Introduction of Hamiltonian

This chapter provides an introduction of Hamiltonian. Topics include introduction of Hamiltonian; energy conservation; conservative system; free fall motion; simple harmonic motion; simple pendulum motion, momentum and Hamilton Equations.

What Is Hamiltonian

Hamiltonian on Free Fall Motion

Hamiltonian on Simple Harmonic Motion

Hamiltonian on Simple Pendulum Motion

What Is Momentum

Relation of Momentum and Hamiltonian

Hamiltonian in Cartesian Coordinates

Relation of Momentum and Potential Energy

Hamilton Equations in Cartesian Coordinates

Takeaways:

• Hamiltonian is defined as the total of kinetic and potential energy: H = T + V.
• Hamiltonian is a constant on a conservative system.
• Constant Hamiltonian on free fall motion gives a = g.
• Constant Hamiltonian on simple harmonic motion gives m*a = -k*x.
• Constant Hamiltonian on simple pendulum motion gives l*θ" = -g*sin(θ).
• Momentum p is defined as p = m*v.
• Momentum p is related to potential energy V as ∂V/∂r = -p'.
• Hamilton Equations are partial derivatives of Hamiltonian against position and momentum: ∂H/∂r = -p', ∂H/∂p = r'.