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Second variation of the oscillator action on a finite interval: both signs for $T > π / ω$ and the value at $T = π / ω$

Statement

Let $ω > 0$ and $T > 0$ , let $U : = R$ , and consider the time-dependent Lagrangian chart over $[0, T] \times U$ with $O : = R \times R \times R$ and $L (q, v) : = \frac{1}{2} (v^{2} - ω^{2} q^{2})$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Write $S$ for its action and $δ^{2} S$ for the second variation of The fixed-endpoint second variation of a time-dependent action at a curve. Then:

the zero curve $q_{0} \equiv 0$ is a $C^{2}$ admissible curve and is stationary for $S$ ;
for every integer $n \geq 1$ the map $η_{n} (t) : = \sin (n π t / T)$ is a variation of $q_{0}$ , and

$δ^{2} S (q_{0}) [η_{n}, η_{n}] = \frac{T}{2} ((\frac{n π}{T})^{2} - ω^{2});$

moreover $S (ε η_{n}) = \frac{ε^{2}}{2} δ^{2} S (q_{0}) [η_{n}, η_{n}]$ for every real $ε$ ; 3. if $T > π / ω$ , then $δ^{2} S (q_{0}) [η_{1}, η_{1}] < 0$ and $δ^{2} S (q_{0}) [η_{n}, η_{n}] > 0$ for every integer $n > ω T / π$ . Consequently, for every real $δ_{0} > 0$ there exist variations $η$ of $q_{0}$ with $\sup_{t \in [0, T]} (∣ η (t) ∣ + ∣ η^{'} (t) ∣ + ∣ η^{''} (t) ∣) < δ_{0}$ and $S (η) < 0$ , and variations with the same bound and $S (η) > 0$ ; hence the zero curve is neither a local minimiser nor a local maximiser of $S$ , in the sense that every neighbourhood of $q_{0}$ for the supremum distance $\sup_{[0, T]} (∣ q - q_{0} ∣ + ∣ q^{'} - q_{0}^{'} ∣ + ∣ q^{''} - q_{0}^{''} ∣)$ contains curves whose action is below $S (q_{0})$ and curves whose action is above $S (q_{0})$ ; 4. if $T = π / ω$ , then $δ^{2} S (q_{0}) [η_{1}, η_{1}] = 0$ , while $δ^{2} S (q_{0}) [η_{n}, η_{n}] > 0$ for every integer $n \geq 2$ .

Facts & Assumptions

Given: The parameters $ω > 0$ and $T > 0$ , the chart data $(O, L)$ above, the zero curve $q_{0} \equiv 0$ , the action $S$ , the second variation $δ^{2} S$ , and the maps $η_{n} (t) = \sin (n π t / T)$ for integers $n \geq 1$ .

[F1]

Chart and stationarity (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression): the Euler-Lagrange expression of this chart along a $C^{2}$ curve $q$ is $E (q) (t) = - ω^{2} q (t) - q^{''} (t)$ , and $q$ is stationary for $S$ if and only if $E (q) = 0$ on $[0, T]$ .

[F2]

Second variation formula (The second variation of a time-dependent action: twice differentiability of the fixed-endpoint family and the Hessian integral formula, The fixed-endpoint second variation of a time-dependent action at a curve): for every $C^{2}$ curve $q$ on $[0, T]$ and every variation $η$ , $δ^{2} S (q) [η, η] = \int_{0}^{T} (η^{'} (t)^{2} - ω^{2} η (t)^{2}) d t,$ because the second partials of $L$ are the constants $\partial^{2} L / \partial q^{2} = - ω^{2}$ , $\partial^{2} L / \partial q \partial v = 0$ and $\partial^{2} L / \partial v^{2} = 1$ .

[F3]

Trigonometric data (The derivatives of sine and cosine are cosine and minus sine, Sine and cosine defined by their real power series, The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ): $\sin$ and $\cos$ are differentiable with $\sin^{'} = \cos$ and $\cos^{'} = - \sin$ , and $\sin 0 = 0$ ; for every real $u$ the addition formulas and the Pythagorean identity give $\cos^{2} u = \frac{1}{2} (1 + \cos 2 u)$ and $\sin^{2} u = \frac{1}{2} (1 - \cos 2 u)$ ; the zero set of $\sin$ is exactly $π Z$ . Hence for every integer $n \geq 1$ the map $η_{n} (t) = \sin (n π t / T)$ is $C^{2}$ on an open interval containing $[0, T]$ , with $η_{n}^{'} (t) = (n π / T) \cos (n π t / T)$ and $η_{n} (0) = η_{n} (T) = \sin (n π) = 0$ .

[F4]

Integration (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 $\int_{a}^{b} f$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): continuous real functions on $[0, T]$ are Darboux integrable; if $G$ is differentiable with derivative $G^{'}$ and $G^{'}$ is integrable then $\int_{0}^{T} G^{'} = G (T) - G (0)$ ; and the integral of a finite linear combination of integrable functions is the corresponding combination of integrals.

[F5]

Archimedean property (Every complete ordered field is Archimedean): for every real $x$ there is a natural number $n$ with $x < n$ .

Proof

Proof technique: direct; compute the second variation of the oscillator family with elementary trigonometric integrals, obtain the exact action along the family, and read off the two signs and the degenerate case. [given]

1.1givenF1F4algebra

Stationarity of the zero curve: $q_{0}$ is a $C^{2}$ admissible curve with values in $U = R$ , and $E (q_{0}) (t) = - ω^{2} \cdot 0 - 0 = 0$ for every $t$ , so $q_{0}$ is stationary by [F1]; also $S (q_{0}) = \int_{0}^{T} 0 d t = 0$ by [F4].

1.2givenF2F3

Variation and second variation: for every integer $n \geq 1$ , $η_{n}$ is a variation of $q_{0}$ by [F3], so [F2] and [F3] give $δ^{2} S (q_{0}) [η_{n}, η_{n}] = \int_{0}^{T} ((\frac{n π}{T})^{2} \cos^{2} (\frac{n π t}{T}) - ω^{2} \sin^{2} (\frac{n π t}{T})) d t .$

2.1step 1.2F3F4algebra

Evaluation of the integrals: by [F3] and [F4], $\int_{0}^{T} \cos^{2} (\frac{n π t}{T}) d t = \frac{1}{2} \int_{0}^{T} (1 + \cos \frac{2 n π t}{T}) d t = \frac{T + 0}{2} = \frac{T}{2},$ and likewise $\int_{0}^{T} \sin^{2} (n π t / T) d t = T / 2$ , because the FTC gives $\int_{0}^{T} \cos (2 n π t / T) d t = \frac{T}{2 n π} [\sin (2 n π t / T)]_{0}^{T} = \frac{T}{2 n π} (\sin (2 n π) - \sin 0) = 0$ by the vanishing of $\sin$ at integer multiples of $π$ from [F3]. Substituting into step 1.2 gives the second-variation value of clause 2.

3.1step 1.2step 2.1F4algebra

Exact action along the family: since $L$ is homogeneous of degree two in $(q, v)$ , for every real $ε$ the integrand of $S (ε η_{n})$ is $L (ε η_{n}, ε η_{n}^{'}) = ε^{2} L (η_{n}, η_{n}^{'})$ ; by linearity of the integral [F4] and step 1.2, $S (ε η_{n}) = ε^{2} S (η_{n}) = \frac{ε^{2}}{2} δ^{2} S (q_{0}) [η_{n}, η_{n}]$ , which is the exact identity of clause 2.

4.1step 1.1step 2.1step 3.1F3F5algebra

Signs for $T > π / ω$ : if $T > π / ω$ , then $π / T < ω$ , so clause 2 gives $δ^{2} S (q_{0}) [η_{1}, η_{1}] < 0$ ; by [F5] choose an integer $n$ with $n > ω T / π$ , so $(n π / T)^{2} > ω^{2}$ and $δ^{2} S (q_{0}) [η_{n}, η_{n}] > 0$ . For any real $δ_{0} > 0$ pick $ε > 0$ with $ε (1 + n π / T + (n π / T)^{2}) < δ_{0}$ ; then $∣ η_{n} ∣ \leq 1$ , $∣ η_{n}^{'} ∣ \leq n π / T$ and $∣ η_{n}^{''} ∣ \leq (n π / T)^{2}$ by [F3], so the variations $ε η_{1}$ and $ε η_{n}$ satisfy the displayed supremum bound, while step 3.1 gives $S (ε η_{1}) = \frac{ε^{2}}{2} δ^{2} S (q_{0}) [η_{1}, η_{1}] < 0$ and $S (ε η_{n}) > 0$ ; with $S (q_{0}) = 0$ from step 1.1, the zero curve is neither a local minimiser nor a local maximiser.

5.1step 2.1algebra∎

The degenerate case: if $T = π / ω$ , substitution into clause 2 gives $δ^{2} S (q_{0}) [η_{1}, η_{1}] = \frac{T}{2} (ω^{2} - ω^{2}) = 0$ , while for every integer $n \geq 2$ the value is $\frac{T}{2} (n^{2} ω^{2} - ω^{2}) > 0$ .

Remarks

The computation is a finite-time statement about the action functional on $[0, T]$ with fixed endpoints. For every $T > π / ω$ it shows that the zero stationary curve is neither a local minimum nor a local maximum; nothing here asserts anything about the physical least-action principle, and the results are mathematical properties of the model A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations.
The bound $n > ω T / π$ is the quantisation of the statement that a sufficiently high-frequency sine variation has positive second variation; at $T = π / ω$ the displayed second-variation value in direction $η_{1}$ is zero, consistent with the endpoint non-uniqueness of The undamped oscillator: Dirichlet non-uniqueness at $T = n π / ω$ despite unique initial-value solutions.

Second variation of the oscillator action on a finite interval: both signs for $T > π / ω$ and the value at $T = π / ω$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Second variation of the oscillator action on a finite interval: both signs for T>π/ωT>π/ω and the value at T=π/ωT=π/ω

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Second variation of the oscillator action on a finite interval: both signs for $T > π / ω$ and the value at $T = π / ω$