Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles can be defined by
is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, and V is the potential energy of the system.
Kinetic energy is the energy of the system’s motion, and vk2 = vk · vk is the magnitude squared of velocity, equivalent to the dot product of the velocity with itself. The kinetic energy is a function only of the velocities vk, not the positions rk nor time t, so T = T(v1, v2, …).
The potential energy of the system reflects the energy of interaction between the particles, i.e. how much energy any one particle will have due to all the others and other external influences. For conservative forces (e.g. Newtonian gravity), it is a function of the position vectors of the particles only, so V = V(r1, r2, …). For those non-conservative forces which can be derived from an appropriate potential (e.g. electromagnetic potential), the velocities will appear also, V = V(r1, r2, …, v1, v2, …). If there is some external field or external driving force changing with time, the potential will change with time, so most generally V = V(r1, r2, …, v1, v2, …, t).