Problem 2 - Calculus Of Variations, With Applications To Physics And Engineering

Question

(a) Use Green's theorem (30) of 2-14 to show that a function \(\psi\) which satisfies Laplace's equation in \(R\) and $$ \iint_{B} \psi \frac{\partial \psi}{\partial n} d S=0 $$ where \(B\) is the boundary surface of \(R\), is necessarily a constant. If, further, \(\psi=0\) anywhere on \(B\), it thus follows that \(\psi=0\) identically in \(R\). (b) Use part \((a)\) to prove that \(\nabla^{2} \phi=0\) in \(R\), with \(\phi\) prescribed everywhere on \(B\), is sufficient to determine \(\phi\) uniquely. HrNT: Assume the two solutions \(\phi=\phi_{1}\) and \(\phi=\phi_{2}\), so that, with \(\psi=\phi_{2}-\phi_{1}\), we have \(\nabla^{2} \psi=0\) in \(R, \psi=0\) on \(B\).

Step-by-Step Solution

Answer

First, we use Green's theorem to prove that \( \psi \) is constant throughout its domain. Then, we proved that \( \psi = 0 \) everywhere because of the given condition \( \psi = 0 \) on the boundary. Lastly, we proved that Laplace's equation and boundary condition is sufficient to uniquely determine function \( \phi \).

Step 1: Prove \( \psi \) is a constant

First, note that the Laplace's equation for \( \psi \) can be written as \( \nabla . (\nabla \psi) = 0 \). Then, from the divergence theorem (an extension of Green's theorem to three dimensions), the given integral condition becomes: \[ \iiint_{V} \nabla . (\psi \nabla \psi) dV = 0\] where \( V \) is the volume enclosed by \( R \). On expanding the divergence, this gives \( \iiint_{V} \psi \nabla . \nabla \psi + \nabla \psi . \nabla \psi dV = 0 \). Since \( \psi \) satisfies Laplace's equation, \( \nabla . \nabla \psi = 0 \) and we are left with \( \iiint_{V} \nabla \psi . \nabla \psi dV = 0 \). Now, for \( \nabla \psi . \nabla \psi = 0 \) to hold everywhere in \( V \), it must be the case that \( \nabla \psi = 0 \). This means \( \psi \) does not change, so \( \psi \) is constant throughout \( R \).

Step 2: Prove \(\psi = 0\)

The condition that \( \psi = 0 \) somewhere on the boundary \( B \) now implies that \( \psi = 0 \) everywhere in \( R \), as \( \psi \) is constant.

Step 3: Prove uniqueness of \( \phi \)

In the second part, consider two solutions \( \phi = \phi_1 \) and \( \phi = \phi_2 \). We know that with \( \psi = \phi_2 - \phi_1 \), both functions solve Laplace's equation in \( R \) and \( \psi = 0 \) on \( B \). From what we have proved in part (a), it follows that \( \psi \) must be zero everywhere in \( R \). Therefore, \( \phi_1 = \phi_2 \), which shows that \( \phi \) is uniquely determined by Laplace's equation and its boundary values.

Key Concepts

Laplace's Equation

Laplace's Equation is a fundamental concept in mathematics and physics, appearing frequently in areas such as electrostatics, fluid dynamics, and more. This equation is given by \( abla^2 \psi = 0 \). It states that the divergence of the gradient of a potential function \( \psi \) is zero. In simple terms, this means that the function \( \psi \) is in a form of equilibrium with no net "flow" at any point.

When dealing with Laplace's Equation, the function \( \psi \) as a potential must be smooth and continuous which assures that the field it represents does not have any sinks or sources within the region \( R \).

The beauty of Laplace's Equation lies in its simplicity and the wide array of physical phenomena it describes, such as gravitational and electric potentials when no charge or mass is present within \( R \). Understanding the behavior of \( \psi \) to remain constant under these conditions is key to understanding many naturally occurring systems.

Divergence Theorem

The Divergence Theorem (also known as Gauss's Theorem) establishes a vital relationship between the flow of a vector field through a closed surface and the behavior of the field inside the surface. Mathematically, it is expressed as:

\[ \iiint_{V} abla \cdot \mathbf{F} \, dV = \iint_{B} \mathbf{F} \cdot \mathbf{n} \, dS \]

Here, \( V \) is the volume enclosed by surface \( B \), and \( \mathbf{F} \) is a vector field. This theorem helps in transforming volume integrals into surface integrals and vice versa.

In the context of this problem, the divergence theorem helps us show how the function \( \psi \) has a constant behavior across the region when \( abla \cdot \mathbf{F} = 0 \). This transformation is critical in simplifying complex physical domain problems by analyzing their boundary surfaces. When applying Green's Theorem in two dimensions, it becomes an integral component in analyzing equations like those present in fluid dynamics and electromagnetics.

Boundary Conditions

Boundary Conditions are essential in finding solutions to partial differential equations like Laplace's Equation. They specify the values that the solution must take on the boundary of the domain.

These conditions mean that the values of a function are predetermined along the boundary surfaces, turning an otherwise broad problem into one with specific solutions. A common boundary condition is the Dirichlet condition, where the potential function \( \phi \) is specified on the boundary.

In our scenario, \( \psi = 0 \) anywhere on the boundary \( B \). This implies that at least at one point on the boundary, the effect potential is nullified, guiding the entire potential function \( \psi \) to be zero in the entire region \( R \). Thus, boundary conditions crucially shape the solution's uniqueness and provide leverage in solving these equations effectively.

Uniqueness of Solutions

The Uniqueness of Solutions for equations, particularly Laplace's Equation, tells us that under certain conditions, a differential equation's solution will remain unequivocally the same.

For Laplace’s Equation, if the boundary conditions on the surface \( B \) surrounding the region \( R \) are determined, then there exists only one unique solution to the equation within the region. This is fundamental when addressing problems in fields like electrostatics, where the potential must be the same if the conditions outside a charge-free region remain unchanged.

Unique solutions ensure that for given boundaries, the differential equation does not yield conflicting potential results. When two potential functions that satisfy the same conditions are posed, and by assuming one solution as the difference \( \psi = \phi_2 - \phi_1 \), the uniqueness principle ensures that \( \psi \) must equate to zero, proving that both functions are actually identical everywhere in \( R \). This property is essential in proving the feasibility of solutions and their importance in real-life applications.