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

Takeaways:

• 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'.