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tensor

  

geodesic_eqns

  

generate the Euler-Lagrange equations for the geodesic curves

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

geodesic_eqns(coord, param, Cf2)

Parameters

coord

-

list of coordinate names

param

-

name of the variable to parametrize the curves with

Cf2

-

Christoffel symbols of the second kind

Description

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][GeodesicEquations] and Physics[Geodesics] instead.

• 

The function geodesic_eqns(coord, Tau, Cf2) generates (but does not solve) the Euler-Lagrange equations of the geodesics for a metric with Christoffel symbols of the second kind Cf2 and coordinate variables coord.  The equations are written in terms of the coordinate variable names as functions of the given parameter Tau. They are returned in the format of a list of equations.

• 

Cf2 should be indexed using the cf2 indexing function provided by the tensor package.  It can be computed using the Christoffel2 routine.

Examples

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][GeodesicEquations] and Physics[Geodesics] instead.

withtensor:

Determine the geodesic equations for the Poincare half-plane. The coordinates are:

coordu,v

coordu,v

(1)

The metric is:

g_comptsarraysymmetric,sparse,1..2,1..2,1,1=1v2,2,2=1v2:

gcreate1,1,evalg_compts

gtablecompts=1v2001v2,index_char=−1,−1

(2)

ginvinvertg,detg:

d1gd1metricg,coord:d2gd2metricd1g,coord:

Cf1Christoffel1d1g:

Cf2Christoffel2ginv,Cf1:

displayGRChristoffel2,Cf2

The Christoffel Symbols of the Second Kind

non-zero components :

{1,12}=1v

{2,11}=1v

{2,22}=1v

(3)

Now generate the geodesic equations:

eqnsgeodesic_eqnscoord,t,Cf2

eqnsⅆ2ⅆt2ut2ⅆⅆtutⅆⅆtvtv=0,ⅆ2ⅆt2vt+ⅆⅆtut2vⅆⅆtvt2v=0

(4)

How about Euclidean 3-space in Cartesian coordinates?

coordx,y,z

coordx,y,z

(5)

g_comptsarraysymmetric,sparse,1..3,1..3,1,1=1,2,2=1,3,3=1:

gcreate1,1,evalg_compts

gtablecompts=100010001,index_char=−1,−1

(6)

ginvinvertg,detg:

d1gd1metricg,coord:d2gd2metricd1g,coord:

Cf1Christoffel1d1g:

Cf2Christoffel2ginv,Cf1:

displayGRChristoffel2,Cf2

The Christoffel Symbols of the Second Kind

non-zero components :

None

(7)

eqnsgeodesic_eqnscoord,t,Cf2

eqnsⅆ2ⅆt2xt=0,ⅆ2ⅆt2yt=0,ⅆ2ⅆt2zt=0

(8)

mapeval,subsxt=at+b,yt=ct+e,zt=ft+h,eqns

0=0

(9)

and in spherical-polar coordinates?

coordr,θ,φ

coordr,θ,φ

(10)

The metric is:

g_comptsarraysymmetric,sparse,1..3,1..3,1,1=1,2,2=r2,3,3=r2sinθ2:

gcreate1,1,evalg_compts

gtablecompts=1000r2000r2sinθ2,index_char=−1,−1

(11)

ginvinvertg,detg:

d1gd1metricg,coord:d2gd2metricd1g,coord:

Cf1Christoffel1d1g:

Cf2Christoffel2ginv,Cf1:

displayGRChristoffel2,Cf2

The Christoffel Symbols of the Second Kind

non-zero components :

{1,22}=r

{1,33}=rsinθ2

{2,12}=1r

{2,33}=sinθcosθ

{3,13}=1r

{3,23}=cosθsinθ

(12)

Now generate the geodesic equations:

eqnsgeodesic_eqnscoord,t,Cf2

eqnsⅆ2ⅆt2φt+2ⅆⅆtrtⅆⅆtφtr+2cosθⅆⅆtθtⅆⅆtφtsinθ=0,ⅆ2ⅆt2rtrⅆⅆtθt2rsinθ2ⅆⅆtφt2=0,ⅆ2ⅆt2θt+2ⅆⅆtrtⅆⅆtθtrsinθcosθⅆⅆtφt2=0

(13)

See Also

DifferentialGeometry[Tensor][GeodesicEquations]

dsolve

Physics[Geodesics]

tensor(deprecated)

tensor(deprecated)[Christoffel2]

tensor(deprecated)[indexing]