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The conjugate time of the harmonic model: the complete integral's mixed derivative is unbounded and the characteristic family refocuses at $ω t \in π Z$

Physical scope

A worked warning inside the declared harmonic model of the framework's force branch: the explicit complete integral of the oscillator's Hamilton-Jacobi equation is defined only on the time strips between the conjugate times $ω t \in π Z$ ; its mixed derivative $S_{x α} = - m ω \csc (ω t)$ is unbounded as those times are approached; and the characteristic family through a fixed event refocuses there (all members with the same initial position pass through the same point a half-period later), so the map from the family's label to the position degenerates. The example asserts no measured value and no claim about any apparatus; it exhibits why the Hamilton-Jacobi statements of this page are local in time and carry no claim across a caustic.

Example

Adopt the harmonic model of The harmonic (linear restoring) model: a declared linear force law with parameter $k > 0$ as in the companion example. For the CM12-CM14 interfaces, declare an ambient one-label potential presentation on $Q_{a m b} = R^{3}$ , time window $R$ , mass operator $m I_{3}$ , potential $U_{a m b} (y) = k y_{1}^{2} / 2$ and residual force $d = 0$ . Its smooth immersion $X : R \to Q_{a m b}$ , $X (x) = (x, 0, 0)$ has mass metric $G = D X^{T} m I_{3} D X = m$ and substituted potential $k x^{2} / 2$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance). The transverse equations are $0 = 0$ , so scalar harmonic motions reconstruct ambient motions. The displayed scalar Hamiltonian is the generalized-coordinate chart Hamiltonian of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions clause (b).

Take one label of mass $m > 0$ , spring constant $k > 0$ , $ω = \sqrt{k / m} > 0$ , $T = 2 π / ω$ , branch Hamiltonian $H (x, p) = p^{2} / (2 m) + m ω^{2} x^{2} / 2$ , and the family

$S (t, x, α) = \frac{m ω}{2} [(x^{2} + α^{2}) \cot (ω t) - 2 x α \csc (ω t)]$ on the open strips where $\sin (ω t) \neq 0$ . The example verifies:

(nondegeneracy between conjugate times) the mixed derivative is $S_{x α} (t) = - \frac{m ω}{\sin (ω t)},$ so $S$ is a complete integral of the model's Hamilton-Jacobi equation at every point of each time strip $ω t \in (k π, (k + 1) π)$ , and $∣ S_{x α} (t) ∣ \to + \infty$ as $ω t \to k π$ for every $k \in Z$ ; the family has a pole there and offers no complete integral across it.
(degeneration of the characteristic family) the characteristic curves of $S$ are $x (t) = α \cos (ω t) - \frac{β}{m ω} \sin (ω t), β = S_{α} (t, x (t), α),$ and the derivative of the position with respect to the Jacobi label $β$ at fixed $(α, t)$ is $\frac{\partial x}{\partial β} (t) = - \frac{\sin (ω t)}{m ω}, \frac{\partial x}{\partial β} (t) = \frac{1}{S_{x α} (t)} (\sin (ω t) \neq 0),$ so it vanishes exactly at the conjugatetimes $ω t \in π Z$ , where the map from the label $β$ to the position degenerates while $S_{x α}$ is unbounded.
(refocusing of the model's motions) independently, the harmonic motions through a fixed event $(t_{0}, x_{0})$ , labelled by their initial momentum $q$ , $x (t; q) = x_{0} \cos (ω (t - t_{0})) + \frac{q}{m ω} \sin (ω (t - t_{0})),$ all satisfy $x (t_{1}; q) = - x_{0}$ at the conjugate time $t_{1} = t_{0} + π / ω$ , for every $q$ , and $\partial x (t; q) / \partial q = 0$ at $t = t_{1}$ : the family of model solutions through the event refocuses at $(t_{1}, - x_{0})$ , so no label of the family can be recovered from the position there.

All data $m, k$ are declared model parameters and no measured value is asserted.

Facts & Assumptions

Given: The harmonic model data $m > 0$ , $k > 0$ , the constants $ω, T$ , the Hamiltonian $H$ , the function $S$ on the strips $\sin (ω t) \neq 0$ , and the conjugate time $t_{1} = t_{0} + π / ω$ .

[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): the motions are exactly the curves $x (t) = A \cos (ω (t - t_{0})) + B \sin (ω (t - t_{0}))$ with reals $A, B$ ; the branch Hamiltonian is the displayed $H$ ; and the least positive period of a nonconstant motion is $T$ .

[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 $g (c) \neq 0$ , The zero sets of sine and cosine and the least positive common period 2 pi, Parity and the Pythagorean identity for sine and cosine, Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c \in I$ with $f^{'} (c) \neq 0$ , then the inverse $g$ is differentiable at $f (c)$ with $g^{'} (f (c)) = 1 / f^{'} (c)$ ; and if $f^{'} (c) = 0$ then $g$ is not differentiable at $f (c)$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): $(\sin)^{'} = \cos$ , $(\cos)^{'} = - \sin$ , $(\cot)^{'} = - \csc^{2}$ , $(\csc)^{'} = - \csc \cot$ ; the chain rule and algebra of derivatives apply; $\sin (k π) = 0$ and $\cos (k π) = (- 1)^{k}$ for integers $k$ ; and the one-variable inverse-function rule holds where the derivative is nonzero.

[L3]

Complete-integral definitions and characteristics (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 complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical $(β, α)$ labels): a $C^{3}$ solution $S$ of $S_{t} + H (x, S_{x}) = 0$ is a complete integral at a point exactly when $S_{x α} \neq 0$ there, and its characteristic curves with $p = S_{x}$ solve the Hamilton equations with $β = S_{α}$ constant.

[L4]

Hamilton equations in one dimension (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): for the branch Hamiltonian the field is $X_{H} = (H_{p}, - H_{x})$ , so $x^{'} = p / m$ and $p^{'} = - m ω^{2} x$ along solutions.

Verification

technique · direct; differentiate the displayed family, evaluate the mixed derivative, solve the characteristics explicitly, and exhibit the refocusing of the model's solutions at the conjugate time

1.1L2L3algebra

Mixed derivative: by [L2], differentiating the displayed $S$ gives $S_{x} = m ω (x \cot (ω t) - α \csc (ω t))$ and hence $S_{x α} = - m ω \csc (ω t) = - m ω / \sin (ω t)$ ; this is finite and nonzero exactly when $\sin (ω t) \neq 0$ , and $∣ S_{x α} ∣ \to + \infty$ as $ω t \to k π$ , $k \in Z$ , because $\sin (k π) = 0$ . Direct substitution gives $S_{t} + H (x, S_{x}) = \frac{m ω^{2} x^{2}}{2} (1 + \cot^{2} (ω t) - \csc^{2} (ω t)) = 0$ on the strips by [L2]. Thus by [L3], $S$ is a complete integral at every point of the strips and has no finite nondegenerate extension across the conjugate times. This is claim 1.

2.1L1L2L3L4step 1.1algebra

Characteristic curves: by [L3], solving $β = S_{α} (t, x, α) = m ω (α \cot (ω t) - x \csc (ω t))$ for $x$ gives $x (t) = α \cos (ω t) - \frac{β}{m ω} \sin (ω t),$ and with $p = S_{x}$ the pair solves $x^{'} = p / m$ , $p^{'} = - m ω^{2} x$ ; explicitly $x^{'} (t) = - α ω \sin (ω t) - β m^{- 1} \cos (ω t)$ and $\partial_{β} x (t) = - \sin (ω t) / (m ω)$ , which vanishes exactly at $ω t \in π Z$ ; moreover $(\partial_{β} x) (t) = 1 / S_{x α} (t)$ by step 1.1. At those times the label $β$ has no first-order effect on the position, so the map from the Jacobi label to the position degenerates while the mixed derivative is unbounded: this is the conjugate time of the family. This is claim 2.

3.1L1L2step 2.1algebra

Refocusing: by [L1] the curve $x (t; q) = x_{0} \cos (ω (t - t_{0})) + q m^{- 1} ω^{- 1} \sin (ω (t - t_{0}))$ is the motion of the model with $x (t_{0}) = x_{0}$ and $x^{'} (t_{0}) = q / m$ . At $t_{1} = t_{0} + π / ω$ , [L2] gives $\cos π = - 1$ and $\sin π = 0$ , so $x (t_{1}; q) = - x_{0}$ for every $q$ , and $\partial x (t; q) / \partial q = (m ω)^{- 1} \sin (ω (t - t_{0})) = 0$ at $t = t_{1}$ . Hence every member of the family of model solutions through $(t_{0}, x_{0})$ passes through the single point $(t_{1}, - x_{0})$ , and the initial-momentum label cannot be recovered from the position at $t_{1}$ ; by the inverse-function rule of [L2] this is the same degeneration as in step 2.1. This is claim 3.

4.1step 1.1step 2.1step 3.1∎

Claims 1-3 are steps 1.1, 2.1 and 3.1 respectively; the example is verified.

Remarks

The failure is a caustic, not an accident of the chart. The half-period refocusing in step 3.1 shows that the obstruction is invariant: at the conjugate time the endpoint-position map of the family of extremals is not locally invertible, so no smooth complete integral based on that family can satisfy the nondegeneracy $\det S_{q α} \neq 0$ across the caustic. The companion scope remark records the same warning for the page.
No empirical content. The conjugate time $π / ω$ is a property of the ideal harmonic model; the example asserts no measured period or frequency and no claim about a real oscillator.

The conjugate time of the harmonic model: the complete integral's mixed derivative is unbounded and the characteristic family refocuses at $ω t \in π Z$

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The conjugate time of the harmonic model: the complete integral's mixed derivative is unbounded and the characteristic family refocuses at ωt∈πZωt∈πZ

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The conjugate time of the harmonic model: the complete integral's mixed derivative is unbounded and the characteristic family refocuses at $ω t \in π Z$