Hamilton’s principle states that the development in time for a mechanical system is such that the integral of the difference between the kinetic and the potential energy is stationary. More specific we haver:

**Hamilton’s principle:** The motion of a mechanical system from time *t _{0}* to

is stationary for the functions *y _{1}(t), y_{2}(t),…,y_{n(}t),* describing the development in time for the system.

Here

that is, the difference between the kinetic energy *T* and the potential energy *V*.

**Remark 1:** Hamilton’s principle is often mathematically expressed as

The quantity

is called the *action integral*.

**Remark 2:** Hamilton’s principle mean that among all curves between the end points, the motion will occur along the curve that gives an extreme value (actually stationary value) to the action integral.

**Remark 3:** By using calculus of variations (see chapter 3) we see that *y _{i}(t)*must fulfill the Euler equations

In the special case when we are dealing with mechanical systems these equations are called *Lagrange’s equations*.