0 (integrability) F ( z ) Second condition rewrites as polynomial divisibility 3 − z 2 F ′′ ( z ) � F ′ � � � numer | numer − 6 ⇒ a 3 = 0 F ( z ) F Thierry Combot University of Burgundy, Dijon France Integrability conditions
If p = 4 , k ∞ = 3 , k 0 = − 1, then a 3 � =0 , a 1 � =0 , a 2 � =0 , a 2 2 − 4 a 1 a 3 � =0(multiplicity & asymptotics) 3 − z 2 F ′′ ( z ) = 0 for F ′ ( z ) = 0 , F ( z ) � = 0 , z � = 0 (integrability) F ( z ) Second condition rewrites as polynomial divisibility 3 − z 2 F ′′ ( z ) � F ′ � � � numer | numer ⇒ a 2 = 0 F F ( z ) ⇒ The output is [ a 1 , a 2 ] , [ a 1 , a 3 ] , [ a 2 , a 3 ] , [ a 2 ] , [ a 3 ] Thierry Combot University of Burgundy, Dijon France Integrability conditions
Implementation in Maple 17. Main tool: Elimination in polynomial ideals through Fgb Testing the condition on eigenvalues ⇔ Euclidean division of uni- variate polynomials Thierry Combot University of Burgundy, Dijon France Integrability conditions
The collinear three body problem: Three bodies on the line interacting by gravity. After reduction, potential of the form 1 1 1 V = + + a 1 q 1 + a 2 q 2 a 3 q 1 + a 4 q 2 a 5 q 1 + a 6 q 2 Thanks to the algorithm Theorem The collinear three body problem with non zero masses is non-integrable. Thierry Combot University of Burgundy, Dijon France Integrability conditions
Conclusion Other integrability conditions rely on eigenvalues and higher order derivatives of V ⇒ these can be easily included in the algorithm Most of the computation time on the elimination ⇒ search a better representation to improve times Implementation of additional conditions ⇒ probably only zero di- mensional ideal would appear ⇒ improved times Thierry Combot University of Burgundy, Dijon France Integrability conditions
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