Introduction of Generalized Coordinates
This chapter provides an introduction of Generalized Coordinates. Topics include introduction of generalized position, velocity and momentum; Lagrange Equations and Hamilton Equations in generalized coordinates; Legendre Transformation.
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 Generalized Momentum
What Is Legendre Transformation
Hamilton Equations in Generalized Coordinates
- Generalized position q and generalized velocity q' can be transformed
to Cartesian position r and Cartesian velocity r' by
a transformation function r(t) = r(q(t)).
- Hamilton's Principle in Generalized Coordinates stays in the same form as
d(S[q(t)]) / d(q(t)) = 0.
- Lagrange Equations in Generalized Coordinates stay in the same form as
d(∂L/∂q')/dt = ∂L/∂q.
- Generalized momentum is defined as:
p = ∂L/∂q'.
- Legendre Transformation
H = p∙q' - L
gives conversion between Lagrangian L and Hamiltonian H.
- Hamilton Equations in Generalized Coordinates stay in the same form as
∂H/∂q = -p', ∂H/∂p = q'.
Table of Contents
About This Book
Introduction of Space
Introduction of Frame of Reference
Introduction of Time
Introduction of Speed
Newton's Laws of Motion
Introduction of Special Relativity
Time Dilation in Special Relativity
Length Contraction in Special Relativity
The Relativity of Simultaneity
Introduction of Spacetime
Minkowski Spacetime and Diagrams
Introduction of Hamiltonian
Introduction of Lagrangian
►Introduction of Generalized Coordinates
Phase Space and Phase Portrait
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