**Physics Notes - Herong's Tutorial Notes** - v3.24, by Herong Yang

What Is Legendre Transformation

This section provides a quick introduction to Legendre Transformation, which is an equation to transform Lagrangian L to Hamiltonian H.

**What Is Legendre Transformation?**
Legendre Transformation is an equation to transform Lagrangian L
to Hamiltonian H as shown below:

H = p∙q' - L # H is the Hamiltonian # p is the generalized momentum # q' is the generalized velocity # ∙ is the dot product # L is the Lagrangian

From Legendre Transformation, we can get a nice definition of the kinetic energy: T = 0.5*p∙q'.

H = p∙q' - L or: T + V = p∙q' - (T - V) # Since H = T + V, L = T - V # T is the kinetic energy # V is the potential energy or: T = p∙q' - T # Cancel out V or: T = 0.5*p∙q' (C.5)

Table of Contents

Introduction of Frame of Reference

Introduction of Special Relativity

Time Dilation in Special Relativity

Length Contraction in Special Relativity

The Relativity of Simultaneity

Minkowski Spacetime and Diagrams

►Introduction of Generalized Coordinates

Generalized Coordinates and Generalized Velocity

Simple Pendulum Motion in Generalized Coordinates

Hamilton's Principle in Generalized Coordinates

Lagrange Equations in Generalized Coordinates

Lagrange Equations on Simple Pendulum

►What Is Legendre Transformation

Hamilton Equations in Generalized Coordinates