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The velocity-Verlet step matches the exact flow through order two: local truncation error $O (h^{3})$ on a convex tube

Statement

Let $d \geq 1$ and let $U$ , $M$ , $V$ , $A$ , $B$ , $H = A + B$ , $D_{h}$ , $K_{h}$ and $V_{h}$ be as in The splitting Hamiltonian $H = A + B$ , the drift and kick maps, and the velocity-Verlet map. Let $W \subseteq U$ be open, bounded and convex, let $K \subseteq W$ be nonempty compact, and let $r > 0$ satisfy $K + {\overline{B}}_{r} (0) \subseteq W$ (Open ball, closed ball and sphere in a metric space). Put

$X_{H} (q, p) : = (M^{- 1} p, - \nabla V (q)), M_{0} : = \sup_{W} ∥ X_{H} ∥, L_{1} : = \sup_{W} ∥ D X_{H} ∥, L_{2} : = \sup_{W} ∥ D^{2} X_{H} ∥,$

the derivatives being the total derivatives with operator norms (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, $C^{k}$ Euclidean maps and diffeomorphisms, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder); assume all three numbers are finite (for example, this holds when $\overline{W} \subseteq U$ ). Let $Φ_{t}$ be the local flow of $X_{H}$ (The local flow of an autonomous first-order system).

Then there exist $h_{0} > 0$ and $C < \infty$ , depending only on $d$ , $r$ , $∥ M^{- 1} ∥$ , $\sup_{K} ∥ p ∥$ , $M_{0}$ , $L_{1}$ and $L_{2}$ , such that for every $(q, p) \in K$ and every $0 < ∣ h ∣ \leq h_{0}$ :

$Φ_{h} (q, p)$ is defined and lies in $W$ , and $∥ Φ_{h} (q, p) - (q, p) - h X_{H} (q, p) - \frac{h^{2}}{2} D X_{H} (q, p) X_{H} (q, p) ∥ \leq C ∣ h ∣^{3} .$
$V_{h} (q, p)$ is defined and lies in $W$ , and its two block components satisfy $V_{h} (q, p) = (q + h M^{- 1} p - \frac{h^{2}}{2} M^{- 1} \nabla V (q) + ρ_{1}, p - h \nabla V (q) - \frac{h^{2}}{2} H_{V} (q) M^{- 1} p + ρ_{2})$ with $∥ ρ_{1} ∥, ∥ ρ_{2} ∥ \leq C ∣ h ∣^{3}$ ; equivalently $V_{h}$ and $Φ_{h}$ have the same expansion through order two, $X_{H} (q, p) = (M^{- 1} p, - \nabla V (q)), D X_{H} (q, p) X_{H} (q, p) = (- M^{- 1} \nabla V (q), - H_{V} (q) M^{- 1} p) .$
Consequently $∥ V_{h} (q, p) - Φ_{h} (q, p) ∥ \leq C ∣ h ∣^{3}$ for every $(q, p) \in K$ and $0 < ∣ h ∣ \leq h_{0}$ .

Facts & Assumptions

Given: The data of The splitting Hamiltonian $H = A + B$ , the drift and kick maps, and the velocity-Verlet map; an open bounded convex $W \subseteq U$ , a nonempty compact $K \subseteq W$ and $r > 0$ with $K + {\overline{B}}_{r} (0) \subseteq W$ ; the finite constants $M_{0}, L_{1}, L_{2}$ of the Statement; the local flow $Φ_{t}$ of $X_{H}$ ; the abbreviations $z = (q, p)$ .

[F1]

Field and derivatives. The coordinate Hamiltonian-field convention $X_{G} = J \nabla G$ of The splitting Hamiltonian $H = A + B$ , the drift and kick maps, and the velocity-Verlet map gives $X_{H} (q, p) = J \nabla H (q, p) = (M^{- 1} p, - \nabla V (q))$ ; here $(M^{- 1})^{T} = M^{- 1}$ by that definition. Its total derivative is the block matrix $D X_{H} (q, p) = (\begin{array}{cc} 0 & M^{- 1} \\ - H_{V} (q) & 0 \end{array})$ with $H_{V}$ the Hessian of $V$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The Hessian matrix and critical points of a scalar field). Since $V \in C^{3}$ , $X_{H}$ is $C^{2}$ , and the finiteness of $M_{0}, L_{1}, L_{2}$ is an explicit hypothesis, satisfied in particular when $\overline{W} \subseteq U$ by continuity and the extreme value theorem ( $C^{k}$ Euclidean maps and diffeomorphisms, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F2]

Quantitative local existence and uniqueness. If $F$ is continuous and locally Lipschitz on the open $D \subseteq R \times R^{n}$ , $(t_{0}, x_{0}) \in D$ , and $h, r > 0$ satisfy $[t_{0} - h, t_{0} + h] \times \overline{B} (x_{0}, r) \subseteq D$ , $∥ F ∥ \leq M$ and $Lip$ -constant $L$ on that cylinder with $h M \leq r$ and $L h < 1$ , then there is exactly one solution through $(t_{0}, x_{0})$ on $[t_{0} - h, t_{0} + h]$ with graph in the cylinder; any two solutions through the same data agree on every common interval containing $t_{0}$ (Picard-Lindelöf local existence and uniqueness for first-order systems, The local flow of an autonomous first-order system). A $C^{1}$ field on $W$ has Lipschitz constant $L_{1}$ on $W$ : for $x, y \in W$ the segment lies in $W$ by convexity, and the mean value estimate $∥ F (x) - F (y) ∥ \leq L_{1} ∥ x - y ∥$ follows by integrating the derivative along the segment.

[F3]

Vector-valued FTC and norm inequality. For a $C^{1}$ curve $u : [a, b] \to R^{n}$ with continuous derivative, $u (b) - u (a) = \int_{a}^{b} u^{'}$ ; and $∥ \int_{a}^{b} u ∥ \leq \int_{a}^{b} ∥ u ∥$ for continuous $u$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

[F4]

Chain rule. $D (g \circ f) (x) = D g (f (x)) \circ D f (x)$ for totally differentiable composites (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F5]

Hessian symmetry and the integral form of the mean value estimate. $H_{V} (q)^{T} = H_{V} (q)$ , and for $C^{1}$ $H_{V}$ and points $q, q^{'} \in Q : = {pr}_{q} (W) \subseteq Π$ , where ${pr}_{q} (q, p) = q$ and $Q$ is open and convex, one has $\nabla V (q^{'}) - \nabla V (q) = \int_{0}^{1} H_{V} (q + t (q^{'} - q)) (q^{'} - q) d t$ by [F3] applied to $t \mapsto \nabla V (q + t (q^{'} - q))$ and [F4] (The Hessian of a $C^{2}$ scalar field is symmetric, The Hessian matrix and critical points of a scalar field).

Proof

technique · direct; bound the second derivative of the exact-flow deviation by the fundamental theorem of calculus, and expand the three Verlet factors exactly, bounding the potential-gradient remainder by the Lipschitz constant of the Hessian

1.1givenF1F2F3algebra

Uniform short existence. Put $P = \sup_{K} ∥ p ∥$ and $M_{1} = ∥ M^{- 1} ∥$ . Choose $0 < h_{0} \leq 1$ so that $h_{0} M_{0} \leq r / 8$ , $h_{0} L_{1} \leq 1 / 2$ , and $h_{0} M_{1} (P + M_{0}) \leq r / 8$ , ignoring a restriction whose left side is identically zero. Then [F2] on the ball $\overline{B} (z, r / 2) \subseteq W$ supplies the flow from each $z \in K$ for $∣ t ∣ \leq h_{0}$ . Its displacement is at most $M_{0} ∣ t ∣$ by the integral equation and [F3].

2.1givenstep 1.1F2F3F4algebra

Deviation of the exact flow. Define $g (t) : = Φ_{t} (z) - z - t X_{H} (z) - \frac{t^{2}}{2} D X_{H} (z) X_{H} (z)$ for $∣ t ∣ \leq h_{0}$ . By [F4] and the flow equation $\partial_{t} Φ_{t} = X_{H} (Φ_{t})$ (The local flow of an autonomous first-order system) the curve $g^{'}$ has derivative $g^{''} (t) = D X_{H} (Φ_{t} (z)) X_{H} (Φ_{t} (z)) - D X_{H} (z) X_{H} (z)$ ; writing $Δ_{t} : = Φ_{t} (z) - z$ and using the mean value estimate for $X_{H}$ on the convex $W$ (so that $∥ X_{H} (Φ_{t}) - X_{H} (z) ∥ \leq L_{1} ∥ Δ_{t} ∥$ , [F2]), $∥ g^{''} (t) ∥ \leq ∥ D X_{H} (Φ_{t}) - D X_{H} (z) ∥ ∥ X_{H} (Φ_{t}) ∥ + ∥ D X_{H} (z) ∥ ∥ X_{H} (Φ_{t}) - X_{H} (z) ∥ \leq (L_{2} M_{0} + L_{1}^{2}) ∥ Δ_{t} ∥ \leq (L_{2} M_{0} + L_{1}^{2}) M_{0} ∣ t ∣ .$ Since $g (0) = g^{'} (0) = 0$ , two applications of [F3] give $∥ g^{'} ∥ \leq \frac{1}{2} (L_{2} M_{0} + L_{1}^{2}) M_{0} t^{2}$ and $∥ g (h) ∥ \leq \frac{1}{6} (L_{2} M_{0} + L_{1}^{2}) M_{0} ∣ h ∣^{3}$ . This proves clause 1 with $C_{1} : = \frac{1}{6} (L_{2} M_{0} + L_{1}^{2}) M_{0} + 1$ .

2.2givenstep 1.1F1algebra

Intermediate points. Put $p_{1} = p - h \nabla V (q) / 2$ , $q_{2} = q + h M^{- 1} p_{1}$ , and $p_{2} = p_{1} - h \nabla V (q_{2}) / 2$ . The first kick displacement is at most $r / 16$ , and the drift displacement is at most $r / 8$ , so $(q_{2}, p_{1}) \in B (z, r / 2) \subseteq W$ . The final kick also has displacement at most $r / 16$ since the field is bounded there. Thus every intermediate point and the endpoint lie in $W$ , legitimating all three evaluations. The block formula for $D X_{H} X_{H}$ follows from [F1].

3.1givenstep 2.2F1F2F3F5algebra

Expansion of the potential gradient. With $w : = q_{2} - q$ , [F5] gives $\nabla V (q_{2}) = \nabla V (q) + \int_{0}^{1} H_{V} (q + t w) w d t = \nabla V (q) + H_{V} (q) w + R$ with $∥ R ∥ \leq \frac{1}{2} L_{V} ∥ w ∥^{2}$ , where $L_{V} : = \sup_{q \in Q} ∥ D H_{V} (q) ∥ \leq L_{2}$ is a Lipschitz constant of $H_{V}$ on the convex position set $Q = {pr}_{q} (W)$ . Indeed, each $q \in Q$ has a lift $(q, p) \in W$ , and $D H_{V} (q)$ is the momentum-output, position-input block of $D^{2} X_{H} (q, p)$ , giving its operator-norm bound by $L_{2}$ . Integrating this derivative along segments in $Q$ gives $∥ H_{V} (q + t w) - H_{V} (q) ∥ \leq t L_{2} ∥ w ∥$ , and then [F3] gives the stated remainder. Both endpoints $q, q_{2}$ lie in $Q$ by step 2.2, and $∥ w ∥ \leq r / 8$ .

4.1givenstep 2.2step 3.1F1algebra

Exact algebra and remainders. The position identity is exact: $q_{2} = q + h M^{- 1} p - h^{2} M^{- 1} \nabla V (q) / 2$ , so $ρ_{1} = 0$ . In the momentum identity use step 3.1 with $w = h M^{- 1} p - h^{2} M^{- 1} \nabla V (q) / 2$ . This gives $p_{2} = p - h \nabla V (q) - h^{2} H_{V} (q) M^{- 1} p / 2 + ρ_{2}$ , where $ρ_{2} = h^{3} H_{V} (q) M^{- 1} \nabla V (q) / 4 - h R / 2$ . Its norm is at most $∣ h ∣^{3} (L_{1} M_{1} M_{0} + L_{2} M_{1}^{2} (P + M_{0})^{2}) / 4$ , since $∥ H_{V} (q) ∥ \leq L_{1}$ and $∥ D H_{V} (q) ∥ \leq L_{2}$ on $Q$ are block restrictions of $D X_{H}, D^{2} X_{H}$ at any lift $(q, p) \in W$ .

5.1step 2.1step 4.1algebra∎

Conclusion. Step 2.1 bounds the exact-flow remainder and step 4.1 bounds the Verlet remainder. Their terms through degree two agree, so the triangle inequality gives the cubic local defect after enlarging $C$ to bound both expansions and their difference. All constants use only the stated finite bounds and tube data.

Remarks

The convexity of $W$ is used only to have the segment between any two points of $W$ inside $W$ , so that mean value estimates for $X_{H}$ and $H_{V}$ apply; the constants are unoptimised, and no sharpness of the cubic order is claimed here. The oscillator example on the companion page supplies an exact quadratic invariant and an original-energy comparison.
The hypotheses are open conditions on the data: the convex tube is an explicit hypothesis; a compact subset of a general nonconvex $U$ need not admit one (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). No physical interpretation of $M$ , $V$ or $h$ enters the argument.

The velocity-Verlet step matches the exact flow through order two: local truncation error $O (h^{3})$ on a convex tube

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

The velocity-Verlet step matches the exact flow through order two: local truncation error O(h3)O(h3) on a convex tube

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

The velocity-Verlet step matches the exact flow through order two: local truncation error $O (h^{3})$ on a convex tube