In a mechanical system, there are two types of energy that need to be considered. The first is the energy of motion, also called the kinetic energy:
Clearly, for a free particle, this is the only type of energy, and it is constant.
Newton's second law of motion
tells us nothing about the nature of the forces. In general, the forces may depend on both position and velocity:
The velocity dependence can be important when describing frictional
forces, which take the form 
and are responsible for
giving rise to damped motion.
However, if we restrict ourselves to forces that depend only on position, F = F(x), then some important properties of the system result. First. suppose there exists a function, U(x), such that
The function, U(x), is called the potential energy and it gives rise to forces that are known as conservative forces. The potential energy is energy of ``potential motion'' that is realized by conversion to kinetic energy. An example is the motion of an object in a gravitational field. If the object is lifted to a height h above the ground, then, when released, it will fall toward the ground accelerating toward the ground, and gradually increasing its velocity, hence its kinetic energy. We say that, at any height, h, the particle has potential energy, mgh, where g is the acceleration of the Earth's gradivational field and m is the object's mass, that is converted to kinetic energy as the particle accelerates toward the ground. This example suggests that there is a quantity that is constant or conserved over the motion of an object, namely, the total energy:
In order to see that Newton's second law , 
,
conserves the total energy, we only need to show that dE/dt = 0:
where the last line follows from the fact that 
.
In general, the energy of an N-particle in three dimensions is
where the N-particle potential energy function gives rise to interatomic forces via
It is left as an exercise to show that dE/dt = 0 if
Newton's equations of motion 
are obeyed.
An elegant formulation of Newtonian mechanics can be developed if one considers a different combination of the two energyes, known as the classical Lagrangian, which, for a single particle in one-dimension, is defined to be
Why K-U? This is not an arbitrary form! In fact, the difference K-U comes directly from an exact quantum mechanical treatment of the system, as developed in the statistical mechanics course. In particular, in the Feynman path integral formulation of quantum mechanics, it will be seen that this form falls out quite naturally as the classical limit of quantum mechanics.
Given the Lagrangian function, the equations of motion can be derived directly from the so called Euler-Lagrange equation:
Thus, for 
, we have
which is just Newton's second law for the system.
For an N-particle system in three dimensions, the Lagrangian is given by
where the shorthand notation is 
and 
. Again,
substituting into the Euler-Lagrange equation gives
which, again, is Newton's second law for the systems and constitutes a set of 3N coupled second order differential equations.