FOUNDATIONS OF QUANTUM MECHANICS

VI. 5. THE HAMILTON - JACOBI EQUATION 137
Substitution of (VI. 32) in (VI. 29) yields
S γ =
∫
γ
( 3n∑
j=1
)
p j ˙q j − H (⃗q, ⃗p, t) dt =
3n∑
j=1
∫
γ
p j dq j −
∫
γ
H (⃗q, ⃗p, t) dt, (VI. 34)
and variation of S γ in this form yields the 2n Hamiltonian equations of motion,
˙q j = ∂H
∂p i
,
ṗ j = − ∂H
∂q i
. (VI. 35)
Now consider the action S γ along a real path γ 0 , i.e., a path satisfying the equations of motion,
and form its differential,
dS(⃗q, ⃗q 0 , t − t 0 ) =
3n∑
j=1
(p j dq j − p 0j dq 0j ) − H (⃗q, ⃗p, t) dt. (VI. 36)
Comparison with
dS(⃗q, ⃗q 0 , t − t 0 ) =
3n∑
j=1
( ∂S
∂q j
dq j +
∂S )
dq 0j + ∂S dt (VI. 37)
∂q 0j ∂t
and using requirement (VI. 30) shows that
H (⃗q, ⃗p, t) = − ∂S
∂t ,
p j = ∂S
∂q j
,
p 0j = − ∂S
∂q 0j
, (VI. 38)
and therefore
∂S
(
∂t + H ⃗q, ∂S )
∂⃗q , t
= 0. (VI. 39)
This is (VI. 7), the Hamilton - Jacobi equation, as discussed on p. 129. The technique to solve the
mechanical equations of motion by means of this equation is especially due to Jacobi. Without discussing
this technique in detail, we mention the following.
For definite q 0 and t 0 it is possible to consider the action S as a function on configuration space. It
can be shown that the paths satisfying the equations of motion are always perpendicular to the hyperplanes
of constant S, hence the frequently quoted analogy with optics; paths are comparable to rays
of light, and planes of constant S to wave fronts. If, for one moment in time, the values S are given
over the complete configuration space, the Hamilton - Jacobi equation determines how they evolve in
the course of time. The problem to find the paths of the particles is thus reduced to constructing the
curves which are normal to the planes of constant S.
◃ Remark
Schrödinger originally based his derivation of wave mechanics on the idea that wave mechanics is to
classical mechanics as wave optics is to ray optics, and with the just mentioned wave fronts and the
Hamilton - Jacobi equation he came to his wave mechanics. ▹