ReLie: A Reduce Program for Lie Group Analysis of Differential Equations
Abstract
:1. Introduction
2. Basic Elements of the Theory
2.1. Lie Point Symmetries
2.2. Q-Conditional Symmetries
2.3. Contact Transformations
2.4. Variational Symmetries
2.5. Approximate Symmetries
2.6. Equivalence Transformations
2.7. Lie Remarkable Equations
3. The Program ReLie
in ``relie.red’’ $
3.1. Computing Lie Point Symmetries of Differential Equations
- the list diffeqs of the left-hand sides of the differential equations to be studied which are assumed with zero right-hand sides;
- the list leadders of some derivatives appearing in the differential equations: When computing the invariance conditions, the elements in the list leadders are removed by solving the differential equations with respect to them.
- 1.
- the first one is a list of conditions that are still unsolved (in this case the list is empty since no condition remains unsolved);
- 2.
- the second one is a list giving the solution to the determining equations, i.e., the expressions of the infinitesimals;
- 3.
- the third one is a list containing the parameters involved in the solution (in this casek_1andk_2);
- 4.
- the fourth one is a list of expressions which can not vanish (in this case the list is empty).
- relie(1) is equivalent to calling relieinit();
- relie(2) is equivalent to calling in sequence relieinit() and relieinv();
- relie(3) is equivalent to calling in sequence relieinit(), relieinv() and reliedet();
- relie(4) is equivalent to calling in sequence relieinit(), relieinv(), reliedet() and reliesolve().
3.1.1. An Example of Group Classification
3.1.2. Commutator Table
3.2. Computation of Conditional Symmetries
- {1, xi_x2, xi_x3, eta_u, eta_v} if nonclassical = 1;
- {0, 1, xi_x3, eta_u, eta_v} if nonclassical = 2;
- {0, 0, 1, eta_u, eta_v} if nonclassical = 3.
- qcond:={1,2} if the invariant surface conditions of both dependent variables have to be used;
- qcond:={1} if the invariant surface condition of the first dependent variable has to be used;
- qcond:={2} if the invariant surface condition of the second dependent variable has to be used.
3.3. Computation of Contact Symmetries
3.4. Computation of Variational Symmetries and Associated Conservation Laws
3.5. Computation of Approximate Symmetries
3.6. Computation of Equivalence Transformations
- the list arbelem of the arbitrary elements involved in the differential equations;
- the integer arborder denoting the highest order of the derivatives of the arbitrary elements with respect to their arguments;
- the integer zorder characterizing the variables the arbitrary elements depend on; for instance, if zorder is 0, then the arbitrary elements depend at most on the independent and dependent variables; if zorder is 1, then the arbitrary elements depend at most on the independent, dependent variables and first order derivatives, ….
3.7. Inverse Lie Problem
4. Inside ReLie: Global Variables and Routines
4.1. Input Variables
- approxorder: Maximum order of approximate symmetries of equations containing a small parameter; by default it is 0, i.e., exact symmetries; the small parameter involved in the approximate symmetries must be denoted by epsilon; when looking for approximate symmetries the user has to define the rule let epsilon**(approxorder+1)=0;
- arbelem: List of the arbitrary elements (only for equivalence transformations); by default it is an empty list;
- arborder: Maximum order of derivatives of arbitrary elements (only for equivalence transformations); by default it is , i.e., point symmetries;
- contact: Set to 1 for contact symmetries; by default it is 0;
- diffeqs: List of the left-hand sides of differential equations (with vanishing right-hand sides);
- freepars: List of arbitrary constants or functions involved in the differential equations (for group classification problems); by default it is an empty list;
- generalequiv: Set to 1 for general equivalence transformations where all infinitesimals depend on independent and dependent variables and arbitrary elements; the default value is 0, meaning that the infinitesimals of independent and dependent variables do not depend upon the arbitrary elements;
- jetorder: Maximum order of derivatives in differential equations;
- lagrangian: A list with only one element corresponding to the Lagrangian (it is necessary to set variational to 1);
- leadders: List of the leading derivatives; diffeqs are solved with respect to them;
- nonclassical: Set to a value between 1 and the number of independent variables (only for conditional symmetries); by default it is 0, i.e., classical symmetries;
- nonpolyders: List of derivatives not occurring in polynomial form in the differential equations; by default it is an empty list;
- nonzeropars: List of arbitrary constants or functions involved in the differential equations that can not vanish; by default it is an empty list;
- qcond: List of indexes of dependent variables whose invariant surface conditions have to be used for computing conditional symmetries;
- uvar: List of the dependent variables;
- variational: Set to 1 if variational symmetries of a Lagrangian are needed; by default it is 0;
- xvar: List of the independent variables;
- zorder: Maximum order of derivatives of uvar with respect to xvar the elements in arbelem depend on (only for equivalence transformations); if zorder is set to 0, the arbitrary elements depend on xvar and uvar; zorder cannot exceed jetorder.
4.2. Output Variables
- allinfinitesimals: List of two lists; the first sublist is the list of the infinitesimals, in order, of the independent variables, dependent variables and arbitrary elements (the latter in the case of equivalence transformations); the second sublist is the list of various terms (constants and functions) involved in the expression of infinitesimals;
- allminors: List of minors of a given order extracted from a matrix; returned by the function minors(m,k), where m is a matrix and k a positive integer, or by the function inverselie(k) that takes the jetorder-th distribution of generators as the matrix from which the minors of order k are extracted;
- arbconst: List of arbitrary constants involved in the expression of the symmetries;
- arbfun: List of arbitrary functions involved in the symmetries;
- cogenerators: List of the functions entering the definition of variational symmetries and corresponding to the infinitesimal generators (produced by reliegen());
- commtable: Table of commutators of a list of vector fields;
- deteqs: List of the determining equations (produced by reliedet());
- distribution: Matrix of the jetorder-th distribution of a list of generators (produced by reliedistrib()), i.e., a matrix where each row is the prolonged vector field evaluated on one of the provided infinitesimal generators;
- fluxes: List of the fluxes of the conservation law corresponding to a Lie generator (computed by relieclaw());
- generators: List of the infinitesimal generators of the finite Lie algebra admitted by the differential equations at hand (produced by reliegen()); the list generators may also been obtained by calling generatealgebra(k), where k can be 1 (algebra of isometries), 2 (algebra of affine transformations) or 3 (algebra of projective transformations); of course, it is necessary to set jetorder, xvar and uvar before calling generatealgebra(k);
- invcond: List of the invariance conditions of the differential equations at hand (produced by relieinv());
- nzcomm: List of non-zero commutators of a list of vector fields;
- prolongation: List of two lists: The first one is the list of the coordinates of the jet space, the second one the list of the corresponding infinitesimals (produced by relieprol());
- splitsymmetries: List of lists: The k-th element is a list containing the infinitesimals corresponding to the k-th element of generators (produced by reliegen()); the list splitsymmetries may also been obtained by calling generatealgebra(k), where k can be 1 (algebra of isometries), 2 (algebra of affine transformations) or 3 (algebra of projective transformations); of course, it is necessary to set jetorder, xvar and uvar before calling generatealgebra(k); the list splitsymmetries is used internally by the functions reliedistrib(), inverselie() and testrank();
- symmetries: List of the infinitesimals admitted by the differential equations at hand (produced by reliesolve()).
4.3. Intermediate Variables
- jet: List of three lists: Indices for computing the infinitesimals and their prolongations, coordinates of jet space and their internal representation;
- jetapprox: List of two lists: List of independent variables and expansions of dependent variables and their derivatives, and list of their internal representation (only for approximate symmetries);
- jetequiv: List of three lists: Indices for computing the infinitesimals and their prolongations for arbitrary elements, arbitrary elements, and their internal representation (only for equivalence transformations);
- jetsplit: List of two lists: Indices for computing the infinitesimals and their prolongations, list of independent variables, zeroth order dependent variables and their derivatives (only for approximate symmetries);
- solutiondedv: Solution of the differential equations specified in diffeqs with respect to leadders; for conditional symmetries, the invariant surface conditions and their needed differential consequences are solved too;
- steprelie: Integer that stores the status of the computation; 0: No computation done; 1: relieinit() has been called; 2: relieinv() has been called; 3: reliedet() has been called; 4: reliesolve() has been called;
- zvar: List of two lists: The first one is the list of the variables arbelem depend on, the second one the corresponding infinitesimals (only for equivalence transformations).
4.4. Functions
- abelian(gens): Checks if the generators gens span an Abelian Lie algebra;
- commutatortable(gens): Returns the commutator table of the generators gens;
- essentialpars(gens,vars): Takes a list of generators gens of a multiparameter Lie group of transformations for the variables vars, and returns the generators which are not linearly independent;
- generatealgebra(k): Once jetorder, xvar and uvar have been properly assigned, this function returns a list of generators spanning the algebra of isometries (for k = 1), affine algebra (for k = 2), projective algebra (for k = 3);
- inverselie(k): Computes all the minors of order k of the jetorder-th distribution generated by the list of vector fields contained in generators;
- liebracket(gen1,gen2): Returns the Lie bracket of the generators gen1 and gen2;
- newordering(lis,ind): Returns a list of the elements in lis reordered according to the permutation ind of the integers , where n is the length of list lis;
- nonzerocommutators(gens): Returns nzcomm, a list of non-zero commutators of generators gens;
- offprintcrack(): Prevents reliesolve() to display the steps needed for solving determining equations (this is the default configuration);
- onprintcrack(): Sets a variable used in CRACK package (in turn used in the function reliesolve()) in order to display the steps needed for solving determining equations;
- relieclaw(k): Returns fluxes, a list of the components of the fluxes of the conservation law corresponding to the k-th Lie generator (obtained after calling reliegen());
- reliedet(): Splits the invariant conditions providing the list deteqs of determining equations;
- reliedistrib(): Returns the matrix distribution, i.e., a matrix whose rows are the prolonged vector fields evaluated in the list splitsymmetries (computed by the function reliegen(), or by the function generatealgebra(), or suitably assigned by the user);
- reliegen(k,lis): Returns the list generators; k is an integer (less or equal to the length of symmetries) and lis a list that can be empty; if lis is made by as many values as the number of arbitrary constants occurring in symmetries, generators consists of a list of vector fields, where each vector field is obtained replacing the i-th parameter by the i-th element in the list lis (or 1 if lis is empty or its length is different from the number of arbitrary constants entering symmetries) and the other parameters are replaced by 0; if the list lis is {-1}, then generators is a list with only one element containing the components of the infinitesimals in their general form, i.e., the linear combinations of all admitted generators; the function produces also the list splitsymmetries whose k-th element is a list containing the infinitesimals corresponding to the k-th element of generators;
- relieinit(): If input data have been correctly defined, the function initializes the objects for doing the computation;
- relieinv(): Computes the invariance conditions; returns invcond;
- relieprol(): Returns the prolongation of a general vector field;
- reliesolve(): Solves the determining equations for the infinitesimals, and returns the list symmetries; in group classification problems (but also in the case of conditional symmetries), the list symmetries may contain different solutions for the infinitesimals according to the values of freepars; as a default reliesolve() does not display the steps made to obtain the solution of determining equations; the user can see these steps by calling onprintcrack(); this is suggested when reliesolve() seems to use too much time to complete its execution; the list symmetries contains a list of the sets of solutions of determining equations; each element of this list in turn is a list of four elements: The first one is a list of conditions (possibly empty) that remained unsolved; the second one is a list giving the solution to the determining equations, i.e., the expressions of the infinitesimals; the third one is a list containing the parameters involved in the solution; the fourth one is a list of expressions which can not vanish (this list can be empty);
- solvable(gens): Checks if the generators gens span a solvable Lie algebra;
- testrank(gens): Returns the rank of the jetorder-th distribution generated by the generators gens.
- allcoeffs(lis1,lis2): Returns the list of coefficients of lis1 (list of polynomials) with respect to the variables in list lis2;
- bincoeff(n,k): Returns ;
- combnorep(n,k): Returns the combinations without repetition of k elements chosen in ;
- combrep(n,k): Returns the combinations with repetition of k elements chosen in ;
- delzero(lis): Returns a list containing all non-zero elements in the list lis;
- dependence(lis1,lis2): Declares that the elements in the list lis1 depend on the variables in the list lis2;
- dlie(obj,var): Returns the usual Lie derivative of obj with respect to var;
- dlieapprox(obj,var): Returns the Lie derivative of obj with respect to var in the context of approximate symmetries;
- dliestar(obj,var): Returns the additional Lie derivative used for equivalence transformations;
- kroneckerdelta(k1,k2): Returns 1 if , 0 otherwise;
- letterlist(obj,n): obj is a symbol, n a positive integer;for instance, letterlist(x,4) builds the list ;
- letterlistvar(obj,lis): obj is a symbol, lis a list;for instance, letterlistvar(xi_,{x,y}) builds the list {xi_x,xi_y};
- listletter(lis,ch): lis is a list, ch a symbol;for instance, listletter({u_,v_},x) builds the list {u_x,v_x};
- membership(elem,lis): Returns the number of occurrences of the element elem in the list lis;
- minors(m,k): m is a matrix and k is a positive integer: Returns the list of minors of order k of the matrix m;
- nodependence(lis1,lis2): Removes the dependence of the objects in the list lis1 upon the variables in the list lis2;
- removeelement(lis,elem): Removes the element elem from the list lis;
- removemultiple(lis1,lis2): Removes from the list lis1 the elements of the list lis2;
- scalarmult(obj,lis): Returns a list whose k-th element is the product of the scalar obj and the k-th element of list lis;
- scalarproduct(lis1,lis2): Returns the sum of the products element by element of two lists with the same number of elements;
- sumlist(lis1,lis2): Returns a list summing element by element the two lists with the same length;
- zerolist(n): Returns a list of n zeros, the empty list if .
5. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Oliveri, F. ReLie: A Reduce Program for Lie Group Analysis of Differential Equations. Symmetry 2021, 13, 1826. https://doi.org/10.3390/sym13101826
Oliveri F. ReLie: A Reduce Program for Lie Group Analysis of Differential Equations. Symmetry. 2021; 13(10):1826. https://doi.org/10.3390/sym13101826
Chicago/Turabian StyleOliveri, Francesco. 2021. "ReLie: A Reduce Program for Lie Group Analysis of Differential Equations" Symmetry 13, no. 10: 1826. https://doi.org/10.3390/sym13101826