The splitting Hamiltonian $H=A+B$, the drift and kick maps, and the velocity-Verlet map

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.