Introduction of Lagrangian

This chapter provides an introduction of Lagrangian Mechanics. Topics include introduction of Lagrangian; Action, Hamilton Principle, Stationary Action, Lagrange Equations, examples of free fall motion; simple harmonic motion, simple pendulum motion.

What Is Lagrangian

Action - Integral of Lagrangian

Action - Functional of Position Function x(t)

Hamilton's Principle - Stationary Action

What Is Lagrange Equation

Other Proofs of the Lagrange Equation

Lagrange Equation on Free Fall Motion

Lagrange Equation on Simple Harmonic Motion

Lagrangian in Cartesian Coordinates

Lagrange Equations in Cartesian Coordinates

Takeaways:

• Lagrangian Function is defined as the difference of kinetic and potential energy: H = T - V.
• Action is defined as the integral of the Lagrangian function L(t) between two time instances, t1 and t2.
• Hamilton's Principle states that the true position function x(t) of a system is a stationary point of the Action S[x(t)] functional.
• Lagrange Equations state that the time derivative of the partial derivative Lagrangian L against velocity equals to the partial derivative Lagrangian L against position: d(∂L/∂r')/dt = ∂L/∂r.

Table of Contents