Quantum Mechanics for Mathematicians

which we will write in terms of their position and velocity vectors as

γ(t) = (q(t),q ̇(t))

one can define a functional on the space of such paths:

Definition(Action).The actionSfor a pathγis

S[γ] =

∫t 2

t 1

L(q(t),q ̇(t))dt

The fundamental principle of classical mechanics in the Lagrangian formal-
ism is that classical trajectories are given by critical points of the action func-
tional. These may correspond to minima of the action (so this is sometimes
called the “principle of least action”), but one gets classical trajectories also for
critical points that are not minima of the action. One can define the appropriate
notion of critical point as follows:

Definition(Critical point forS).A pathγis a critical point of the functional
S[γ]if

δS(γ)≡

d

ds

S(γs)|s=0= 0

where
γs: [t 1 ,t 2 ]→Rd

is a smooth family of paths parametrized by an intervals∈(−,), withγ 0 =γ.

We’ll now ignore analytical details and adopt the physicist’s interpreta-
tion ofδSas the first-order change inSdue to an infinitesimal changeδγ=
(δq(t),δq ̇(t)) in the path.
When (q(t),q ̇(t)) satisfy a certain differential equation, the pathγwill be a
critical point and thus a classical trajectory:

Theorem(Euler-Lagrange equations).One has

δS[γ] = 0

for all variations ofγwith endpointsγ(t 1 )andγ(t 2 )fixed if

∂L

∂qj

(q(t),q ̇(t))−

d

dt

(

∂L

∂q ̇j

(q(t),q ̇(t))

)

= 0

forj= 1,···,d. These are called the Euler-Lagrange equations.

Proof.Ignoring analytical details, the Euler-Lagrange equations follow from the
following calculations, which we’ll just do ford= 1, with the generalization to