Equations of Motion

We consider a system of $N$ particles moving under the influence of a potential function $U$  [19,20]. Particles are described by their positions ${\bf R}$ and momenta ${\bf P} = M {\bf V}$ . The union of all positions (or momenta) $\{ {\bf R}_1, {\bf R}_2 , \ldots , {\bf R}_N \}$ will be called ${\bf R}^N$ (${\bf P}^N$ ). The potential is assumed to be a function of the positions only; $U({\bf R}^N)$ .

The Hamiltonian ${\cal H}$ of this system is

$${\cal H}({\bf R}^N,{\bf P}^N) = \sum_{I=1}^N {{\bf P}_I^2 \over 2 M_I} + U({\bf R}^N) \enspace .$$ (1)

The forces on the particle are derived from the potential

$$F_I({\bf R}^N) = - {\partial U({\bf R}^N) \over \partial {\bf R}_I } \enspace .$$ (2)

The equations of motion are according to Hamilton's equation

$${\bf\dot R}_I = {\partial {\cal H} \over \partial {\bf P}_I} = {{\bf P}_I \over M_I}$$ (3)

$${\bf\dot P}_I = -{\partial {\cal H} \over \partial {\bf R}_I} = -{\partial U \over \partial {\bf R}_I} = F_I({\bf R}^N) \enspace ,$$ (4)

from which we get Newton's second law

$$M_I {\bf\ddot R}_I = F_I({\bf R}^N) \enspace .$$ (5)

The equations of motion can also be derived using the Lagrange formalism. The Lagrange function is

$${\cal L}({\bf R}^N,{\bf\dot R}^N) = \sum_{I=1}^N {1 \over 2}{M_I {\bf\dot R}_I^2 } - U({\bf R}^N) \enspace ,$$ (6)

and the associated Euler-Lagrange equation

$${d \over dt } { \partial {\cal L} \over \partial {\bf\dot R}_i} = { \partial {\cal L} \over \partial {\bf R}_i}$$ (7)

leads to the same final result. The two formulations are equivalent, but the ab initio molecular dynamics literature almost exclusively uses the Lagrangian techniques.