Infinite Limits

Lagrangian mechanics is a powerful mathematical framework that provides a systematic and elegant approach to understanding the dynamics of physical systems. Developed by Joseph-Louis Lagrange in the late 18th century, it offers an alternative formulation to Newton's laws of motion, providing deeper insights into the fundamental principles governing the behaviour of objects. In this post, we will embark on a comprehensive journey into the realm of Lagrangian mechanics, covering its principles, applications, and significance in the field of physics. Joseph Louis Lagrange Image Credit: Encyclopaedia Britannica The Lagrangian Formulation: At the heart of Lagrangian mechanics lies the concept of the Lagrangian, denoted by L. The Lagrangian is a mathematical function that encapsulates the kinetic and potential energies of a system (KE-PE), providing a comprehensive description of its dynamics. By defining the Lagrangian, we can derive the equations of motion for a system using the princi...

  Newton's method, also known as the Newton-Raphson method, is an algorithm used to find the roots of a polynomial function. It was developed in the late 1600s. For a function f(x), finding the value of x for which f(x) = 0 is the goal of Newton's algorithm. Image Credit: GeoGebra Here's an algorithm you can use: Choose an initial guess for the root, let's call it x₀. Calculate the function value and its derivative at the current guess. These are denoted as f(x₀) and f'(x₀), respectively. Use the formula: x₁ = x₀ - f(x₀) / f'(x₀) to update the guess. This formula is derived from the tangent line approximation to the function at x₀, assuming it is differentiable. Repeat the last two sentences until the desired level of accuracy is achieved or until a specified number of iterations have been performed. You will know your answer is correct as the result of every iteration converges to the same value. What to know about the algorithm: Newton's algorithm typical...

Hey! Happy pi day to all my readers! Here's something cool: e iπ = -1                                                                      (by Euler’s identity)  Taking natural logarithm on both sides, ln (-1) = iπ   Hence, π = ln(-1)/i     Here’s a simple question for pi day:    What does the series above converge at?   Anyways, here’s pi to 30 decimal places: 3.141592653589793238462643383279. Forgot to post, so just adding a Happy Tau Day Here! - 6/28

What is the Rocket Equation? The rocket equation is a fundamental principle in physics, based on Newton’s Third Law of Motion) that describes the motion of rockets in space. It is a mathematical formula that calculates the velocity of a rocket based on its mass, the mass of the propellant it carries, and the speed at which it expels that propellant. Credit: NASA   The equation shows that as the mass of the rocket decreases (as it expels propellant), its velocity increases. This is because the rocket's momentum is conserved, and as the mass decreases, the velocity must increase to maintain momentum. The rocket equation is important because it determines the amount of propellant required to achieve a certain velocity or distance in space. It also shows that there are limits to the performance of rockets, as increasing the propellant mass requires a corresponding increase in the initial mass of the rocket, which makes it heavier and more difficult to launch. Credit: The Planetary Soci...

  Technical analysis is a trading methodology used to evaluate securities and financial markets based on statistical trends and market activity. Technical analysts examine past market data, including price and volume, to forecast future market behavior and make investment decisions. In this blog, we will explore the basic concepts and tools used in technical analysis. The Basics: Price charts The cornerstone of technical analysis is the price chart. Technical analysts use charts to visualize historical prices and volume of a security over a specific period. These charts are available in various formats, including line charts, bar charts, and candlestick charts. Line charts are the simplest type of chart, displaying the security's closing prices over a given period. Bar charts add more detail by showing the open, high, low, and close prices of the security in a given time frame. Candlestick charts, the most popular type of chart, show the same information as a bar chart but are easi...