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The harmonic model worked through: an explicit complete integral, its characteristic motion, and the action-angle chart with $I = \oint p d x$ and period-one frequency $1 / T$

Physical scope

A worked instance of the Hamilton-Jacobi and action-angle items inside the declared scalar harmonic model, identified with the generalized-coordinate presentation of the explicit straight-line immersion into a three-dimensional force model: for one label of mass $m > 0$ and spring constant $k > 0$ , the explicit family $S$ is a complete integral of the model's Hamilton-Jacobi equation wherever $\sin (ω t) \neq 0$ ; its characteristic curves are the harmonic motions with $β = S_{α}$ constant; and the explicit chart $(θ, I)$ is canonical with $H = I / T$ , $\dot{I} = 0$ , $\dot{θ} = 1 / T$ and $I = \oint p d x$ , in the period-one convention, with the radian rescaling recovering the sources' chart. The example asserts no measured value, no claim that a real oscillator realizes the model, and no global extension of $S$ across the conjugate times $ω t \in π Z$ .

Example

Adopt the harmonic model of The harmonic (linear restoring) model: a declared linear force law with parameter $k > 0$ : one label of mass $m > 0$ on $R$ with spring constant $k > 0$ , force law $F (x) = - k x$ , potential $U (x) = \frac{1}{2} k x^{2}$ and no residual force. For the CM12-CM14 presentation interfaces, declare the ambient one-label model on $Q_{a m b} = R^{3}$ with mass operator $m I_{3}$ , time window $R$ , potential $U_{a m b} (y) = k y_{1}^{2} / 2$ , $d = 0$ and smooth force $F_{a m b} (y, v) = (- k y_{1}, 0, 0)$ (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions). The generalized-coordinate map $X : R \to Q_{a m b}$ , $X (x) = (x, 0, 0)$ is a smooth immersion with length-valued coordinate $x$ , $G = D X^{T} m I_{3} D X = m$ and substituted potential $k x^{2} / 2$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). Thus its chart Hamiltonian $\tilde{H}$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions clause (b) is exactly the scalar Hamiltonian below; write it as $H$ . A scalar harmonic motion reconstructs an ambient motion $X \circ x$ since the longitudinal equation is $m x^{''} = - k x$ and both transverse equations are $0 = 0$ . Write

$ω : = \sqrt{\frac{k}{m}} > 0, T : = \frac{2 π}{ω},$

so that $U (x) = \frac{1}{2} m ω^{2} x^{2}$ and the branch Hamiltonian in the generalized-coordinate chart is

$H (x, p) = \frac{p^{2}}{2 m} + \frac{m ω^{2}}{2} x^{2}$

(Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions), with $[H] = μ ℓ^{2} τ^{- 2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). On the open set

$V : = {(t, x, α) \in R^{3} : \sin (ω t) \neq 0}$

define

$S (t, x, α) : = \frac{m ω}{2} [(x^{2} + α^{2}) \cot (ω t) - 2 x α \csc (ω t)] .$ The example checks the following claims of this page on this instance.

(complete integral) $S$ is of class $C^{\infty}$ on $V$ and solves the model's Hamilton-Jacobi equation $S_{t} + H (x, S_{x}) = 0$ on $V$ ; the mixed derivative is $S_{x α} = - m ω \csc (ω t)$ , so $S$ is a complete integral of the model's Hamilton-Jacobi equation at every point of $V$ (The Hamilton-Jacobi equation of a declared potential presentation: the class of action-dimension functions $S$ with $S_{t} + H (t, q, S_{q}) = 0$ in one momentum chart, A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants).
(characteristics) for every $(t_{0}, x_{0}, α_{0}) \in V$ and every real $β$ , the curve $x (t) = α_{0} \cos (ω t) - \frac{β}{m ω} \sin (ω t)$ satisfies $S_{α} (t, x (t), α_{0}) = β$ , and with $p (t) : = S_{x} (t, x (t), α_{0})$ the pair $(x, p)$ solves $x^{'} = H_{p}$ , $p^{'} = - H_{x}$ on every interval on which $\sin (ω t) \neq 0$ ; hence $x$ is a motion of the harmonic model with $p = m x^{'}$ and $S_{α}$ is constant along it (A complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical $(β, α)$ labels).
(action-angle chart) the map $(θ, I) ⟼ (x, p) = (\sqrt{\frac{I}{π m ω}} \sin (2 π θ), \sqrt{\frac{I m ω}{π}} \cos (2 π θ))$

for $I > 0$ and $θ \in R / Z$ is a $C^{\infty}$ parametrization of the punctured phase plane; local real lifts of $θ$ give charts with ${x, p} = 1$ for the presentation's bracket, i.e. a canonical chart of Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts with $ω_{f o r m} = d θ \land d I$ in the period-one convention of Action and angle coordinates, and in it $H = h (I) = \frac{I}{T},$ so that every motion satisfies $\dot{I} = 0$ , $\dot{θ} = 1 / T$ , by the canonical Hamiltonian field of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian clause (a).
(action normalization) the action coordinate is the period integral $I = \oint p d x = \int_{0}^{T} p (t) x^{'} (t) d t$ along one period of the oscillator motion, in the period-one normalization discussed in the remark of Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H = h (I)$ and linear motion on the tori.
(radian convention) with $θ^{r a d} : = 2 π θ$ and $I^{r a d} : = I / (2 π)$ the chart becomes $x = \sqrt{2 I^{r a d} / (m ω)} \sin θ^{r a d}$ , $p = \sqrt{2 I^{r a d} m ω} \cos θ^{r a d}$ with $H = ω I^{r a d}$ and ${\dot{θ}}^{r a d} = ω$ , which is the radian chart of the sources (Period-one to radian action-angle rescaling: $θ^{r a d} = 2 π θ$ and $I^{r a d} = I / (2 π)$ preserve $ω$ ).

All data $m, k$ are declared model parameters; the example asserts no measured value and no claim about a real oscillator.

Facts & Assumptions

Given: The harmonic model data $m > 0$ , $k > 0$ , the ambient potential presentation and immersion $X$ just declared, the constants $ω, T$ , the Hamiltonian $H$ , the open set $V$ and the function $S$ displayed above.

[L1]

Model and Hamiltonian (The harmonic (linear restoring) model: a declared linear force law with parameter $k > 0$ , The motions of the harmonic model are the sinusoids $A \cos (ω (t - t_{0})) + B \sin (ω (t - t_{0}))$ : amplitude and phase, the least period, the conserved energy $k R^{2} / 2$ and the elliptical phase curves, Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces): scalar motions of the model are the solutions of $m x^{''} = - k x$ , and are exactly the curves $x (t) = A \cos (ω (t - t_{0})) + B \sin (ω (t - t_{0}))$ with reals $A, B$ ; the inverse Legendre branch is $(x, p) \mapsto p / m$ , equivalently $p = m x^{'}$ , and the branch Hamiltonian is the displayed $H$ . The least positive period of $x$ is $T$ when $(A, B) \neq (0, 0)$ , by clause 3 of The motions of the harmonic model are the sinusoids $A \cos (ω (t - t_{0})) + B \sin (ω (t - t_{0}))$ : amplitude and phase, the least period, the conserved energy $k R^{2} / 2$ and the elliptical phase curves.

[L2]

Calculus ( $C^{k}$ Euclidean maps and diffeomorphisms, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Double-angle and quadratic power-reduction identities, The zero sets of sine and cosine and the least positive common period 2 pi, Parity and the Pythagorean identity for sine and cosine, If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): the derivatives $(\sin)^{'} = \cos$ , $(\cos)^{'} = - \sin$ , $(\cot)^{'} = - \csc^{2}$ , $(\csc)^{'} = - \csc \cot$ hold wherever the functions are defined; the chain rule and the algebra of derivatives apply; the power-reduction identity $\cos^{2} u = (1 + \cos 2 u) / 2$ and the Pythagorean identity $\sin^{2} u + \cos^{2} u = 1$ hold; $\sin (k π) = 0$ for integers $k$ ; and a continuous $u : [a, b] \to R$ with a primitive $W$ satisfies $\int_{a}^{b} u = W (b) - W (a)$ .

[L3]

Complete-integral definitions (The Hamilton-Jacobi equation of a declared potential presentation: the class of action-dimension functions $S$ with $S_{t} + H (t, q, S_{q}) = 0$ in one momentum chart, A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants): a $C^{3}$ solution $S$ of $S_{t} + H (x, S_{x}) = 0$ with $S_{x α} \neq 0$ at a point is a complete integral there.

[L4]

Characteristics theorem (A complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical $(β, α)$ labels): a complete integral $S$ produces curves with $p = S_{x}$ solving the Hamilton equations and with $β = S_{α}$ constant.

[L5]

Canonical chart and field (Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts, The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian clause (a), Action and angle coordinates, The Euclidean inverse function theorem): a smooth coordinate map with nonzero Jacobian has smooth local inverses. In coordinates $(θ, I)$ with two-form $d θ \land d I$ , the defining equation $ι_{X_{H}} ω_{f o r m} = d H$ reads $\dot{θ} = H_{I}$ , $\dot{I} = - H_{θ}$ . The radian rescaling preserves the two-form and transforms the Hamiltonian as $\tilde{h} (I^{r a d}) = h (2 π I^{r a d})$ .

Verification

technique · direct; compute the partial derivatives, verify the Hamilton-Jacobi equation and the characteristic equations by substitution, check the Poisson bracket of the explicit chart, and evaluate the period integral

1.1L2L3givenalgebra

Derivatives of $S$ : with $c : = \cos (ω t)$ and $s : = \sin (ω t)$ , [L2] gives $S_{x} = m ω (x \cot (ω t) - α \csc (ω t))$ and $S_{t} = \frac{m ω^{2}}{2} [- (x^{2} + α^{2}) \csc^{2} (ω t) + 2 x α \csc (ω t) \cot (ω t)] .$ Substituting $p = S_{x}$ into $H$ , $S_{t} + H (x, S_{x}) = \frac{m ω^{2}}{2} [- (x^{2} + α^{2}) \csc^{2} + 2 x α \csc \cot + (x \cot - α \csc)^{2} + x^{2}] = \frac{m ω^{2}}{2} x^{2} (- \csc^{2} + \cot^{2} + 1) = 0,$ because $\cot^{2} - \csc^{2} = - 1$ ; and $S_{x α} = - m ω \csc (ω t) \neq 0$ on $V$ . Hence $S$ is a $C^{\infty}$ complete integral of the model's Hamilton-Jacobi equation at every point of $V$ , which is claim 1. Dimensions: $[m ω x^{2}] = μ τ^{- 1} ℓ^{2} = [S]$ .

2.1L1L2L4step 1.1algebra

The characteristic motion: put $x (t) : = α_{0} \cos (ω t) - β (m ω)^{- 1} \sin (ω t)$ and $p (t) : = S_{x} (t, x (t), α_{0})$ . By [L2], $x^{'} (t) = - α_{0} ω \sin (ω t) - β m^{- 1} \cos (ω t)$ and $\frac{p (t)}{m} = ω [x (t) \frac{c}{s} - α_{0} \frac{1}{s}] = ω [(α_{0} c - \frac{β}{m ω} s) \frac{c}{s} - \frac{α_{0}}{s}] = - ω α_{0} s - \frac{β}{m} c = x^{'} (t),$ using $α_{0} (c^{2} - 1) / s = - α_{0} s$ , so $p = m x^{'}$ and $x^{'} = H_{p} = p / m$ , and $p^{'} (t) = - m ω^{2} α_{0} c + β ω s = - m ω^{2} x (t) = - H_{x}$ . For the constants, differentiating the displayed $S$ in $α$ gives $S_{α} (t, x, α) = m ω (α \cot (ω t) - x \csc (ω t))$ , so $S_{α} (t, x (t), α_{0}) = m ω [α_{0} \cot (ω t) - (α_{0} c - \frac{β}{m ω} s) \csc (ω t)] = m ω α_{0} (\cot - c \csc) + β = β,$ since $\cot u - \cos u \csc u = 0$ . Hence the characteristic curve lies on the level set $S_{α} = β$ , it is a motion of the model with $p = m x^{'}$ by [L1], and $β = S_{α}$ is constant along it, which is claim 2.

3.1L2L5step 2.1algebra

The canonical chart: write $x = \sqrt{I / (π m ω)} \sin (2 π θ)$ and $p = \sqrt{I m ω / π} \cos (2 π θ)$ ; by [L2], $x_{θ} p_{I} - x_{I} p_{θ} = \cos^{2} (2 π θ) + \sin^{2} (2 π θ) = 1,$ the computation being: $x_{θ} = 2 π \sqrt{I / (π m ω)} \cos$ , $p_{I} = \frac{1}{2} \sqrt{m ω / π} I^{- 1 / 2} \cos$ give $x_{θ} p_{I} = \cos^{2}$ ; and $x_{I} = \frac{1}{2} (π m ω I)^{- 1 / 2} \sin$ , $p_{θ} = - 2 π \sqrt{I m ω / π} \sin$ give $x_{I} p_{θ} = - \sin^{2}$ . The energy fixes $I = 2 π H / ω > 0$ , and the normalized pair $(x \sqrt{π m ω / I}, p \sqrt{π / (I m ω)})$ fixes $θ$ modulo one, so the parametrization is bijective onto the punctured phase plane; the nonzero Jacobian makes its inverse smooth in local angle lifts. Hence ${x, p} = 1$ in the chart, i.e. the coordinates are canonical for the presentation's bracket and $ω_{f o r m} = d θ \land d I$ . Substituting into $H$ , $H = \frac{p^{2}}{2 m} + \frac{m ω^{2}}{2} x^{2} = \frac{I ω}{2 π} (\cos^{2} + \sin^{2}) = \frac{I ω}{2 π} = \frac{I}{T} = h (I) .$ Writing $X_{H} = a \partial_{θ} + b \partial_{I}$ , contraction gives $a d I - b d θ = d H = T^{- 1} d I$ , so $a = 1 / T$ , $b = 0$ by [L5]. Hence $\dot{I} = 0$ and $\dot{θ} = 1 / T$ along the scalar harmonic motions, which is claim 3.

4.1L2step 3.1algebra

The period integral: parametrize the motion of step 3.1 with $I > 0$ , $θ_{0} = 0$ by $x (t) = A \sin (ω t)$ , $A : = \sqrt{I / (π m ω)}$ , so that $p (t) = m x^{'} (t) = m A ω \cos (ω t)$ and the least positive period is $T$ by [L1]. Then by [L2], $\oint p d x = \int_{0}^{T} p (t) x^{'} (t) d t = m A^{2} ω^{2} \int_{0}^{T} \cos^{2} (ω t) d t = m A^{2} ω^{2} \cdot \frac{T}{2} = m A^{2} ω π = I,$ because $\cos^{2} (ω t) = (1 + \cos 2 ω t) / 2$ , the integral of $\cos (2 ω t)$ over $[0, T]$ vanishes being an increment of $\sin (2 ω t) / (2 ω)$ at $\sin (2 π k) = 0$ , and $m A^{2} ω π = m \frac{I}{π m ω} ω π = I$ . This is claim 4.

4.2L5step 3.1algebra

Radian rescaling: the map $[u] \mapsto [2 π u]$ from $R / Z$ to $R / (2 π Z)$ is a smooth bijection with smooth inverse $[v] \mapsto [v / (2 π)]$ , as checked in local real lifts. Thus $θ^{r a d} = 2 π θ$ and $I^{r a d} = I / (2 π)$ satisfy $d θ^{r a d} \land d I^{r a d} = (2 π) d θ \land d I / (2 π) = d θ \land d I$ and keep the two-form and give $H = h (2 π I^{r a d}) = 2 π I^{r a d} / T = ω I^{r a d}$ and ${\dot{θ}}^{r a d} = 2 π \dot{θ} = 2 π / T = ω$ ; inverting the rescaling in the chart of step 3.1 gives $x = \sqrt{2 I^{r a d} / (m ω)} \sin θ^{r a d}$ and $p = \sqrt{2 I^{r a d} m ω} \cos θ^{r a d}$ , the source's chart. This is claim 5.

5.1step 1.1step 2.1step 3.1step 4.1step 4.2∎

Claims 1-5 are verified by steps 1.1, 2.1, 3.1, 4.1 and 4.2 respectively; the example is verified.

Remarks

Where the construction breaks. The family $S$ is defined only for $\sin (ω t) \neq 0$ , and the mixed derivative $S_{x α} = - m ω \csc (ω t)$ is unbounded as $ω t \to π Z$ ; the complete integral is a local object between the conjugate times of the oscillator and is not continued across them. The companion example computes the focusing of the characteristic family at those times.
No empirical content. The oscillator data are declared model parameters, and the displayed identities are statements about the model; no measured frequency, amplitude or period is asserted.

The harmonic model worked through: an explicit complete integral, its characteristic motion, and the action-angle chart with $I = \oint p d x$ and period-one frequency $1 / T$

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The harmonic model worked through: an explicit complete integral, its characteristic motion, and the action-angle chart with I=∮p dxI=∮pdx and period-one frequency 1/T1/T

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The harmonic model worked through: an explicit complete integral, its characteristic motion, and the action-angle chart with $I = \oint p d x$ and period-one frequency $1 / T$