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Mathematics for Physics

107 Derivation of Hamiltonian

A derivation of the Hamiltonian that is used to obtain \(H(p,q)=\displaystyle\sum_{i}p_{i}\dfrac{dq_{i}}{dt}-L\) from the Lagrangian by a Legendre transform, where a Legendre transform is a change from velocity variables to momentum variables.

1. Start from a Lagrangian that depends on position and velocity. This principle is used to treat \(H\) as the Legendre transform of \(L\).

The Lagrangian is

\[ L = L\left(q,\dfrac{dq}{dt}\right) \]

where

  • \(L\) is the Lagrangian.
  • \(q\) is a generalized coordinate.
  • \(\dfrac{dq}{dt}\) is the generalized velocity.

2. The conjugate momentum is the slope of \(L\) in velocity. This principle is used to introduce \(p\) as the new independent variable.

The conjugate momentum is

\[ p = \dfrac{\partial L}{\partial \left(\dfrac{dq}{dt}\right)} \]

where

  • \(p\) is the conjugate momentum.
  • \(L\) is the Lagrangian.

3. The Hamiltonian is \(p\) times velocity minus \(L\), with velocity expressed in terms of \(p\). This principle is used to write \(H\) as a function of \(p\) and \(q\) alone.

The Hamiltonian is

\[ H(p,q) = p\,\dfrac{dq}{dt}(p) - L\left(q,\dfrac{dq}{dt}(p)\right) \]

where

  • \(H\) is the Hamiltonian.
  • \(p\) is the conjugate momentum.
  • \(L\) is the Lagrangian.

4. For \(L=\dfrac{1}{2}m(\mathrm{d}q/\mathrm{d}t)^{2}-U(q)\) one obtains \(H=p^{2}/2m+U(q)\). This principle is used to recover the energy \(T+U\) and the quantum operator \(P^{2}/2m+V(X)\).

The simple Hamiltonian is

\[ H(p,q) = \dfrac{p^{2}}{2m} + U(q) \]

where

  • \(m\) is the mass.
  • \(U(q)\) is the potential energy.

5. Matching the differential of \(H\) yields Hamilton’s equations. This principle is used to write first-order equations for \(q\) and \(p\).

Hamilton’s equations are

\[ \dfrac{dq_{i}}{dt} = \dfrac{\partial H}{\partial p_{i}},\qquad \dfrac{dp_{i}}{dt} = -\dfrac{\partial H}{\partial q_{i}} \]

where

  • \(q_{i}\) is a generalized coordinate.
  • \(p_{i}\) is the conjugate momentum.
  • \(H\) is the Hamiltonian.

Note: These principles are the Lagrangian starting point, conjugate momentum, the Legendre transform, \(H=T+U\) for a standard kinetic term, and Hamilton’s equations.

107.1 References

  1. MIT OpenCourseWare. 8.223 Classical Mechanics II, Lecture 15: Introduction to Hamiltonian Mechanics (IAP 2017). — Legendre transform \(H = p\dot{q}-L\), simple case \(H=T+U\), and Hamilton’s equations.