Physics Notes - Herong's Tutorial Notes - v3.24, by Herong Yang
Relation of Momentum and Potential Energy
This section describes the Relation of Momentum and Potential Energy.
With Hamiltonian expressed in terms of position r and momentum p, we can derive the relation of Momentum and Potential Energy by applying the Law of Conservation of Energy:
H = constant T + V = constant (H.2) or: dH/dt = 0 (H.3) or: d(0.5*|p|2/m + V(r))/dt = 0 # H.12 applied or: p/m ∙ dp/dt + ∂V(r)/∂r ∙ dr/dt = 0 # The chain rule for derivatives applied # ∙ is the vector dot product or: p/m ∙ p' + ∂V(r)/∂r ∙ r' = 0 # p' = dp/dt and r' = dr/dt applied or: r' ∙ p' + ∂V(r)/∂r ∙ r' = 0 # H.13, p = m*r' applied or: p' + ∂V(r)/∂r = 0 # Cancel out r'. # It become 3 equations, one in each dimension. or: ∂V/∂r = -p' (H.14)
Very nice. the law of conservation of energy says that the potential energy change in space equals to the negative change of momentum!
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
Hamiltonian on Free Fall Motion
Hamiltonian on Simple Harmonic Motion
Hamiltonian on Simple Pendulum Motion
Relation of Momentum and Hamiltonian
Hamiltonian in Cartesian Coordinates
►Relation of Momentum and Potential Energy
Hamilton Equations in Cartesian Coordinates
Introduction of Generalized Coordinates