The cyclic spring chain, its discrete energy, and the finite-regularity periodic comparison class

Let $N\ge 2$ be an integer, let $a>0$ and $c>0$ be real parameters, and let $L:=Na$. Indices are read modulo $N$: $z_{j+N}:=z_j$.

(a) The chain. A state of the chain is a pair $(q,p)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$, where $q=(q_1,\dots ,q_N)$ are the positions and $p=(q_1^{\mathrm{\prime }},\dots ,q_N^{\mathrm{\prime }})$ the velocities of the $N$ nodes. The cyclic spring-chain system is the first-order system

$q_j^{\mathrm{\prime }}=p_j,p_j^{\mathrm{\prime }}=\frac{c^2}{a^2}(q_{j+1}-2q_j+q_{j-1})(j\in \mathbb{Z}\mathrm{/}N),$

equivalently the second-order system $q_j^{\mathrm{\prime }\mathrm{\prime }}=\frac{c^2}{a^2}(q_{j+1}-2q_j+q_{j-1})$ for a $C^2$ curve $q:I\to {\mathbb{R}}^N$. Its discrete energy is

$E(q,p):=\frac{1}{2}{\sum }_{j=1}^N(p_j^2+\frac{c^2}{a^2}(q_{j+1}-q_j{)}^2)\mathrm{.}$

(b) Discrete norms and differences. For $z,w\in {\mathbb{R}}^N$ define the scaled inner product and norm

$⟨z,w{⟩}_a:=a{\sum }_{j=1}^N{z}_j{w}_j,\mathrm{\parallel }z{\mathrm{\parallel }}_a:=⟨z,z{⟩}_a^{1\mathrm{/}2},$

and for the discrete difference $(D_{+}z{)}_j:=(z_{j+1}-z_j)\mathrm{/}a$. The discrete Laplacian is $({\mathrm{\Delta }}_{a}z{)}_j:=(z_{j+1}-2z_j+z_{j-1})\mathrm{/}{a}^2$.

(c) The comparison class. Let $I\subseteq \mathbb{R}$ be a nonempty open interval and let $u:I\times \mathbb{R}\to \mathbb{R}$ be a function that is $C^2$ in the first variable and $C^4$ in the second, is $L$-periodic in $x$ ($u(t,x+L)=u(t,x)$ for all $(t,x)$), and satisfies

$u_{tt}(t,x)=c^2{u}_{xx}(t,x)((t,x)\in I\times \mathbb{R})\mathrm{.}$

For a compact subinterval $[0,T]\subseteq I$ put $M_T:=\sup \{\mathrm{\mid }{\mathrm{\partial }}_x^{4}u(t,x)\mathrm{\mid }:(t,x)\in [0,T]\times \mathbb{R}\}$ (finite when continuous derivatives are bounded on the compact time interval, as assumed in the items that use it), and for $a=L\mathrm{/}N$ define the sampled initial data

$q_j(0):=u(0,ja),q_j^{\mathrm{\prime }}(0):=u_t(0,ja)(j=1,\dots ,N)\mathrm{.}$

• This is a purely mathematical definition. The mechanical reading — $N$ equal masses $m$, nearest-neighbour springs of stiffness $k$ with $c^2=k{a}^2\mathrm{/}m$, a closed chain — is a model interpretation attached by a consuming physical item; the mathematical results of this collection use only the ODE system and the comparison class.

• The cyclic indexing (index $N+1$ is index $1$) makes the chain closed and makes summation by parts in the discrete variable exact; it is part of the model, not a boundary condition. Other boundary conditions are separate models and are not covered by this definition.

• The comparison class requires a globally defined periodic solution with the regularity in clause (c); the items using it establish existence of the chain solutions and compare the two evolutions on a compact time interval. No uniqueness or continuation statement for $u$ is part of this definition.