It is a delightful fact that one can get both the fundamental equation of classical mechanics (Newton’s Second Law) and the fundamental equation of quantum mechanics (Schrödinger’s equation) by solving very simple variational problems based on the familiar conservation of mechanical energy equation

In the present note I want to briefly set these out emphasising the common underlying structure provided by the conservation of mechanical energy and the calculus of variations. The kinetic energy will be taken to be

where is the particle’s velocity, is its momentum, and is its mass. The potential energy will be regarded as some function of only.

To obtain Newton’s Second Law we find the stationary path followed by the particle with respect to the functional

The function is usually termed the `Lagrangian’ in classical mechanics. The functional is usually called the `action’. The Euler-Lagrange equation for this calculus of variations problem is

and this is Newton’s Second Law in disguise! We have

and

so substituting these into the Euler-Lagrange equation we get Newton’s Second Law, .

To obtain Schrödinger’s equation we introduce a function

where is again the momentum of the particle and is the reduced Planck’s constant from quantum mechanics. (Note that has units of length mass time so we need to remove these by dividing by which has the same units. The function in quantum mechanics is dimensionless). We then have

and differentiating both sides gives

so

Therefore we can write the kinetic energy as

and putting this into the conservation of mechanical energy equation gives

We now find the stationary path followed by the particle with respect to the functional

The Euler-Lagrange equation for this calculus of variations problem is

and this is Schrödinger’s equation in disguise! We have

and

so substituting these into the Euler-Lagrange equation we get

and this is the (time-independent) Schrödinger equation for a particle of mass with fixed total energy in a potential on the line .