LESSON NOTES · 01

From energy to trajectories

Prerequisites: Gradients and ordinary differential equations

1. A classical nuclear model

Classical MD assigns positions and momenta to nuclei or coarse-grained particles. A potential model supplies forces. A force field is a physical approximation; an integrator is a numerical approximation to the chosen equations. Born–Oppenheimer MD can use electronic calculations for a potential surface while propagating nuclei classically; it does not automatically include nuclear zero-point motion or tunneling.

2. Derive Newton’s equations

Let \(R=(r_1,\ldots,r_N)\) be the configuration and \(p\) the momenta. Assume a separable time-independent Hamiltonian and smooth potential. Hamilton’s equations give velocity and force directly.

\[ H(R,p)=\sum_i\frac{p_i^2}{2m_i}+U(R),\quad\dot r_i=\frac{\partial H}{\partial p_i}=\frac{p_i}{m_i},\quad\dot p_i=-\nabla_iU=F_i,\quad m_i\ddot r_i=F_i. \]

3. What is conserved?

Differentiate \(H\) along the exact trajectory. Position and momentum contributions cancel, proving exact isolated-dynamics energy conservation. This does not prove finite-step conservation. Translation-invariant \(U\) gives momentum conservation because internal pair forces cancel. External fields, thermostats and time-dependent potentials can change these conclusions.

\[ \frac{dH}{dt}=\sum_i\left[\nabla_iU\cdot\dot r_i+\frac{p_i}{m_i}\cdot\dot p_i\right]=0. \]

Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.

Worked oscillator trajectory

For \(U=kx^2/2\), use \(x(0)=A,v(0)=0\). Position and velocity oscillate, kinetic and potential energy exchange, and the phase-space orbit is an ellipse. Neither energy contribution is individually constant. These are exact analytic model curves, not simulation output.

\[ \omega=\sqrt{k/m},\quad x=A\cos\omega t,\quad v=-A\omega\sin\omega t,\quad E=kA^2/2,\quad\frac{p^2}{mkA^2}+\frac{x^2}{A^2}=1. \]

Check yourself and model limits

Why can integration accuracy not repair a wrong force field?

Answer and reasoning

It converges more accurately to the wrong model equations.

If \(k\) quadruples at fixed mass?

Answer and reasoning

\(\omega\) doubles, so the period halves. Classical trajectories require a justified dynamical model before interpreting physical-time correlations.

References and further reading

• GROMACS 2026.0: Molecular dynamics algorithms

Derivations and numerical examples here are original teaching constructions, not copied passages or reported research data.

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