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Equations of motions for Euclidean and ...

Posted: panke 145 Product: Maple 13

March 06 2016

2 10

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

	Calling Sequence

Parameters

parameterSequence	-
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

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})

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

(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

	Calling Sequence

Parameters

parameterSequence	-
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

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})

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

-(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) ] )