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Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 10 1 Summary: Curvilinear Coordinates 1. Summary of Integra
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Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 10
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Summary: Curvilinear Coordinates 1. Summary of Integral Theorems 2. Generalized Coordinates 3. Cartesian Coordinates: Surfaces of Constant Coordinate 4. Cylindrical Coordinates: Volume Element 5. Spherical Coordinates: Transformations between Coordinates
Suggested Reading: Griffiths: Chapter 1, Sections 1.4.1 - 1.4.2, pages 38-45. Weber and Arfken, Chapter 2, Sections 2.1-2.3 and 2.5, pages 96-121 and 126-136. Kreyszig, Chapter 8, Section 8.12, pages 486494. Wangsness, Chapter 1, Sections 1.16 - 1.17, pages 28-34.
Physics 2460 Electricity and Magnetism I, Fall 2007, Lecture 10
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Introduction to Curvilinear Coordinates Up until now the discussion has been restricted to the use of Cartesian coordinates defined by coordinate axes x, y and z with the orthonormal baˆ Often the symmetry of the probˆ k]. sis set [ˆi, j, lem suggests the use of a different set of coordinates. For example, problems dealing with a central potential such as Newtonian gravitation or electrostatics, both of which vary as ∼ 1/r, suggest that we should use a coordinate system where one of the coordinates is the radial distance from the origin ‘r’. Thus, we should use spherical polar coordinates rather than Cartesian coordinates. Since the ground state of the hydrogen atom is spherically symmetric, it seems logical that we should use spherical coordinates to describe its solution. Classic texts in theoretical physics, such as Morse and Feshbach (“Methods of Theoretical Physics”, McGraw-Hill, New York, 1953) define as many as 11 different sets of specialized coordinates as well as giving a description of how to produce any generalized set of coordinates. We will restrict ourselves to a very limited discussion of the general problem and then
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introduce the use of cylindrical and spherical polar coordinates. We will be particularly interested in showing how one can transform between sets of coordinates and how the differential operators for the gradient, divergence, curl and Laplacian appear in these other coordinate systems. All three of these coordinate systems (Cartesian, Cylindrical and Spherical) have orthonormal basis vectors ˆ [ρ, ˆ zˆ], and[ˆ ˆ φ]). ˆ ˆ k], (specifically, [ˆi, j, ˆ φ, r, θ,
Generalized Coordinates: We can define a coordinate transformation by the set of equations q1 = q1(x, y, z) q2 = q2(x, y, z) q3 = q3(x, y, z) The equations of physics should remain valid if we change coordinate systems. Thus, we often use the symmetry of a problem to determine our choice of coordinate systems. The two most common coordinate systems (besides Cartesian) are cylindrical and spherical coordinates. These are examples of a more general class of curvilinear coordinate systems.
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The sketch below shows the surfaces of constant coordinate and the unit vectors at an arbitrary point P : [x, y, z] for the Cartesian coordinate system
Recall that the unit vector for any coordinate always points in the direction of increasing coordinate. Of course, for Cartesian coordinates
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the surfaces of constant coordinate are defined by x = const, y = const or z = const. The unit vectors are normal to the corresponding surface of constant coordinate. The volume element for Cartesian coordinates is
∆V = ∆x∆y∆z or, in the limit as these quantities approach 0, dV = dxdydz.
Cylindrical Polar Coordinates A corresponding diagram for cylindrical coordinates is shown below
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Again, the surfaces of constant coordinate ρ = const, φ = const and z = const are shown, but in this case the surface of constant ρ is curved (looking down at the x,y-plane (or the ρ, φ-plane if you prefer) the surfaces of constant ρ are right circular cylinders concentric with the z-axis. The surface of constant φ is the semi-infinite plane beginning
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at the z-axis and extending to infinity at an angle φ relative to the x-axis (which is taken as the reference point). Note: it is common to draw the Cartesian axes even when using cylindrical or spherical coordinates as these provide points of reference for defining the various angles used. The unit vectors, as always, point in the direction of increasing coordinate. Thus, ρˆ points perpendicularly outward from the z-axis and is always parallel to the x,y-plane. The direction of φˆ depends on the value of φ. At φ = 0 it points perpendicular to the x-axis whereas at φ = π/2 it points in a direction perpendicular to the y-axis. In every case it is perpendicular to the z-axis and ˆ zˆ] are ˆ [ρ, ˆ φ, is perpendicular to the direction of ρ. always mutually orthogonal. The volume element at a point P : [ρ, φ, z] can be calculated by considering the length of the arc made when φ is changed by an amount ∆φ. Since the radius of the surface at constant ρ at P is just ρ, then change φ by ∆φ results in an arc of length ρ · ∆φ (φ is measured in radians). The volume element then is just ∆V = ∆ρ · ρ∆φ · ∆z
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or, in the limit as these lengths approach 0, dV = ρ dρ dφ dz Surface area elements will depend on which surface is being considered. For example, the surface area element at constant z would be dA = ρ dρ dφ while that at constant ρ would be dA = ρ dφ dz. What is the area element for a surface at constant φ? The Cartesian components [x1, x2, x3] of a vector ~r = [ρ, φ, z] are given by the transformation equations x1 = ρ cos φ x2 = ρ sin φ x3 = z where we use the notation [x1, x2, x3] to denote the Cartesian components instead of the usual [x, y, z].
Spherical Polar Coordinates In the case of spherical coordinates, the surfaces of constant ‘r’ are spheres centred at the origin, those of constant θ are cones forming the angle θ with respect to the z-axis and those of constant φ are semi-infinite planes beginning at the z-axis and
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extending to infinity at an angle φ relative to the x-axis.
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ˆ φ] ˆ at an arbitrary Again, the unit vectors [ˆ r, θ, point P : [r, θ, φ] point in the direction of increasing coordinate and are mutually orthogonal. Thus, rˆ points radially outward from the origin, the direction of θˆ depends on the value of θ. When θ = 0 the unit vector θˆ points outward from the z-axis in constant φ plane while at θ = π/2 it points in the -z direction (though still in the constant φ plane). The unit vector φˆ is also dependent on φ and behaves just as in the case of cylindrical coordinates. The volume element now depends on two arclengths. As θ is increased by ∆θ it traces out an arc of length r · ∆θ in the constant φ plane, while increasing φ by an amount ∆φ traces out an arc in the x,y-plane of r sin θ · ∆φ. Thus, the volume element is ∆V = (∆r)(r∆θ)(r sin θ∆φ) so, in the limit, dV = r2 sin θ dr dθ dφ Surface area elements again will depend on which surface is being considered. These can be deduced by inspection of the sketch above. For example, for the constant θ cone, the surface area element
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is dA = r sin θdφdr. What are the other possibilities? The coordinate transformation equations are x1 = r sin θ cos φ x2 = r sin θ sin φ x3 = r cos θ
Generalized Coordinates: q1, q2, q3 In general we can write xi = xi(q1, q2, q3) where the qi are our curvilinear coordinates (we will assume that the qi are mutually orthogonal, i.e., qˆi · qˆj = δij ,
i, j = 1, 2, 3
in Cartesian coordinates we write ~r = xˆi + y jˆ + z kˆ = x1ˆi + x2jˆ + x3kˆ but, xi = xi(q1, q2, q3) so that ∂~r ∂x1 ˆ ∂x2 ˆ ∂x3 ˆ = i+ j+ k ∂qi ∂qi ∂qi ∂qi
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This is the change in ~r along the qi-direction. thus, a unit vector in the qi-direction is simply ∂~r/∂qi qˆi ≡ |∂~r/∂qi| or, writing it out explicitly in terms of the Cartesian unit vectors, ∂x2 ˆ ∂x3 ˆ 1 ˆ ( ∂x ) ) i + ( j + ( ∂qi ∂qi ∂qi )k qˆi = q ∂x2 2 ∂x3 2 1 2 ( ∂x ∂qi ) + ( ∂qi ) + ( ∂qi ) We can define the denominator as a sort of scaling parameter s ∂x1 2 ∂x2 2 ∂x3 2 hi = ( ) +( ) +( ) ∂qi ∂qi ∂qi Then, qˆi =
1 ∂x1 ˆ 1 ∂x2 ˆ 1 ∂x3 ˆ ( )i + ( )j + ( )k hi ∂qi hi ∂qi hi ∂qi
or, 1 ∂~r , i = 1, 2, 3 hi ∂qi Since we can express ~r = ~r(q1, q2, q3) then the complete differential ∂~r ∂~r ∂~r d~r = dq1 + dq2 + dq3 (1) ∂q1 ∂q2 ∂q3 = (h1dq1)ˆ q1 + (h2dq2)ˆ q2 + (h3dq3)ˆ q3 qˆi =
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Now the total differential of the scalar function ϕ(q1, q2, q3) is dϕ(q1, q2, q3) =
∂ϕ ∂ϕ ∂ϕ dq1 + dq2 + dq3 (2) ∂q1 ∂q2 ∂q3
In curvilinear coordinates, the gradient of ϕ(q1, q2, q3) is ~ q ϕ = f1 qˆ1 + f2 qˆ2 + f3 qˆ3 (3) ∇ but we still don’t know f1, f2, and, f3. To find these, calculate the directional derivative along d~r. Then, from Eqn. (1) we can define the unit vector dq1 d~r dq2 dq3 qˆ1 + h2 qˆ2 + h3 qˆ3 = h1 δˆ ≡ dr dr dr dr Now, dϕ/dr gives the change in φ going from ~r to ~r + d~r, but this is the definition of the directional ˆ derivative Dδˆϕ along δ, dϕ ˆ ~ q ϕ · δ, ≡ Dδˆϕ = ∇ dr
d~r δˆ ≡ dr
Thus, ~ q ϕ · δˆ dr = ∇ ~ q ϕ · d~r dϕ = ∇ Equating Eqn. (2) with Eqn. (4), ~ q ϕ · d~r = ∂ϕ dq1 + ∂ϕ dq2 + ∂ϕ dq3 ∇ ∂q1 ∂q2 ∂q3
(4)
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But, from Eqn. (1) and Eqn. (3), ~ q ϕ · d~r = f1(h1 dq1) + f2(h2 dq2) + f3(h3 dq3) ∇ so we can identify f1, f2, andf3 as f1 =
1 ∂ϕ , h1 ∂q1
f2 =
1 ∂ϕ h2 ∂q2
f3 =
1 ∂ϕ h3 ∂q2
So, finally, in curvilinear coordinates (q1, q2, q3) the gradient of ϕ is
~ qϕ = ∇
1 ∂ϕ 1 ∂ϕ 1 ∂ϕ qˆ1 + qˆ2 + qˆ3 h1 ∂q1 h2 ∂q2 h3 ∂q3