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)
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