Overview of the LieAlgebrasOfVectorFields Package - Maple Programming Help

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Overview of the LieAlgebrasOfVectorFields Package

 

Calling Sequence

Description

List of LieAlgebrasOfVectorFields Package Commands

List of LAVF object Methods

List of Distribution object Methods

List of VFPDO object Methods

Compatibility

Calling Sequence

LieAlgebrasOfVectorFields[command]( arguments )

command( arguments )

Description

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The LieAlgebrasOfVectorFields package provides a suite of commands and methods for the Lie equations which arise as the determining equations for symmetries of differential equations. The package attempts to extract information of Lie algebras of vector fields (LAVF) without having to solve its determining system. To be more specific, the package can extract information including both geometric and algebraic properties of a LAVF directly via its determining system.

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The package deals with the determining system that can be in either un-solved or fully-solved, or any forms in between (i.e. partially solved).

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The main object of this package, LAVF, represents a LAVF consisting of the determining system S and its formal vector field V where its components are (part of) dependent variables of S. Most methods are within the LAVF object including both algebraic & geometric, and covering both finite and infinite dimensional cases. See Overview of the LAVF object for more detail.

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The primary job of this package is to provide constructor for the LAVF object, then many methods become accessible once a LAVF object is constructed.

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The package also provides two other groups of commands for working with vector fields and linear homogeneous PDE. For more detail, see Overview of VectorField commands and Overview of Linear Homogeneous PDE commands.

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The package utilizes several existing Maple packages such as DEtools and PDEtools. In particular, the command DEtools[rifsimp] is used heavily by the package.

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Each command in the LieAlgebrasOfVectorFields package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

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As the underlying implementation of the LieAlgebrasOfVectorFields package is a module, it is also possible to use the form LieAlgebrasOfVectorFields:-command to access a command from the package. For more information, see Module Members.

List of LieAlgebrasOfVectorFields Package Commands

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These are the exports of this package:

LAVF

A constructor to create a LAVF object.

DisplayStructure

Display commutation relation structure in various forms.

Distribution

A constructor to create a Distribution object.

EliminationLAVF

Constructing a LAVF object by eliminating some dependent variables of DEs system

IsLinearizable

Checking if an ODE system can be transformed into a linear one.

SymmetryLAVF

Constructing a LAVF object for the symmetry determining equations of a given list of DEs.

VFPDO

A constructor to create a VFPDO object.

List of LAVF object Methods

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The following is a list of available methods for a LAVF object.

AdjointMatrix

AreCommuting

AreSame

AreSameSpace

Centraliser

Centre

CleanDependencies

Copy

DChange

DerivedAlgebra

DerivedSeries

GetDeterminingSystem

GetIDBasis

GetSpace

GetVectorField

Hypercentre

ImplicitForm

Intersection

InvariantCount

Invariants

IsAbelian

IsCommutative

IsFiniteType

IsFlat

IsIdeal

IsInvariant

IsLieAlgebra

IsNilpotent

IsPerfect

IsReductive

IsSemiSimple

IsSolvable

IsSubspace

IsTransitive

IsTrivial

 

KillingForm

KillingOrthogonal

KillingPolynomial

KillingRadical

LAVFSolve

LieProduct

LowerCentralSeries

NilpotentRadical

NilRadical

Normaliser

OrbitDimension

OrbitDistribution

ParametricDerivatives

ProjectToSpace

SetIDBasis

SolutionDimension

SolvableRadical

StructureCoefficients

StructureConstants

Transporter

UpperCentralSeries

VectorSpaceSum

 

 

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The following Maple builtins are extended to allow LAVF object: has, hastype, indets, type. See LAVF Object Overloaded Builtins for more detail

List of Distribution object Methods

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The following is a list of available commands in a Distribution object.

AreSameSpace

CauchyDistribution

Codimension

DChange

DerivedDistribution

Dimension

GetAnnihilator

GetSpace

GetVectorFields

Integrals

Intersection

IsIntegrable

IsInvariant

IsInvolutive

IsSubspace

IsTrivial

VectorSpaceSum

 

 

 

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The following Maple builtins functions are extended so that they work for a Distribution object: type, has, hastype, indets, convert. See Distribution Object Overloaded Builtins for more detail.

List of VFPDO object Methods

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The following is a list of available methods for a VFPDO object.

AreSameSpace

GetSystemCount

GetSpace

 

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A VFPDO object can also act as a differential operator. See VFPDO Object as Operator for more detail.

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The following Maple builtins are extended to allow VFPDO object: has, hastype, indets, type. See VFPDO Object Overloaded Builtins for more detail.

Compatibility

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The LieAlgebrasOfVectorFields package was introduced in Maple 2020.

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For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

LAVF (Object overview)

Distribution (Object Overview)

VFPDO (Object Overview)

Vector Field (Commands)

Linear Homogeneous PDE (Commands)