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LESSON NOTES · 01

1. From energy to trajectories

Position in the course: Lesson 1 of 11. Complete the preceding derivation and use the explained exercises to check understanding.

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.

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

From energy to trajectories — schematic under the stated model assumptions

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

4. 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.

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

5. 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.

6. From a conserved quantity to a useful diagnostic

For central pair forces, angular momentum is conserved because each pair torque is \((r_i-r_j)\times F_{ij}=0\). Translation invariance gives total momentum conservation; rotation invariance gives angular momentum conservation. A periodic box does not always admit the same global angular-momentum interpretation as a free finite cluster. Boundary conditions belong to the model, so test only conservation laws the model actually possesses.

For a time-dependent potential, the cancellation in the energy derivative leaves \(dH/dt=\partial U/\partial t\). Driving can therefore do work while the force remains an exact negative gradient at each instant. In thermostat dynamics an additional bath term changes the physical energy. Before labeling a drift as numerical error, state which isolated or extended energy is supposed to be conserved.

Worked check: let \(U(x,t)=k[x-a(t)]^2/2\). Its force is \(-k[x-a(t)]\), while explicit power injection is \(\partial_tU=-k[x-a(t)]\dot a\). For a moving trap, a changing particle energy need not indicate a broken integrator. The same distinction matters under changing fields or deformed cells.

Exercise: for \(U(r_1,r_2)=u(|r_1-r_2|)\), prove \(F_1=-F_2\). The gradients of the separation have opposite signs, so the total internal force vanishes. A learned force model that predicts independent vectors without preserving energy or symmetry may violate this relation even when average force error is small. Validate symmetry, energy gradients and units separately before trusting a trajectory.

7. References and further reading

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

Quantum mechanics · Monte Carlo · Quantum Monte Carlo · CP2K · Quantum-Espresso

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