Overview of the JetCalculus package

Description


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Jet spaces play a fundamental role in the geometric approach to the calculus of variations and to differential equations. The JetCalculus package is a specialized package for symbolic computations on jet spaces and is fully compatible with the other packages and commands in DifferentialGeometry.

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This package contains commands for prolonging both vector fields and transformations to jet spaces and for calculating EulerLagrange equations for variational problems with any number of independent and dependent variables and any number of derivatives.

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Jet spaces admit a very important generalization of the de Rham complex which is called the variational bicomplex  so named because one of the differentials in the variational bicomplex can be identified with the EulerLagrange operator for the calculus of variations. The JetCalculus packages provides the full functionality needed for computations within the theoretical framework of the variational bicomplex, including the horizontal and vertical exterior derivative operators, the associated homotopy operators and the integration by parts operator.

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This package can be used in conjunction with the PDEtools package for the systematic analysis of symmetries of differential equations, for the study of conservation laws and integrable evolution equations, for the study of invariant variational problems, and for the inverse problem to the calculus of variations. It will also be of interest to those working in the areas of integrable systems and exterior differential systems.

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The JetCalculus package is a subpackage of DifferentialGeometry. Each command in the JetCalculus package can be accessed by using either the long form or the short form of the command name in the command calling sequence.



List of the JetCalculus commands


The following is a list of available commands
A brief description of the subpackage's commands is as follows
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AssignVectorType: assign a type (projectable, point, contact, ...) to a vector.

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EulerLagrange: calculate the EulerLagrange equations for a Lagrangian.

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Noether: find the conservation law for the EulerLagrange equations from a given symmetry of the Lagrangian

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ProjectedPullback: pullback a differential biform of type (r, s) by a transformation to a differential biform of type (r, s).

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Prolong: prolong a jet space, vector field, transformation, or differential equation to a higher order jet space.

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TotalDiff: take the total derivative of an expression, a differential form or a contact form.

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TotalJacobian: find the Jacobian of a transformation using total derivatives.

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VerticalHomotopy: apply the vertical homotopy operator to a biform on a jet space.

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ZigZag: lift a dHclosed form on a jet space to a dclosed form.







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