Suppose that a Hamiltonian is slowly time-varying, for example, a one-dimensional harmonic oscillator with a changing frequency:
The action J of a classical orbit is the area enclosed by the orbit in phase space:
Since J is an integral over a full period, it is only a function of the energy. When the Hamiltonian is constant in time, and J is constant in time, the canonically conjugate variable  increases in time at a steady rate:
So the constant  can be used to change time derivatives along the orbit to partial derivatives with respect to  at constant J. Differentiating the integral for J with respect to J gives an identity that fixes :
The integrand is the Poisson bracket of x and p. The Poisson bracket of two canonically conjugate quantities, like x and p, is equal to 1 in any canonical coordinate system. So
and  is the inverse period. The variable  increases by an equal amount in each period for all values of J' it is an angle variable.