Hamiltonian mechanics

Hamiltonian mechanics was invented in 1833 by Hamilton. Like Lagrangian mechanics, it is a re-formulation of classical mechanics.

In Lagrangian mechanics, the equations of motion are dependent on generalized coordinates {q_j | j=1,...N} and matching generalized velocities {q_j′ | j=1,...N}. Abusing the notation, we write the Lagrangian as L(q_j, q_j′, t), with the subscripted variables understood to represent all N variables of that type. Hamiltonian mechanics aims to replace the generalized velocity variables with generalized momentum variables, also known as conjugate momenta. For each generalized velocity, there is one corresponding conjugate momentum, defined as:

<math>p_j = {\partial L \over \partial q'_j}</math>.

In Cartesian coordinates, the generalized momenta are precisely the physical linear momenta. In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum. For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.

The Hamiltonian is the Legendre transform[?] of the Lagrangian:

<math>H\left(q_j,p_j,t\right) = \sum_i q'_i p_i - L(q_j,q'_j,t)</math>.

If the transformation equations defining the generalized coordinates are independent of t, it can be shown that H is equal to the total energy E = T + V.

Each side in the definition of H produces a differential:

<math>\begin{matrix}

dH &=& \sum_i \left[ \left({\partial H \over \partial q_i}\right) dq_i + \left({\partial H \over \partial p_i}\right) dp_i + \left({\partial H \over \partial t}\right) dt \right] \\

  &=& \sum_i \left[ q'_i dp_i + p_i dq'_i - \left({\partial L \over \partial q_i}\right) dq_i - \left({\partial L \over \partial q'_i}\right) dq'_i - \left({\partial L \over \partial t}\right) dt \right]

\end{matrix}</math>.

Substituting the previous definition of the conjugate momenta into this equation and matching coefficients, we obtain the equations of motion of Hamiltonian mechanics, known as the canonical equations of Hamilton:

<math>

{\partial H \over \partial q_j} = - p'_j, \qquad {\partial H \over \partial p_j} = q'_j, \qquad {\partial H \over \partial t } = - {\partial L \over \partial t} </math>

Hamilton's equations are first-order differential equations, and thus easier to solve than Lagrange's equations, which are second-order. However, the steps leading to the equations of motion are more onerous than in Lagrangian mechanics - beginning with the generalized coordinates and the Lagrangian, we must calculate the Hamiltonian, express each generalized velocity in terms of the conjugate momenta, and replace the generalized velocities in the Hamiltonian with the conjugate momenta. All in all, there is little labor saved from solving a problem with Hamiltonian mechanics rather than Lagrangian mechanics. Ultimately, it will produce the same solution as Lagrangian mechanics and Newton's laws of motion.

The principal appeal of the Hamiltonian approach is that it provides the groundwork for deeper results in the theory of classical mechanics.

(To Be Done: The relationship between the classical Hamiltonian and constants of motion, and the Hamilton-Jacobi equation: these will clarify the relationship between the classical and quantum Hamiltonian.)