JetCalculus[Prolong] - prolong a jet space, vector field, transformation, or differential equation to a higher order jet space
Calling Sequences
Prolong(k)
Prolong(X, k)
Prolong(k)
Prolong( k)
Parameters
k - a non-negative integer
X - a vector field defined on a fiber bundle or the jet space of a fiber bundle
- a transformation, defined on a fiber bundle or the jet space of a fiber bundle
- a differential equation, defined in terms of standard jet space coordinates
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Description
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Let be a fiber bundle, with base dimension and fiber dimension and let be the -th jet bundle. The Prolong command will take a geometry object defined, either on or on and extend or lift that object to a higher order jet space . The lifting or prolongation procedures considered here require only algebraic operations and differentiations. There are 4 different types of prolongation which can be performed by the command Prolong.
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1. Prolongation of Jet Spaces. Suppose that the command DGsetup has been used to initialize a jet space . This means that the standard jet space coordinates , ..., are protected. The coordinate vector fields, coordinate 1-forms, and contact forms to order are initialized and protected. The command Prolong(k), where with extend these protections and definitions to order . The result is same as making a call to DGsetup to initialize the jet space but is slightly faster since Prolong command only needs to define and protect the coordinates,vectors and 1 -forms from order to
2. Prolongation of Vector Fields. Let be a vector field on We say that preserves the contact ideal on if for any contact form the Lie derivative is also a contact form. Let be a projectable, point, contact, evolutionary, total,or generalized vector field with values in the tangent space E. (See AssignVectorType for the definitions of these types of vector fields.) Then, for each , there is a unique vector field on which preserves the contact ideal on and which projects pointwise to This vector field Z is called the prolongation of to order . and is denoted by . The explicit formula for vector field prolongation is given below. The second calling sequence Prolong(X, k) computes the prolongation of the vector field to order
3. Prolongation of Transformations. Let and be two fiber bundles. We say that a transformation is a generalized contact transformation if for every contact form on , the pullback is a contact form on . Let be a projectable transformation, a point transformation, a contact transformation, a differential substitution or a generalized differential substitution. These maps are defined as mappings from tofor the appropriate values of (See AssignTransformationType for the definitions of these different types of transformations.) Then, for each , there is a unique generalized contact transformation which covers . This transformation is called the prolongation of to order and it denoted by .The third calling sequence Prolong(k) computes the prolongation of to order
4. Prolongation of Differential Equations. A system of -th order differential equations can defined as the zero set of a collection of functions . The th order prolongation of denote by is the system of ( +)-th order differential equations defined as the zero set of the functionsand all their total derivatives to order . The fourth calling sequence Prolong(Delta, k) computes the prolongation of a system of differential equations to order . Use the command DifferentialEquationData to convert a list of functionsinto a differential equation data structure that can be passed to the Prolong command. The result is a new differential equation data structure representing the prolongation of the differential equations.
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If a vector field, transformation or differential equation has been prolonged to a certain order using Prolong, then the prolonged objects may themselves be prolonged to a higher order using Prolong.
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The command Prolong is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form Prolong(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-Prolong(...).
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Details
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If is a generalized vector field on , then the -th prolongation of X is the vector field
where .
For further details see the either of the two books by P. J. Olver.
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Examples
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Example 1. Prolongation of Jet Spaces
Define the jet space where with coordinates
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Display the jet coordinates, the coordinate vector fields, the 1-forms, and the contact 1-forms.
E1 >
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E1 >
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E1 >
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E1 >
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Prolong the jet space to .
Again display the jet coordinates, the coordinate vector fields, the 1-forms, and the contact 1-forms.
E1 >
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E1 >
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E1 >
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E1 >
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Example 2. Prolongation of Vector Fields
Define the jet space where with coordinates .
E1 >
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Define an arbitrary point vector field on .
E2 >
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E2 >
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Prolong to order 1--this agrees with the standard prolongation formula found in all texts.
E2 >
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Define the infinitesimal generator for a rotation in the - plane.
E2 >
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Prolong to order 1.
E2 >
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Prolong to order to 2--we can achieve the same result by prolonging .
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E2 >
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Define the jet space , where with coordinates .
E2 >
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Define a vector field whose flow simultaneously scales the coordinates .
E3 >
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E3 >
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Example 3. Prolongation of Transformations
Define the jet space where with coordinates .
E3 >
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Define a projectable transformation from to .
F >
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Prolong phi1 to order 2.
E2 >
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Define a differential substitution from J to
E2 >
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Prolong phi2 to order 2.
E2 >
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Example 4. Prolongation of Differential Equations
Define a second order ode on (coordinates
E2 >
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Calculate the second order prolongation of DE1. Note that the list of jet variables to be solved for is also prolonged.
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Define a system of over-determined partial differential equations in 2 independent variables and 1 dependent variable .
E2 >
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E4 >
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The second prolongation of DE2 is an overdetermined system of Frobenius type--all the 4-th order derivatives of which can be solved for.
E4 >
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