Harmonic oscillator

The harmonic oscillator is a classical system whose restoring force is proportional to displacement. Its positive-energy trajectories are closed ellipses, and a canonical change to action and angle turns their motion into a uniform rotation. This record fixes the one-degree-of-freedom, undamped, undriven model, summarizes its exact observables, and identifies the domain on which its action–angle description is valid. The oscillator lesson derives both solution methods and matches arbitrary initial data.

Required background. Position, momentum, derivatives, and a finite-dimensional Hamiltonian. The Hamiltonian-bracket bridge develops the canonical equations. Helpful background. The Liouville–Arnold reference explains regular levels and compact invariant tori.

The classical quadratic Hamiltonian

Let the phase space be $\mathbb R^2$ with coordinates $(q,p)$, where $q$ is displacement and $p$ momentum. Fix a mass $m\gt0$ and angular frequency $\omega_0\gt0$. The Hamiltonian and bracket are

$$H(q,p)=\frac{p^2}{2m}+\frac{m\omega_0^2q^2}{2}, \qquad \{f,g\}=f_qg_p-f_pg_q.$$

Time is real, and $df/dt=\{f,H\}$ for observables without explicit time dependence. The equivalent symplectic convention is $\omega=dq\wedge dp$, $\iota_{X_H}\omega=dH$. It gives

$$\dot q=\frac pm, \qquad \dot p=-m\omega_0^2q, \qquad \ddot q+\omega_0^2q=0.$$

These signs agree with Cannas da Silva, January 2006 revision, §§ 18.1–18.2, pp. 105–107, PDF. The force constant is $k=m\omega_0^2\gt0$. With length, mass and time units, $[p]=$ mass × length/time, $[H]=$ energy, and $[\omega_0]=$ inverse time. Radians are dimensionless.

Specify arbitrary real initial data $q(0)=q_0$, $p(0)=p_0$. There is no spatial boundary or periodic identification of $q$: the configuration space is the real line. Periodicity below is a property of the trajectory. The linear equations have a unique solution for all real times.

Exact solution and observables

The initial-value solution is

$$\begin{aligned} q(t)&=q_0\cos(\omega_0t)+\frac{p_0}{m\omega_0}\sin(\omega_0t),\\ p(t)&=p_0\cos(\omega_0t)-m\omega_0q_0\sin(\omega_0t). \end{aligned}$$

Differentiation checks both Hamilton equations, and evaluation at $t=0$ recovers both initial data. The conserved energy, displacement amplitude and maximal momentum magnitude are

$$\begin{aligned} E&=\frac{p_0^2}{2m}+\frac{m\omega_0^2q_0^2}{2},\\ A&=\sqrt{\frac{2E}{m\omega_0^2}}, \qquad p_{\max}=\sqrt{2mE}=m\omega_0A. \end{aligned}$$

For $E\gt0$, the orbit has minimal period $T=2\pi/\omega_0$. The period is independent of amplitude, a property called isochrony. At $E=0$, the solution is the equilibrium $q=p=0$; it has no distinguished minimal positive oscillation period.

Over one nontrivial period, the mean position and momentum vanish. The time averages of kinetic and potential energy are both $E/2$, because the averages of $\sin^2$ and $\cos^2$ are both $1/2$. These are time averages along a deterministic orbit; no statistical ensemble is assumed.

Positive action and a periodic angle

For $E\gt0$, define the action using a closed orbit $C_E$ oriented in the direction of physical time:

$$I=\frac1{2\pi}\oint_{C_E}p\,dq =\frac E{\omega_0}\gt0.$$

The canonical parameterization used here is

$$q=\sqrt{\frac{2I}{m\omega_0}}\sin\theta, \qquad p=\sqrt{2m\omega_0I}\cos\theta, \qquad \theta\in\mathbb R/2\pi\mathbb Z.$$

On a smooth local angle branch, $\{\theta,I\}=1$ and $dq\wedge dp=d\theta\wedge dI$. Thus

$$H=\omega_0I, \qquad I(t)=I_0, \qquad \theta(t)=\theta_0+\omega_0t\pmod{2\pi}.$$

The action has energy × time units; the angle is dimensionless. For nonzero initial data,

$$I_0=\frac E{\omega_0}, \qquad \theta_0=\operatorname{atan2}(m\omega_0q_0,p_0)\pmod{2\pi},$$

where $\operatorname{atan2}(Y,X)$ retains the signs of both sine and cosine. The lesson checks the bracket and reconstructs both $q(t)$ and $p(t)$, including initial data for which a one-argument inverse tangent gives the wrong quadrant.

With $q$ horizontal and $p$ vertical, physical motion is clockwise: at $q=0,p\gt0$, displacement increases. Along the stated parameterization, $p\,dq=2I\cos^2\theta\,d\theta$, proving the positive integral. Torrielli 2016, § 2.2, pp. 6–7, equations (2.23)–(2.32), v1 PDF uses a counterclockwise cycle for the unit oscillator and obtains action $-E$. Reversing the cycle and changing the angle together gives the positive-action convention above.

Regular circles and the equilibrium

The sole integral $H$ suffices for one-degree-of-freedom Liouville integrability on the regular set: $\{H,H\}=0$ and

$$dH=m\omega_0^2q\,dq+\frac pm\,dp\ne0 \qquad\text{when }(q,p)\ne(0,0).$$

Every positive-energy level is a compact connected ellipse, topologically a circle. It is therefore a one-dimensional Liouville torus; compare Cannas da Silva, § 18.4, pp. 110–111, Theorem 18.12, PDF. Here the explicit transformation proves the action–angle representation directly.

The transformation is globally defined on the punctured phase plane if the angle is circle-valued. A single continuous real-valued angle cannot cover an entire closed cycle. At the origin, $dH=0$, $I=0$, and all angle values represent the same point; there is no invertible angle coordinate there. The original solution remains smooth.

Limits and related regimes

For fixed $m,q_0,p_0,t$, the limit $\omega_0\to0^+$ of the direct solution is

$$q(t)\longrightarrow q_0+\frac{p_0}{m}t, \qquad p(t)\longrightarrow p_0.$$

This is the free-particle problem. At nonzero $p_0$, $I_0$ diverges and $T\to\infty$; the closed ellipse becomes a noncompact free-motion level in the limiting problem. The oscillator action–angle coordinates cannot simply be evaluated at $\omega_0=0$.

The quadratic Hamiltonian is exact for the model defined here. It can also arise as an approximation near a stable minimum of a smooth potential with positive second derivative, but the size of neglected nonlinear terms and the resulting error must be assessed for that potential. The quartic action–angle lesson gives an integrable oscillator whose frequency depends on energy and whose minimum has zero second derivative. Its transfer exercise shows when adding a positive quadratic term restores a low-energy harmonic approximation. Damping, driving, a negative force constant, or time-dependent parameters change the equations and their conservation laws.

For several uncoupled positive-frequency oscillators, the same construction applies separately to each pair, giving $H=\sum_j\omega_j I_j$. On a common level with all $I_j\gt0$, the motion lies on a product of circles. A two-oscillator trajectory closes precisely when $\omega_1/\omega_2$ is rational; an irrational ratio gives quasiperiodic motion dense in that two-torus. This comparison concerns the uncoupled model, not an arbitrary interaction between oscillators; see Torrielli, § 2.2, pp. 5–6, equation (2.17), v1 PDF and the Liouville–Arnold reference. Quantum oscillator states and spectra require a different state space and are outside this classical record.

References

• Cannas da Silva, Ana. Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764, Springer, 2001. DOI. Author revision January 2006; MIT-hosted PDF. Locators above refer to its printed pages.
• Torrielli, Alessandro. Lectures on Classical Integrability. arXiv:1606.02946v1 [hep-th], 2016. Version record. Open PDF.