Quantum Mechanics for Mathematicians

Then, instead of working with trajectories characterized at timetby

(q(t),q ̇(t))∈R^2 d

we would like to instead use

(q(t),p(t))∈R^2 d

wherepj=∂∂Lq ̇jand identify thisR^2 d(for example att= 0) as the phase space
of the conventional Hamiltonian formalism.
The transformation

(qj,q ̇k)→

(

qj,pk=

∂L

∂q ̇k

)

between position-velocity and phase space is known as the Legendre transform,
and in good cases (for instance whenLis quadratic in all the velocities) it
is an isomorphism. In general though, this is not an isomorphism, with the
Legendre transform often taking position-velocity space to a lower dimensional
subspace of phase space. Such cases are not unusual and require a much more
complicated formalism, even as classical mechanical systems (this subject is
known as “constrained Hamiltonian dynamics”). One important example we
will study in chapter 46 is that of the free electromagnetic field, with equations of
motion the Maxwell equations. In that case the configuration space coordinates
are the components (A 0 ,A 1 ,A 2 ,A 3 ) of the vector potential, with the problem
arising because the Lagrangian does not depend onA ̇ 0.
Besides a phase space, for a Hamiltonian system one needs a Hamiltonian
function. Choosing

h=

∑d

j=1

pjq ̇j−L(q,q ̇)

will work, provided the relation

pj=

∂L

∂q ̇j

can be used to solve for the velocities ̇qj and express them in terms of the
momentum variables. In that case, computing the differential ofhone finds
(ford= 1, the generalization to higherdis straightforward)

dh=pdq ̇+ ̇qdp−

∂L

∂q

dq−

∂L

∂q ̇

dq ̇

= ̇qdp−

∂L

∂q

dq

So one has
∂h
∂p

= ̇q,

∂h

∂q

=−

∂L

∂q