7 Curvilinear coordinates

Transcription

1 7 Curvilinear coordinates Read: Boas sec. 5.4, 0.8, Review of spherical and cylindrical coords. First I ll review spherical and cylindrical coordinate systems so you can have them in mind when we discuss more general cases. 7.. Spherical coordinates Figure : Spherical coordinate system. The conventional choice of coordinates is shown in Fig.. θ is called the polar angle, φ the azimuthal angle. The transformation from Cartesian coords. is x = r sin θ cos φ y = r sin θ sin φ z = r cos θ. () In the figure the unit vectors pointing in the directions of the changes of the three spherical coordinates r, θ, φ are also shown. Any vector can be expressed in terms of them: A = A xˆx + A y ŷ + A z ẑ = A rˆr + A θ ˆθ + Aφ ˆφ. (2) Note the qualitatively new element here: while both ˆx, ŷ, ẑ and ˆr, ˆθ, ˆφ are three mutually orthogonal unit vectors, ˆx, ŷ, ẑ are fixed in space but ˆr, ˆθ, ˆφ point in different directions according to the direction of vector r. We now ask by how large

2 a distance ds the head of the vector ˆr changes if infinitesimal changes dr, dθ, dφ are made in the three spherical directions: ds r = dr, ds θ = rdθ, ds φ = r sin θdφ, (3) as seen from figure 2 (only the ˆθ and ˆφ displacements are shown). So the total change is The volume element will be Figure 2: Geometry of infinitesimal changes of r. d s = drˆr + rdθˆθ + r sin θdφ ˆφ. (4) dτ = ds r ds θ ds φ = r 2 sin θ dr dθ dφ, (5) and the surface measure at constant r will be d a = ds θ ds φ = ˆr = r 2 sin θdθdφ ˆr. (6) Ex. : Volume of sphere of radius R: R π 2π dτ = r 2 sin θdr dθ dφ = (R 3 /3)(2)(2π) = 4 3 πr3. (7) sphere r=0 θ=0 φ=0 More interesting: gradient, etc. in spherical coordinates: ψ = x = r and î, ĵ, and ˆk can be replaced by x î + y ĵ + r x + θ z ˆk θ x + φ î = sin θ cos φ ˆr + cos θ cos φ ˆθ sin φ ˆφ φ, etc., (8) x ĵ = sin θ sin φ ˆr + cos θ sin φ ˆθ + cos φ ˆφ ˆk = cos θ ˆr sin θ ˆθ. (9) 2

3 Combining all these we find ψ = r ˆr + r θ ˆθ + r sin θ φ ˆφ. (0) Similarly, we find and A = ( ) r 2 A r 2 r + r r sin θ θ (sin θa θ) + A φ r sin θ φ A ˆr r ˆθ r sin θ ˆφ = r 2 sin θ r θ φ A r ra θ r sin θa φ () (2) and 2 ψ = ( r 2 ) + r 2 r r r 2 sin θ ( sin θ ) + θ θ 2 ψ r 2 sin 2 (3) θ φ Cylindrical coordinates I won t belabor the cylindrical coordinates, but just give you the results to have handy. I ve written here the cylindrical radial coordinate as called r, the angle variable θ, like Boas, but keep in mind that a lot of books use ρ and φ. x = r cos θ ; y = r sin θ ; z = z ds r = dr ; ds θ = rdθ ; ds z = dz d l = dr ˆr + rdθ ˆθ + dz ẑ dτ = rdrdθdz ψ = r ˆr + r θ ˆθ + z ẑ A = r r (ra r) + A θ r θ + A z z A = ˆr rˆθ ẑ r r θ z A r ra θ A z (4) 3

4 7..3 General coordinate systems With these specific examples in mind, let s go back the the general case, and see where all the factors come from. We can pick a new set of coordinates q, q 2, q 3, which have isosurfaces which need not be planes nor parallel to each other. Let s just assume that among x, y, z and q, q 2, q 3 there are some relations x = x(q, q 2, q 3 ), ; y = y(q, q 2, q 3 ) ; z = z(q, q 2, q 3 ) (5) which we can find and invert to get q = q (x, y, z) ; q 2 = q 2 (x, y, z) ; q 3 = q 3 (x, y, z) (6) The differentials are then dx = x q dq + x q 2 dq 2 + x q 3 dq 3 (7) It s very useful to know what the measure of distance, or metric, is in a given coordinate system. Of course in Cartesian coordinates, the distance between two points whose coordinates differ by dx, dy, dz is ds, where ds 2 = dx 2 + dy 2 + dz 2. (8) Your book calls ds the arc length. Now if you imagine squaring an equation like (7), you ll get terms like dq 2, but also terms like dq dq 2, etc. So in general, plugging into (8) we expect ds 2 = g dq 2 + g 2 dq dq = ij g ij dq i dq j, (9) and the g ij are called the metric components ( and g itself is the metric tensor). In Einstein s theory of general relativity, the metric components depend on the amount of mass nearby! Most of the coordinate systems we are interested in are orthogonal, i.e. g ij δ ij. Thus we can write ds 2 = (h dq ) 2 + (h 2 dq 2 ) 2 + (h 3 dq 3 ) 2. (20) The h i s are called scale factors, and are for Cartesian coordinates. Now let s look at the change of the position vector r, in our new coordinate system, when we change the coordinates q i by a small amount. We have d r = d s = h dq ˆq + h 2 dq 2 ˆq 2 + h 3 dq 3 ˆq 3. (2) 4

5 We can define the distance changes s, s 2, and s 3 by s i h i q i. Let s make contact with something concrete by comparing with, say, spherical coordinates. The infinitesimal change in the position vector is what s given in (4), so we can identify the scale factors for spherical coordinates as h r =, h θ = r, and h φ = r sin θ. Since we know how to express d r now, we can immediately say how to do line elements for line integrals, v d r = v i h i dq i, (22) i as well as surface and volume integrals: v d a = = v ds 2 ds 3 + v 2 ds ds 3 + v 3 ds ds 2 v h 2 h 3 dq 2 dq 3 + v 2 h h 3 dq dq 3 + v 3 h h 2 dq dq 2 (23) and dτ... = h h 2 h 3 dq dq 2 dq 3..., (24) where the... stands for the integrand. Differential operators in curvilinear coordinates. I am not going to develop all of this here; it s pretty tedious, and is discussed in Boas secs. 9.8 and 9.9. However the basic idea comes from noting that the gradient is the fastest change of a scalar field, so the q component is obtained by dotting into ˆq, i.e. ˆq ψ = s = h q, (25) etc. Note that we are allowed to do the last step because h is a function of q 2 and q 3, but these are held constant during the partial differentiation. Therefore: ψ = i ˆq i h i q i. (26) Similarly v = h h 2 h 3 [ (v h 2 h 3 ) + (v 2 h h 3 ) + ] (v 3 h h 2 ) q q 2 q 3 (27) 5

6 and v = h h 2 h 3 ˆq h ˆq 2 h 2 ˆq 3 h 3 q q 2 q 3 h v h 2 v 2 h 3 v 3 (28) 6