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This procedure calculate the equations of motions for Euclidean space and Minkowski space  with help of the Jacobian matrix.

Procedures
Calculation the equation of motions for Euclidean space and Minkowski space

"EQM := proc(eq, g,xup,xa,xu , eta ,var)"

Calling Sequence
 

EQM(eq, g, xup, xa, xu, eta, var)

Parameters
 

parameterSequence

-

eq, g, xup, xa, xu, eta, var

eq

out

equation of motion

g

out

metric

xup

out

velocitiy vector

xa

in

position vector

xu

in

vector of the independet coortinates

eta

in

signature matrix for Minkowski space

var

in

independet variable
 
> 

``

 Procedur Code
 
> 

restart; with(linalg); EQM := proc (eq, g, xup, xa, xu, eta, var) local J, Jp, xdd, l, xupp, ndim; ndim := vectdim(xu); xup := vector(ndim); xupp := vector(ndim); for l to ndim do xup[l] := diff(xu[l](var), var); xupp[l] := diff(diff(xu[l](var), var), var) end do; J := jacobian(xa, xu); g := multiply(transpose(J), eta, J); g := map(simplify, g); Jp := jacobian(multiply(J, xup), xu); Jp := map(simplify, Jp); xdd := multiply(inverse(g), transpose(J), eta, Jp, xup); xdd := map(simplify, xdd); xdd := map(convert, xdd, diff); eq := vector(vectdim(xupp)); for l to ndim do eq[l] := xupp[l]+xdd[l] = 0 end do end proc
> 

``

Input

 
> 

xa := Vector(3, {(1) = R*sin(`ϕ`)*cos(`ϑ`), (2) = R*sin(`ϕ`)*sin(`ϑ`), (3) = R*cos(`ϕ`)}); xu := Vector(2, {(1) = `ϕ`, (2) = `ϑ`}); eta := Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1})

 

> 

EQM(eq, g, xup, xa, xu, eta, t):

Output EOM
 
> 

for i to vectdim(xu) do eq[i] end do;

diff(diff(`ϕ`(t), t), t)-cos(`ϕ`)*sin(`ϕ`)*(diff(`ϑ`(t), t))^2 = 0

  

diff(diff(`ϑ`(t), t), t)+2*cos(`ϕ`)*(diff(`ϑ`(t), t))*(diff(`ϕ`(t), t))/sin(`ϕ`) = 0

 (5.1)

Output Line-Element
 
> 

ds2 := expand(multiply(transpose(xup), g, xup));

(diff(`ϕ`(t), t))^2*R^2+(diff(`ϑ`(t), t))^2*R^2-(diff(`ϑ`(t), t))^2*R^2*cos(`ϕ`)^2

 (6.1)

Output Metric
 
> 

assume(cos(`ϕ`)^2 = 1-sin(`ϕ`)^2); g := map(simplify, g)

array( 1 .. 2, 1 .. 2, [( 2, 2 ) = (R^2*sin(`ϕ`)^2), ( 1, 2 ) = (0), ( 2, 1 ) = (0), ( 1, 1 ) = (R^2) ] )

 (7.1)
> 

``
> 

``

 

Download bsp_jacobi.mw

Procedures
Calculation the equation of motions for Euclidean space and Minkowski space

"EQM := proc(eq, g,xup,xa,xu , eta ,var)"

Calling Sequence
 

EQM(eq, g, xup, xa, xu, eta, var)

Parameters
 

parameterSequence

-

eq, g, xup, xa, xu, eta, var

eq

out

equation of motion

g

out

metric

xup

out

velocitiy vector

xa

in

position vector

xu

in

vector of the independet coortinates

eta

in

signature matrix for Minkowski space

var

in

independet variable
 
> 

``

 Procedur Code
 
> 

restart; with(linalg); EQM := proc (eq, g, xup, xa, xu, eta, var) local J, Jp, xdd, l, xupp, ndim; ndim := vectdim(xu); xup := vector(ndim); xupp := vector(ndim); for l to ndim do xup[l] := diff(xu[l](var), var); xupp[l] := diff(diff(xu[l](var), var), var) end do; J := jacobian(xa, xu); g := multiply(transpose(J), eta, J); g := map(simplify, g); Jp := jacobian(multiply(J, xup), xu); Jp := map(simplify, Jp); xdd := multiply(inverse(g), transpose(J), eta, Jp, xup); xdd := map(simplify, xdd); xdd := map(convert, xdd, diff); eq := vector(vectdim(xupp)); for l to ndim do eq[l] := xupp[l]+xdd[l] = 0 end do end proc
> 

``

Input

 
> 

t := x[0]/c; xa := Vector(4, {(1) = t, (2) = r*cos(`ϕ`), (3) = r*sin(`ϕ`), (4) = x[3]}); xu := Vector(4, {(1) = x[0], (2) = r, (3) = `ϕ`, (4) = x[3]}); eta := Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1})

 

> 

EQM(eq, g, xup, xa, xu, eta, tau):

Output EOM
 
> 

for i to vectdim(xu) do eq[i] end do;

diff(diff(x[0](tau), tau), tau) = 0

  

diff(diff(r(tau), tau), tau)-(diff(`ϕ`(tau), tau))^2*r = 0

  

diff(diff(`ϕ`(tau), tau), tau)+2*(diff(`ϕ`(tau), tau))*(diff(r(tau), tau))/r = 0

  

diff(diff(x[3](tau), tau), tau) = 0

 (5.1)

Output Line-Element
 
> 

ds2 := expand(multiply(transpose(xup), g, xup));

-(diff(x[0](tau), tau))^2/c^2+(diff(r(tau), tau))^2+(diff(`ϕ`(tau), tau))^2*r^2+(diff(x[3](tau), tau))^2

 (6.1)

Output Metric
 
> 

assume(cos(`ϕ`)^2 = 1-sin(`ϕ`)^2); g := map(simplify, g)

array( 1 .. 4, 1 .. 4, [( 3, 3 ) = (r^2), ( 3, 4 ) = (0), ( 4, 1 ) = (0), ( 1, 1 ) = (-1/c^2), ( 4, 3 ) = (0), ( 4, 2 ) = (0), ( 2, 2 ) = (1), ( 3, 2 ) = (0), ( 3, 1 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 1, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 4 ) = (1), ( 2, 1 ) = (0), ( 1, 3 ) = (0) ] )

 (7.1)
> 

``
> 

``

 

Download bsp_jacobi_minkowski.mw
differential-equation jacobian linalg metric

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