14.2 Development of differential equations and variational principles
2 min read•august 9, 2024
The 18th century saw the birth of , a powerful tool for modeling real-world phenomena. Mathematicians like Euler developed methods to solve these equations, paving the way for advancements in physics and engineering.
Variational principles emerged as a way to find optimal solutions in complex systems. These principles, including the and , revolutionized and laid the groundwork for modern physics.
Differential Equations
Types and Classifications of Differential Equations
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involve functions of one independent variable and their derivatives
contain functions of multiple independent variables and their partial derivatives
require solutions to satisfy specific conditions at the boundaries of a domain
necessitate solutions to meet given initial conditions at a particular point
technique solves differential equations by separating variables to integrate both sides independently
Methods and Applications
Ordinary differential equations model systems changing over time ()
Partial differential equations describe phenomena varying in multiple dimensions ( in a solid)
Boundary value problems apply to steady-state scenarios ( in a metal rod with fixed end temperatures)
Initial value problems represent dynamic systems with known starting conditions ( with given initial velocity)
Separation of variables simplifies complex equations into solvable components (solving the heat equation in one dimension)
Variational Principles
Fundamental Concepts and Equations
optimizes functionals, which are functions of functions
Euler-Lagrange equation derives from minimizing or maximizing functionals in variational problems
Hamilton's principle states that the path of a physical system minimizes the
provides a general form for solutions to the one-dimensional wave equation
Applications in Physics and Engineering
Calculus of variations solves in physics and engineering (finding the shape of a soap film between two rings)
Euler-Lagrange equation determines equations of motion in classical mechanics (deriving the motion of a pendulum)
Hamilton's principle unifies various formulations of classical mechanics and extends to
D'Alembert's solution describes in strings and other media (modeling vibrations in a guitar string)