University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

169

Pages:

5
Author:

Jabin F.

Conversion from L to H: Given L, find H. Write pi for all coordinates. Find bi and qi in terms of pi. If bi cannot be written in terms of pi, then for given L, H is not defined and vice versa. To get H in terms of bi, put pi in terms of bi. To get H in terms of qi, put bi in terms of qi. Conversion from H to L: Given H, find L. Write Hamilton's first equation of motion for all coordinates. Find qi in terms of bi and substitute in H. Find pi in terms of qi and substitute in H. Note: Any general question related to L or H can be solved using equations of motion (EOM). Question: The Lagrangian of a system is L = -m_0c^2 * sqrt(1 - v^2/c^2) - V(q, t). Find H. Solution: pi = ∂L/∂vi = -m_0c^2 * (-1/2) * (1 - v^2/c^2)^(-1/2) * (-2vi/c^2) = m_0vi / sqrt(1 - v^2/c^2) pi = m_0vi / sqrt(1 - v^2/c^2) H = Σ(pi * qi) + L = (m_0vi / sqrt(1 - v^2/c^2)) * vi + (-m_0c^2 * sqrt(1 - v^2/c^2)) - V H = (m_0vi^2 / sqrt(1 - v^2/c^2)) - m_0c^2 * sqrt(1 - v^2/c^2) - V H = (m_0c^2 * v^2 / sqrt(1 - v^2/c^2)) - m_0c^2 * sqrt(1 - v^2/c^2) - V H = m_0c^2 * (v^2 / sqrt(1 - v^2/c^2) - sqrt(1 - v^2/c^2)) - V H = m_0c^2 * (v^2 + c^2 - 1) / sqrt(1 - v^2/c^2) - V H = (m_0c^2 + p^2) / sqrt(m_0^2c^2 + p^2) - V H = c * sqrt(m_0^2c^2 + p^2) - V Note: To avoid calculation mistakes and answer quickly, memorize the following result: H = sqrt(m_0^2c^2 + p^2) + V