EXERCISE 2.37 Toroidal coordinates (p, , ) are useful in situations in which it

Question

EXERCISE 2.37 Toroidal coordinates (p, , ) are useful in situations in which it is desired to describe movement relative to a reference circle, which is the case for mag- netohydrodynamic studies in the fusion reactor known as a tokamak. Let R be the radius of this reference circle. Then the transformation to Cartesian coordinates is x-(R + cos ) cos , y-(R + cos ) sin , z sin . Derive expressions for the unit vectors for this coordinate system and for the derivatives of the unit vectors with respect to each toroidal coordinate. Then obtain the toroidal coordinate expressions for velocity and acceleration. Exercise 2.37

Explanation / Answer

given toroidal coordinates rho, psi, R , theta

x = (R + rho*cos(psi))cos(theta)
y = (R + rho*cos(psi))sin(theta)
z = rho*sin(phi)

let a, b, c be unit vectors in the direction of rho, theta and psi
then
a = (Rcos(tehta) + rho*cos(psi)cos(theta) - Rcos(theta))i + (Rsin(theta) + rho*cos(psi)sin(theta) - Rsin(theta))j + (rho*sin(psi))k/|a|
a = rho(cos(psi)cos(theta))i + cos(psi)sin(theta)j + sin(psi)k)

b = -sin(theta)i + cos(theta)j
c = -cos(theta)sin(psi)i - ain(theta)sin(psi)j + cos(psi)k

hence
in toroidal unit coordiantes
r = (rho, theta, psi)

a = rho(cos(psi)cos(theta))i + cos(psi)sin(theta)j + sin(psi)k)
b = -sin(theta)i + cos(theta)j
c = -cos(theta)sin(psi)i - ain(theta)sin(psi)j + cos(psi)k

a' = rho'((cos(psi)cos(theta))i + cos(psi)sin(theta)j + sin(psi)k)) + rho*(-(cos(psi)sin(tehta)theta' + sin(psi)cos(theta)psi')i +(cos(tehta)cos(psi)theta' - sin(psi)sin(theta)*psi')j + cos(psi)*psi'k))

b' = -cos(theta)theta' i - sin(theta)tehta' j
c' = -(cos(theta)cos(psi)psi' - sin(theta)sin(psi)theta')i - (cos(theta)sin(psi)theta' + sin(theta)cos(psi)psi')j - sin(psi)psi' k

similiarly we can find a", b", c"
in toptoidal coordinates
velocity components are (rho*a' + rho'*a, theta'*b + theta*b', psi'*c + psi*c')
acceleration compoennets (rho*a" + 2rho'*a' + rho"*a, theta"*b + b"*tehta + 2*theta'b', psi"c + c"psi + 2c'psi')

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