Editing Verlet integration (section)

===Equations of motion===
Newton's equation of motion for conservative physical systems is
:<math>\boldsymbol M \ddot{\mathbf x}(t) = F\bigl(\mathbf x(t)\bigr) = -\nabla V\bigl(\mathbf x(t)\bigr),</math>
or individually
:<math>m_k \ddot{\mathbf x}_k(t) = F_k\bigl(\mathbf x(t)\bigr) = -\nabla_{\mathbf x_k} V\left(\mathbf x(t)\right),</math>
where
* <math>t</math> is the time,
* <math>\mathbf x(t) = \bigl(\mathbf x_1(t), \ldots, \mathbf x_N(t)\bigr)</math> is the ensemble of the position vector of <math>N</math> objects,
* <math>V</math> is the scalar potential function,
* <math>F</math> is the negative [[Potential gradient|gradient of the potential]], giving the ensemble of forces on the particles,
* <math>\boldsymbol M</math> is the [[mass matrix]], typically diagonal with blocks with mass <math>m_k</math> for every particle.

This equation, for various choices of the potential function <math>V</math>, can be used to describe the evolution of diverse physical systems, from the motion of [[molecular dynamics|interacting molecules]] to the [[N-body problem|orbit of the planets]].

After a transformation to bring the mass to the right side and forgetting the structure of multiple particles, the equation may be simplified to
:<math>\ddot{\mathbf x}(t) = \mathbf A\bigl(\mathbf x(t)\bigr)</math>
with some suitable vector-valued function <math>\mathbf A(\mathbf x)</math> representing the position-dependent acceleration. Typically, an initial position <math>\mathbf x(0) = \mathbf x_0</math> and an initial velocity <math>\mathbf v(0) = \dot{\mathbf x}(0) = \mathbf v_0</math> are also given.