Symplectic Verlet and analytic backward error

Let $(X,d)$ be a metric space (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $V:X\to X$ map $X$ into itself, and let $L\ge 0$ and $h>0$. Assume

$d(V(x),V(y))\le (1+Lh)d(x,y)\text{for all }x,y\in X,$

which is a Lipschitz hypothesis on $V$ with constant $1+Lh$ (Lipschitz map, $\alpha$-Hölder map for rational $0<\alpha \le 1$, and contraction). Fix $z_0\in X$, put $z_{j+1}:=V(z_j)$ for $j\in \mathbb{N}$, and let $x_0:=z_0$ together with points $x_{j+1}\in X$ satisfying

$d(x_{j+1},V(x_j))\le \eta (j\in \mathbb{N})$

for a number $\eta \ge 0$ ("computational error per step"). Then for every $n\in \mathbb{N}$:

1. if $L>0$, ${\displaystyle d(x_n,z_n)\le \eta \frac{(1+Lh{)}^n-1}{Lh}}$, and if $L=0$, $d(x_n,z_n)\le n\eta$;
2. if moreover $F:X\to \mathbb{R}$ satisfies the Lipschitz bound $\mathrm{\mid }F(x)-F(y)\mathrm{\mid }\le \mathrm{\Lambda }d(x,y)$ for all $x,y\in X$ and some $0\le \mathrm{\Lambda }<\mathrm{\infty }$, then $\mathrm{\mid }F(x_n)-F(z_n)\mathrm{\mid }\le \mathrm{\Lambda }\eta \frac{(1+Lh{)}^n-1}{Lh}(L>0),\mathrm{\mid }F(x_n)-F(z_n)\mathrm{\mid }\le \mathrm{\Lambda }n\eta (L=0);$
3. if $nh\le T$ for some $T>0$ and $L>0$, then $(1+Lh{)}^n\le e^{LT}$ and therefore $d(x_n,z_n)\le \eta \frac{e^{LT}-1}{Lh}\le \eta e^{LT}\frac{T}{h}\mathrm{.}$

• The lemma is a general accumulation statement about a one-step scheme with an external per-step perturbation; it does not model any concrete floating-point arithmetic, and $\eta$ is a hypothesis on the computed points, not a derived roundoff bound. For velocity Verlet, its application requires a metric state set $(X,d)$ on which all stages are defined, $V_h:X\to X$, computed points remaining in $X$, and a positive step $h>0$. Establish separately a numerical-map bound $d(V_h(x),V_h(y))\le (1+L_{\mathrm{m}\mathrm{a}\mathrm{p}}h)d(x,y)$ for every $x,y\in X$. Then, for $nh\le T$ and $L_{\mathrm{m}\mathrm{a}\mathrm{p}}>0$, this lemma contributes $\eta (e^{L_{\mathrm{m}\mathrm{a}\mathrm{p}}T}-1)\mathrm{/}(L_{\mathrm{m}\mathrm{a}\mathrm{p}}h)$ to the finite-time error comparison, or $n\eta$ when $L_{\mathrm{m}\mathrm{a}\mathrm{p}}=0$. The exact-field derivative constant in Finite-time global error of the velocity-Verlet method: $\parallel z_n-{\mathrm{\Phi }}_{nh}(z_0)\parallel \le C(e^{LT}-1)h^2\mathrm{/}L$ bounds the exact flow and does not establish this map bound or invariance of $X$. Combining the estimates requires their hypotheses on a common admissible set; exact-arithmetic symplecticity in The drift, kick and velocity-Verlet maps are canonical: $(DC{)}^{\mathsf{T}}JDC=J$ and $C^{\ast }\omega =\omega$ does not itself provide these error hypotheses.
• No choice principle is used; the induction is finite and the geometric sums are finite sums.

Holomorphic maps. Let $m,p\ge 1$ and let $D\subseteq {\mathbb{C}}^m$ be open. Write each coordinate $z_k=x_k+i{y}_k$ and identify $D$ with the corresponding open subset of ${\mathbb{R}}^{2m}$ by $(z_1,\dots ,z_m)\mapsto (x_1,y_1,\dots ,x_m,y_m)$. A map $F:D\to {\mathbb{C}}^p$ is complex differentiable at $a\in D$ when there is a map $L:{\mathbb{C}}^m\to {\mathbb{C}}^p$ that is $\mathbb{C}$-linear — $L(u+v)=L(u)+L(v)$ and $L(\lambda u)=\lambda L(u)$ for $\lambda \in \mathbb{C}$ (Linear map between vector spaces over the same field, A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$) — with

$F(a+v)=F(a)+L(v)+r(v),\frac{\parallel r(v)\parallel }{\parallel v\parallel }⟶0(v\to 0),$

the norm being the Euclidean norm of ${\mathbb{R}}^{2m}$ and ${\mathbb{R}}^{2p}$ (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder). Such an $L$ is unique by the argument of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions, applied componentwise; it is denoted $DF(a)$. The map $F$ is holomorphic on $D$ when it is complex differentiable at every point of $D$; for $m=p=1$ this is the notion of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions. For holomorphic $F$ and $k\le m$ put ${\mathrm{\partial }}_{z_k}F(a):=DF(a)(e_k)$, the $k$-th coordinate derivative; for $p=1$ this is the partial derivative of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ in the complex coordinate direction.

Real analytic functions. Let $N\ge 1$ and let $V\subseteq {\mathbb{R}}^N$ be open. A function $\phi :V\to \mathbb{R}$ is real analytic at $x\in V$ when there are an open set $W\subseteq {\mathbb{C}}^N$ with $x\in W$ and a holomorphic $F:W\to \mathbb{C}$ with $F(\xi )=\phi (\xi )$ for every $\xi \in W\cap V$; $\phi$ is real analytic on $V$ when it is real analytic at every point of $V$. Thus real analyticity is defined here exactly as admitting a local holomorphic extension.

Polydiscs. For $a\in {\mathbb{C}}^m$ and a polyradius $\rho =({\rho }_1,\dots ,{\rho }_m)$ with $0<{\rho }_k\le \mathrm{\infty }$ put

${\mathrm{\Delta }}_{\rho }(a):=\{z\in {\mathbb{C}}^m:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }<{\rho }_k\text{ for all }k\},$

the open polydisc, which is open and convex (Open ball, closed ball and sphere in a metric space, 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). If every ${\rho }_k$ is finite, its distinguished boundary is

${\mathrm{\Gamma }}_{\rho }(a):=\{z\in {\mathbb{C}}^m:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }={\rho }_k\text{ for all }k\}\mathrm{.}$

When any radius is infinite, no distinguished boundary is defined.

Complex phase domains. Let $d\ge 1$. A complex phase domain is a nonempty open $D\subseteq {\mathbb{C}}^{2d}$ whose points are written $Z=(z,w)$ with $z=(z^1,\dots ,z^d)$, $w=(w_1,\dots ,w_d)\in {\mathbb{C}}^d$. In these coordinates, ${\mathrm{\partial }}_{z^i}$ and ${\mathrm{\partial }}_{w_i}$ denote the coordinate derivatives defined above for the corresponding entries of $Z$. On $D$ define the complexified canonical two-form and the complexified Poisson bracket of holomorphic functions $f,g:D\to \mathbb{C}$ by

${\omega }_{\mathbb{C}}:={\sum }_{i=1}^{d}d{z}^i\wedge d{w}_i,\{f,g\}:={\sum }_{i=1}^d({\mathrm{\partial }}_{z^i}f{\mathrm{\partial }}_{w_i}g-{\mathrm{\partial }}_{w_i}f{\mathrm{\partial }}_{z^i}g),$

and for a holomorphic $f:D\to \mathbb{C}$ its complex Hamiltonian field $X_f:=J\mathrm{\nabla }f$, where the same entries of $J=\left(\begin{array}{cc}{0}_d& I_d\\ -I_d& 0_d\end{array}\right)$ from A phase domain, its canonical one-form and its standard symplectic two-form are regarded as a matrix in $M_{2d}(\mathbb{C})$, and $\mathrm{\nabla }f=({\mathrm{\partial }}_{z^1}f,\dots ,{\mathrm{\partial }}_{z^d}f,{\mathrm{\partial }}_{w_1}f,\dots ,{\mathrm{\partial }}_{w_d}f)$ is the gradient column of coordinate derivatives. If $D_{\mathbb{R}}:=D\cap {\mathbb{R}}^{2d}$ is nonempty, it is open in ${\mathbb{R}}^{2d}$, and the restrictions of ${\omega }_{\mathbb{C}}$ and $J$ to $D_{\mathbb{R}}$ are exactly the standard two-form and matrix of A phase domain, its canonical one-form and its standard symplectic two-form.

Distance domains. For a nonempty $U\subseteq {\mathbb{R}}^{2d}$ and $r>0$ put

$D(r):=\{\zeta \in {\mathbb{C}}^{2d}:\mathrm{dist}(\zeta ,\bar{U})<r\},\mathrm{dist}(\zeta ,\bar{U}):=\inf \{\parallel \zeta -x\parallel :x\in \bar{U}\},$

where $\bar{U}$ is the closure of $U$ in ${\mathbb{R}}^{2d}$ and the norm is the Euclidean norm of ${\mathbb{C}}^{2d}\cong {\mathbb{R}}^{4d}$. Each $D(r)$ is open, contains $U$, and $D(r)\subseteq D(r^{\mathrm{\prime }})$ for $r\le r^{\mathrm{\prime }}$; if $U$ is star-shaped with respect to $0$ (Star-shaped open subsets of Euclidean space), then every $D(r)$ is star-shaped with respect to $0$ as well.

• The definition of holomorphy used here is the $\mathbb{C}$-linear total-derivative definition, matching the one-variable published definition. The equivalence with power-series representability and with continuity plus separate holomorphy is not assumed: it is proved in Iterated Cauchy formula, power-series expansion and Cauchy estimates for functions continuous and separately holomorphic on a polydisc and Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits on this page, together with the differentiability of the coordinate derivatives.

• No choice principle is used. All objects are the standard complexifications of the real canonical objects; no physical interpretation is attached to $z,w$ or to ${\omega }_{\mathbb{C}}$ in this definition.

• To verify the distance-domain claim, first note that $\bar{U}$ is star-shaped about $0$: if $x_j\in U$ converges to $x\in \bar{U}$, then $t{x}_j\in U$ and $t{x}_j\to tx$, so $tx\in \bar{U}$ for every $0\le t\le 1$. If $\zeta \in D(r)$, the inequality $\mathrm{dist}(\zeta ,\bar{U})<r$ supplies $x\in \bar{U}$ with $\parallel \zeta -x\parallel <r$. Thus $tx\in \bar{U}$ and $\mathrm{dist}(t\zeta ,\bar{U})\le \parallel t\zeta -tx\parallel =t\parallel \zeta -x\parallel <r$ for $0\le t\le 1$. Hence $t\zeta \in D(r)$, proving that $D(r)$ is star-shaped with respect to $0$.

Let $d\ge 1$ and let $U\subseteq {\mathbb{R}}^{2d}$ be a nonempty open phase domain with coordinates $z=(q,p)$, $q=(q^1,\dots ,q^d)$ and $p=(p_1,\dots ,p_d)$, standard two-form ${\omega }_U={\sum }_{i}d{q}^i\wedge d{p}_i$ and standard symplectic matrix $J$ (A phase domain, its canonical one-form and its standard symplectic two-form).

Fix $M\in M_d(\mathbb{R})$ and $M^{-1}\in M_d(\mathbb{R})$ with $M{M}^{-1}=M^{-1}M=I_d$. Assume $M$ is symmetric and positive definite, meaning $M^{\mathsf{T}}=M$ (The transpose $A^{\mathsf{T}}$ of a matrix) and $v^{\mathsf{T}}Mv>0$ for every nonzero $v\in {\mathbb{R}}^d$; here $v^{\mathsf{T}}Mv$ is the matrix product of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes. Transposing the two inverse identities and using uniqueness of a two-sided inverse gives $(M^{-1}{)}^{\mathsf{T}}=(M^{\mathsf{T}}{)}^{-1}=M^{-1}$. Fix an open $\mathrm{\Pi }\subseteq {\mathbb{R}}^d$ with $U\subseteq \mathrm{\Pi }\times {\mathbb{R}}^d$ and a function $U_{\mathrm{p}\mathrm{o}\mathrm{t}}:\mathrm{\Pi }\to \mathbb{R}$ of class $C^3$ ($C^k$ Euclidean maps and diffeomorphisms), whose gradient $\mathrm{\nabla }{U}_{\mathrm{p}\mathrm{o}\mathrm{t}}$ and Hessian matrix $H_{U_{\mathrm{p}\mathrm{o}\mathrm{t}}}$ are those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and The Hessian matrix and critical points of a scalar field; write $\mathrm{\nabla }{U}_{\mathrm{p}\mathrm{o}\mathrm{t}}(q)$ as a column vector and abbreviate $U_{\mathrm{p}\mathrm{o}\mathrm{t}}(q)$ by $V(q)$ on $U$.

For any $C^1$ scalar function $G:U\to \mathbb{R}$, use the coordinate convention $X_G:=J\mathrm{\nabla }G=({\mathrm{\partial }}_{p_i}G,-{\mathrm{\partial }}_{q^i}G{)}_{i=1}^d$ for its Hamiltonian field. With ${\omega }_U={\sum }_{i}d{q}^i\wedge d{p}_i$, contraction is ${\iota }_{X_G}{\omega }_U:={\sum }_i((X_G{)}_{q^i}d{p}_i-(X_G{)}_{p_i}d{q}^i)={\sum }_i({\mathrm{\partial }}_{p_i}Gd{p}_i+{\mathrm{\partial }}_{q^i}Gd{q}^i)=dG$.

On $U$ define the kinetic part and potential part

$A(q,p):=\frac{1}{2}{p}^{\mathsf{T}}{M}^{-1}p,B(q,p):=V(q),H:=A+B,$

and the ambient-valued drift and kick formulas

${\widehat{D}}_h(q,p):=(q+h{M}^{-1}p,p),{\widehat{K}}_h(q,p):=(q,p-h\mathrm{\nabla }V(q))((q,p)\in U)\mathrm{.}$

Use these as partial self-maps of $U$ by setting ${\mathcal{D}}_h:=\{(q,p)\in U:{\widehat{D}}_h(q,p)\in U\},D_h:={\widehat{D}}_h{\mathrm{\mid }}_{{\mathcal{D}}_h}:{\mathcal{D}}_h\to U,$ ${\mathcal{K}}_h:=\{(q,p)\in U:{\widehat{K}}_h(q,p)\in U\},K_h:={\widehat{K}}_h{\mathrm{\mid }}_{{\mathcal{K}}_h}:{\mathcal{K}}_h\to U\mathrm{.}$ The velocity-Verlet map of the splitting $H=A+B$ at step $h\in \mathbb{R}$ is the partial composite

$V_h:=K_{h\mathrm{/}2}\circ D_h\circ K_{h\mathrm{/}2}:{\mathcal{V}}_h\to U,$ where ${\mathcal{V}}_h:=\{z\in {\mathcal{K}}_{h\mathrm{/}2}:K_{h\mathrm{/}2}(z)\in {\mathcal{D}}_h\text{ and }{D}_h(K_{h\mathrm{/}2}(z))\in {\mathcal{K}}_{h\mathrm{/}2}\}$. Thus every intermediate input and the final output belongs to $U$.

• The definition is purely mathematical and choice-free; the names drift, kick and velocity-Verlet are conventional, and no physical units are assigned. The coordinate Hamiltonian fields of $A$ and $B$ are $X_A(q,p)=(M^{-1}p,0)$ and $X_B(q,p)=(0,-\mathrm{\nabla }V(q))$ by the convention above. The proposition The drift, kick and velocity-Verlet maps are canonical: $(DC{)}^{\mathsf{T}}JDC=J$ and $C^{\ast }\omega =\omega$ identifies $D_t$ and $K_t$ with the respective time-$t$ local flow values whenever their whole intermediate straight drift or kick segment stays in $U$. Endpoint containment alone, used by the algebraic definitions above, does not imply this on an arbitrary open $U$. When each of the three flow segments is admissible, $V_h$ is the symmetric Strang-type splitting of those local flows; its algebraic canonical-form preservation identity holds on the full declared composition domain. The exact flow of $H$ is denoted ${\mathrm{\Phi }}_t$ in the items of this page.
• For $H(q,p)=\frac{1}{2}{p}^{\mathsf{T}}{M}^{-1}p+V(q)$ and the second-order differential equation $M\ddot{q}=-\mathrm{\nabla }V(q)$, the map $V_h$ is the classical velocity-Verlet (leapfrog) update in one-step form: half a kick, a full drift, half a kick. For $d=1$, $M=(1)$ and $V(q)=q^2\mathrm{/}2$, these formulas reduce to the explicit two-by-two map of Oscillator Verlet: a preserved quadratic form and bounded original-energy oscillation ↗ on the companion page.

Let $m\ge 1$, let $a\in {\mathbb{C}}^m$, let $\rho$ be a polyradius with $0<{\rho }_k<\mathrm{\infty }$, and let $f:{\mathrm{\Delta }}_{\rho }(a)\to \mathbb{C}$ be continuous and holomorphic in each variable separately, that is, for every $k$ and every choice of the other coordinates the slice $\zeta \mapsto f(\dots ,\zeta ,\dots )$ is holomorphic on its open disc (Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket, Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions). For a polyradius $r$ with $r_k<{\rho }_k$ put $M(r):={\sup }_{{\mathrm{\Gamma }}_r(a)}\mathrm{\mid }f\mathrm{\mid }$, a finite number.

Then:

1. Iterated Cauchy formula. For every $z\in {\mathrm{\Delta }}_r(a)$, $f(z)=\frac{1}{(2\pi i{)}^m}{\oint }_{\mathrm{\mid }{\zeta }_1-a_1\mathrm{\mid }=r_1}\cdots {\oint }_{\mathrm{\mid }{\zeta }_m-a_m\mathrm{\mid }=r_m}\frac{f(\zeta )}{{\prod }_k({\zeta }_k-z_k)}d{\zeta }_1\cdots d{\zeta }_m,$ and the iterated integrals in any order of the variables agree.
2. Power-series expansion. For every polyradius $r^{\mathrm{\prime }}$ with $r_k<r_k^{\mathrm{\prime }}<{\rho }_k$, and every $z\in {\mathrm{\Delta }}_r(a)$, $f(z)={\sum }_{\alpha \in {\mathbb{N}}^m}{c}_{\alpha }(z-a{)}^{\alpha },c_{\alpha }=\frac{1}{(2\pi i{)}^m}{\oint }_{{\mathrm{\Gamma }}_{r^{\mathrm{\prime }}}(a)}\frac{f(\zeta )d\zeta }{(\zeta -a{)}^{\alpha +\mathbf{1}}},$ where the coefficients $c_{\alpha }$ do not depend on $r^{\mathrm{\prime }}$, the series converges absolutely and uniformly on every compact subset of ${\mathrm{\Delta }}_{\rho }(a)$, and $\mathrm{\mid }{c}_{\alpha }\mathrm{\mid }\le M(r^{\mathrm{\prime }}){\prod }_k(r_k^{\mathrm{\prime }}{)}^{-{\alpha }_k}$.
3. Holomorphy, derivatives and estimates. The function $f$ is holomorphic on ${\mathrm{\Delta }}_{\rho }(a)$ in the sense of Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket. Its coordinate derivatives ${\mathrm{\partial }}_{z_k}f$ exist everywhere on ${\mathrm{\Delta }}_{\rho }(a)$, are continuous and holomorphic in each variable separately there, and satisfy for every polyradius $s=(s_1,\dots ,s_m)$ with $0<s_{\mathrm{\ell }}<{\rho }_{\mathrm{\ell }}$ for all $\mathrm{\ell }$, $\mathrm{\mid }{\mathrm{\partial }}_{z_k}f(a)\mathrm{\mid }\le \frac{M(s)}{s_k}\mathrm{.}$

• The lemma is stated for functions that are continuous and separately holomorphic, which is the hypothesis produced by the Picard iteration of The flow of a bounded holomorphic field on a polydisc: existence in complex time, displacement bound, holomorphy, and the composition law: slice-wise uniform limits of holomorphic functions are holomorphic by the one-variable Weierstrass theorem, and joint continuity comes from uniform convergence of continuous maps. The conclusion upgrades such a function to a holomorphic map in the $\mathbb{C}$-linear total-derivative sense of Complex phase domains, holomorphic maps on ${\mathbb{C}}^m$, and the complexified canonical form and bracket.
• The exponent $2^{m+1}$ in the interior estimate is not optimised; only the form $C\mathrm{/}{r}_k$ is used on this page.

Let $d\ge 1$ 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 $K+{\bar{B}}_r(0)\subseteq W$ (Open ball, closed ball and sphere in a metric space). Put