Question

In the spherical coordinates \(\rho, \theta, \phi(\rho>0,0 \leq \theta<2 \pi, 0 \leq \phi \leq \pi)\) defined by the equations $$ x=\rho \cos \theta \sin \phi, \quad y=\rho \sin \theta \sin \phi, \quad z=\rho \cos \phi $$ Laplace's equation is $$ \rho^{2} u_{\rho \rho}+2 \rho u_{\rho}+\left(\csc ^{2} \phi\right) u_{\theta \theta}+u_{\phi \phi}+(\cot \phi) u_{\phi}=0 $$ (a) Show that if \(u(\rho, \theta, \phi)=\mathrm{P}(\rho) \Theta(\theta) \Phi(\phi),\) then \(\mathrm{P}, \Theta,\) and \(\Phi\) satisfy ordinary differential equations of the form $$ \begin{aligned} \rho^{2} \mathrm{P}^{\prime \prime}+2 \rho \mathrm{P}^{\prime}-\mu^{2} \mathrm{P} &=0 \\ \Theta^{\prime \prime}+\lambda^{2} \Theta &=0 \\\\\left(\sin ^{2} \phi\right) \Phi^{\prime \prime}+(\sin \phi \cos \phi) \Phi^{\prime}+\left(\mu^{2} \sin ^{2} \phi-\lambda^{2}\right) \Phi &=0 \end{aligned} $$ The first of these equations is of the Euler type, while the third is related to Legendre's equation. (b) Show that if \(u(\rho, \theta, \phi)\) is independent of \(\theta,\) then the first equation in part (a) is unchanged, the second is omitted, and the third becomes $$ \left(\sin ^{2} \phi\right) \Phi^{\prime \prime}+(\sin \phi \cos \phi) \Phi^{\prime}+\left(\mu^{2} \sin ^{2} \phi\right) \Phi=0 $$ (c) Show that if a new independent variable is defined by \(s=\cos \phi\), then the equation for \(\Phi\) in part (b) becomes $$ \left(1-s^{2}\right) \frac{d^{2} \Phi}{d s^{2}}-2 s \frac{d \Phi}{d s}+\mu^{2} \Phi=0, \quad-1 \leq s \leq 1 $$ Note that this is Legendre's equation.

Short answer

Short Answer: To prove that the product of three functions, P(ρ), Θ(θ), and Φ(φ), can satisfy Laplace's equation, we first substitute the product form into the equation, differentiate and obtain three groups, one for each function, and derive the given ODEs for P, Θ, and Φ. For u being independent of θ, we show that Θ must be a constant function, which omits the equation for Θ and affects the equation for Φ. Lastly, by introducing a new variable s and transforming the terms in the equation for Φ, we derive Legendre's equation as required.

Step-by-step solution

Substituting the product form:

Given $$u(\rho, \theta, \phi) = \mathrm{P}(\rho) \Theta(\theta) \Phi(\phi),$$. Let's substitute this form into Laplace's equation: $$\rho^{2} u_{\rho \rho}+2 \rho u_{\rho}+\left(\csc ^{2} \phi\right) u_{\theta \theta}+u_{\phi \phi}+(\cot \phi) u_{\phi}=0$$.

Differentiation and separation:

We differentiate the product form and separate the terms into three groups, one with only derivatives of P, another with only derivatives of Θ, and the last one with only derivatives of Φ. We'll then divide by the product u = PΘΦ.

The ODEs for P, Θ, and Φ:

After separating the variables in the Laplace equation and simplifying the terms, we'll get the ODEs for P, Θ, and Φ given in the problem statement. (b)

u being independent of θ:

We are now given that u is independent of θ, so we can immediately see that Θ must be a constant function. In other words, Θ(θ) is a constant and its derivatives with respect to θ are 0.

Impact on ODEs:

With Θ being constant, the second equation of ODEs in part (a) is omitted, leaving us with the third equation in part (a) having the given form. (c)

Introducing new variable s:

We are now given a new independent variable, s, defined as $$s=\cos\phi$$. We shall find the transformation for the terms in the third equation in part (b) to write them in terms of s instead of φ.

Transformation of terms:

By transforming the terms in the third equation in part (b) using the new variable s and simplifying, we arrive at the Legendre equation given in the problem statement.

Key concepts

Euler's Differential Equation

Euler's differential equation is a form of a second-order ordinary differential equation (ODE) with variable coefficients that are powers of the independent variable. It can be recognized by its standard form: \[ ax^2y'' + bxy' + cy = 0 \] where the coefficients are constants, and the functions y' and y'' are the first and second derivatives of the unknown function y(x).

In our exercise related to Laplace's equation in spherical coordinates, the first equation of the separated variables is of the Euler type with \( \mu^2 \) as the separation constant. This particular Euler equation, \[ \rho^2 P'' + 2\rho P' - \mu^2 P = 0 \], describes the radial part of the solution in the spherical coordinate system and typically admits solutions in the form of power functions or polynomials.

Understanding Euler's equation is crucial in the context of solving Laplace's equation in spherical coordinates, as it governs the radial behavior of the potential function, which is of considerable importance in various physical applications, including electrostatics and fluid dynamics.

Legendre's Equation

Legendre's equation is another type of second-order ODE, encountered frequently in physics, especially when dealing with problems that exhibit spherical symmetry. The general form of Legendre's equation is: \[ (1-x^2)y'' - 2xy' + l(l+1)y = 0 \] where \( y \) is a function of variable \( x \) and \( l \) is typically a non-negative integer, representing the degree of the associated Legendre polynomial.

In the context of the exercise, after substituting \( s = \cos \phi \) and simplifying, the equation for \( \Phi \) becomes Legendre's equation, which is critical because its solutions, known as Legendre polynomials, are fundamental to expanding functions in series of spherical harmonics. This property provides significant insight and usefulness in solving Laplace's equation and in expanding potential functions in various problems of mathematical physics.

Separation of Variables

Separation of variables is a powerful method for solving partial differential equations (PDEs) where the solution is assumed to be the product of functions, each a function of a single independent variable. This approach simplifies the PDE into a set of ordinary differential equations (ODEs) that can, in many cases, be solved independently.

In our exercise, this method is applied to Laplace's equation in spherical coordinates. By assuming \( u(\rho, \theta, \phi) = P(\rho)\Theta(\theta)\Phi(\phi) \), separation of variables allows us to manipulate Laplace's equation into three ODEs, each depending on a single variable – \( \rho \) for the radial function \( P \), \( \theta \) for the angular function \( \Theta \) and \( \phi \) for \( \Phi \) – with each ODE containing a separation constant. This approach greatly simplifies the problem, enabling us to find solutions to the original PDE.

Spherical Harmonics

Spherical harmonics are solutions to Laplace’s equation on the surface of a sphere when expressed in spherical coordinates. They play a vital role in many areas of physics and engineering, serving as the angular portion of the solutions to problems in quantum mechanics, gravitational fields, and wave propagation, to name a few.

Mathematically, spherical harmonics are functions of the angles \( \theta \) and \( \phi \) that form an orthonormal basis for the space of square-integrable functions on the sphere surface. Each spherical harmonic is associated with two quantum numbers: the degree \( l \) and the order \( m \) and is denoted by \( Y_l^m(\theta, \phi) \).

The spherical harmonics emerge naturally when solving Laplace's equation through the separation of variables method. In the context of our problem, the Legendre's equation and the resultant Legendre polynomials are actually the polar angle part of the spherical harmonics when the order m is zero, which shows the interconnection of these concepts in mathematical physics.