mathkaveli

In the previous post, we explored the theoretical differences and similarities between Variations of Parameters and Duhamel's Principle. Now, let’s bring these ideas to life with a simple example, demonstrating how these methods work in practice. We'll aim for clarity and connection, using an example that connects to everyday phenomena like heat distribution. A Problem: Modeling Heat Transfer in a Rod Imagine a metal rod that is heated at one end while the rest is kept cool. Over time, heat spreads along the rod. Mathematically, this situation can be modeled with the heat equation , a partial differential equation (PDE): ∂u/∂t = α ∂²u/∂x² where: u(x, t) is the temperature at position x and time t , α is a constant representing how well the material conducts heat, ∂u/∂t is the rate of change of temperature with time, ∂²u/∂x² is the second derivative of temperature with position, representing how temperature changes spatially. Suppose the rod initially has no hea...

When tackling differential equations, particularly in the context of linear systems, two prominent techniques emerge: Variations of Parameters and Duhamel's Principle. Both methods are effective in finding solutions to linear equations, but they differ significantly in their approaches and applications, especially when it comes to partial differential equations (PDEs). This blog post will delve into these two methods, highlighting their similarities, differences, and the regions where they overlap. Overview of Variations of Parameters Variations of Parameters is a widely used technique for solving non-homogeneous linear differential equations. The fundamental concept involves assuming that the constants in the general solution of the corresponding homogeneous equation are not fixed values but rather functions of the independent variable. This allows for the derivation of a particular solution that accommodates the non-homogeneous part of the equation. The steps in applying Variatio...

The pendulum is a quintessential example of classical mechanics—simple in form yet rich in insights. While traditionally described using Newton’s laws or the Lagrangian approach, the Hamiltonian framework offers a deeper perspective. This approach not only clarifies the pendulum’s dynamics but also bridges the gap to quantum mechanics. Let’s explore how the Hamiltonian formalism reframes the pendulum and what insights it provides. Setting the Stage: Describing the Pendulum A pendulum consists of a mass attached to a string, swinging in a vertical plane under gravity. Its motion can be described by a single variable: the angle θ (measured from the downward vertical). This angle, called the generalized coordinate, encapsulates the system’s state. The pendulum’s energy comprises two parts: Kinetic energy , which depends on the pendulum’s speed as it sweeps along its arc. Potential energy , determined by the height of the mass relative to its lowest point. Together, these define the total ...

The pendulum is a quintessential example of classical mechanics—simple in form yet rich in insights. While traditionally described using Newton’s laws or the Lagrangian approach, the Hamiltonian framework offers a deeper perspective. This approach not only clarifies the pendulum’s dynamics but also bridges the gap to quantum mechanics. Let’s explore how the Hamiltonian formalism reframes the pendulum and what insights it provides. Setting the Stage: Describing the Pendulum A pendulum consists of a mass attached to a string, swinging in a vertical plane under gravity. Its motion can be described by a single variable: the angle θ (measured from the downward vertical). This angle, called the generalized coordinate, encapsulates the system’s state. The pendulum’s energy comprises two parts: Kinetic energy , which depends on the pendulum’s speed as it sweeps along its arc. Potential energy , is determined by the height of the mass relative to its lowest point. Together, these define the tot...