Toward Formal Simplification of Parametric Algebraic Equations via their Lie Point Symmetries Alexandre Sedoglavic Projet ALIEN, INRIA Futurs & LIFL UMR 8022 CNRS, USTL Alexandre.Sedoglavic@lifl.fr June 2006
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Outline Introduction An introductory example Main contribution Lie symmetries Algebraic framework Determining system Rewriting original system in an invariant coordinates set Lie algebra, associated automorphisms and invariants Application of these results Conclusion Purely algebraic systems Using the structure of Lie algebra?
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Outline Introduction An introductory example Main contribution Lie symmetries Algebraic framework Determining system Rewriting original system in an invariant coordinates set Lie algebra, associated automorphisms and invariants Application of these results Conclusion Purely algebraic systems Using the structure of Lie algebra?
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Verhulst’s model Population growth with linear predation � d x d t = x ( a − bx ) − cx , (1) d a d t = d b d t = d c d t = 0 . model’s description • x ( t ) represents a species population at time t ; • d x / d t is its change rate; • ( a − bx ) is a per capita birth rate where a denotes the fertility rate; • b denotes environment carrying capacity; • • c is a predation rate.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Verhulst’s model Population growth with linear predation � d x d t = x ( a − bx ) − cx , (1) d a d t = d b d t = d c d t = 0 . model’s characteristics (modeling standpoint) • only the difference ( a − c ) between fertility and predation rate is significant. These parameters should be lumped together; • models should be expressed in dimensionless form • used units in the analysis are then unimportant; • adjectives small and large have a definite relative meaning; • the number of relevant parameters is reduces to dimensionless groupings that determine the dynamics; using dimensional analysis.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Verhulst’s model Population growth with linear predation � d x d t = x ( a − bx ) − cx , (1) d a d t = d b d t = d c d t = 0 . model’s characteristics (modeling standpoint) • only the difference ( a − c ) between fertility and predation rate is significant. These parameters should be lumped together; • models should be expressed in dimensionless form • used units in the analysis are then unimportant; • adjectives small and large have a definite relative meaning; • the number of relevant parameters is reduces to dimensionless groupings that determine the dynamics; using dimensional analysis.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Verhulst’s model Population growth with linear predation � d x d t = x ( a − bx ) − cx , (1) d a d t = d b d t = d c d t = 0 . model’s characteristics (Lie point symmetries standpoint) • one parameter group of translations � t → t a → a + λ T λ : c → c + λ x → x b → b • 2-parameters group of scalings � t → t /λ 2 a → λ 2 a S λ 1 ,λ 2 : c → λ 2 c x → λ 1 x b → λ 2 b /λ 1
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Verhulst’s model Population growth with linear predation (canonical form) d x d t = x ( 1 − x ) . x = 1 is a stable fixed point (1 − 2 x < 0), x = 0 is unstable (1 − 2 x > 0).
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Verhulst’s model Population growth with linear predation (canonical form) d x d t = x ( 1 − x ) . x = 1 is a stable fixed point (1 − 2 x < 0), x = 0 is unstable (1 − 2 x > 0). Simplification: from original model to canonical one Canonical model is obtained after the change of variables: b t = ( a − c ) t , x = a − c x . Thus x = ( a − c ) / b is a stable fixed point of the original model if the equalities 0 < b < 2 ( a − c ) hold.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion These computations can be done in polynomial time w.r.t. inputs size Theorem (Hubert, Sedoglavic 2006) Let Σ be a differential system bearing on n state variables and depending on ℓ parameters that is coded by a straight-line program of size L. There exists a probabilistic algorithm that determines if a Lie point symmetries group of Σ composed of dilatation and translation exists; in that case, a rational set of invariant coordinates is computed and Σ is rewrite in this set with a reduced number of parameters. The arithmetic complexity of this algorithm is bounded by � �� � O ( n + ℓ + 1 ) L + ( n + ℓ + 1 )( 2 n + ℓ + 1 ) .
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Simple application of Lie’s theory If one • considers parameters θ as constant variables d θ/ d t = 0 i.e. considers extended Lie symmetries; • uses classical Lie’s theory i.e. Lie symmetries and their invariants; • knows what type of symmetries could be used, one can • unify (and extend?) available simplification methods that seem (to me) based on rules of thumb; • obtain simple effective and efficient algorithmic tools.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Simple application of Lie’s theory If one • considers parameters θ as constant variables d θ/ d t = 0 i.e. considers extended Lie symmetries; • uses classical Lie’s theory i.e. Lie symmetries and their invariants; • knows what type of symmetries could be used, one can • unify (and extend?) available simplification methods that seem (to me) based on rules of thumb; • obtain simple effective and efficient algorithmic tools.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Simple application of Lie’s theory If one • considers parameters θ as constant variables d θ/ d t = 0 i.e. considers extended Lie symmetries; • uses classical Lie’s theory i.e. Lie symmetries and their invariants; • knows what type of symmetries could be used, one can • unify (and extend?) available simplification methods that seem (to me) based on rules of thumb; • obtain simple effective and efficient algorithmic tools.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Simple application of Lie’s theory If one • considers parameters θ as constant variables d θ/ d t = 0 i.e. considers extended Lie symmetries; • uses classical Lie’s theory i.e. Lie symmetries and their invariants; • knows what type of symmetries could be used, one can • unify (and extend?) available simplification methods that seem (to me) based on rules of thumb; • obtain simple effective and efficient algorithmic tools.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Simple application of Lie’s theory If one • considers parameters θ as constant variables d θ/ d t = 0 i.e. considers extended Lie symmetries; • uses classical Lie’s theory i.e. Lie symmetries and their invariants; • knows what type of symmetries could be used, one can • unify (and extend?) available simplification methods that seem (to me) based on rules of thumb; • obtain simple effective and efficient algorithmic tools.
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Outline Introduction An introductory example Main contribution Lie symmetries Algebraic framework Determining system Rewriting original system in an invariant coordinates set Lie algebra, associated automorphisms and invariants Application of these results Conclusion Purely algebraic systems Using the structure of Lie algebra?
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Representation of point transformation Vector field representation d z 1 / d ε = g 1 ( z 1 , . . . , z n ) , . . . d z n / d ε = g n ( z 1 , . . . , z n ) . Power series representation z 1 ( 0 ) + g 1 ( z 1 , . . . , z n ) ε + O ( ε 2 ) , z 1 ( ε ) = . . . z n ( 0 ) + g n ( z 1 , . . . , z n ) ε + O ( ε 2 ) . z n ( ε ) =
Introduction Lie symmetries Rewriting original system in an invariant coordinates set Conclusion Representation of point transformation Infinitesimal representation i.e. derivations acting on the field K ( z 1 , . . . , z n ) n g i ( z 1 , . . . , z n ) ∂ � δ = . ∂ z i i = 1 i ∈ N δ i ( z ) / i ! Closed form representation (if any) σ ( z ) = � z 1 → ζ 1 ( z 1 , . . . , z n ) , . . σ . z n → ζ n ( z 1 , . . . , z n ) .
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