cambridge01 The Hamiltonian Formalism The Hamiltonian Formalism From this we compute the momentum conjugate to the position p = ∂L.

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4. The Hamiltonian Formalism - damtp

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Figure 50: Motion in configuration space on the left, and in phase space on the right. ... Or, in other words, g is to be thought of as a function of u and y: g = g(u, y). If we .... From this we compute the momentum conjugate to the position p = ∂L.

4. The Hamiltonian Formalism
We’ll now move onto the next level in the formalism of classical mechanics, due initially to Hamilton around 1830. While we won’t use Hamilton’s approach to solve any further complicated problems, we will use it to reveal much more of the structure underlying classical dynamics. If you like, it will help us understands what questions we should ask.
4.1 Hamilton’s Equations
Recall that in the Lagrangian formulation, we have the function L(qi, ˙ qi,t) where qi (i = 1,...,n) are n generalised coordinates. The equations of motion are
d dt ∂L ∂ ˙ qi − ∂L ∂qi
= 0 (4.1)
These are n 2nd order differential equations which require 2n initial conditions, say qi(t = 0) and ˙ qi(t = 0). The basic idea of Hamilton’s approach is to try and place qi and ˙ qi on a more symmetric footing. More precisely, we’ll work with the n generalised momenta that we introduced in section 2.3.3,
pi =
∂L ∂ ˙ qi
i = 1,...,n (4.2)
so pi = pi(qj, ˙ qj,t). This coincides with what we usually call momentum only if we work in Cartesian coordinates (so the kinetic term is 1 2mi ˙ q2 i ). If we rewrite Lagrange’s equations (4.1) using the definition of the momentum (4.2), they become
˙ pi =
∂L ∂qi
(4.3)
The plan will be to eliminate ˙ qi in favour of the momenta pi, and then to place qi and pi on equal footing.
Figure 50: Motion in configuration space on the left, and in phase space on the right.