The principle of least action is the generic name for
the type of work instigated by
Lagrange in his mechanics
A quantity known as the action (and its not a normal every day type of action
, but proabaly does have something to do with the idea
that this quantity is supposed to be responsible for something happening). For Lagrange
his action is the difference between the kinetic and
potential energy of a body. From this point of view
mechanics is transformed into a study of scalar
quantities rather than vector
fall out in a most beautiful fashion.
A particle in flight under an external field has a value for its Kinetic and Potential energy at every point along the path. Let the path be 'r' then we have the functions
K(r) and V(r) for the kinetic and potential respectively. Create the Langrangian L(r) = K(r) - V(r). We can now integrate this over the path r. Two different paths r and r' will give different values for integrated function L. The principle of least action asserts that physical systems follow the path with gives a minimum value for L(r) over all possible paths r.
The Calculus of Variations is used to determine the path r which gives us the minimum value of the function L(r).
In a conservative field, such as Gravity, the path chosen by this method is the same as that described by Newton's equations of motion! Ta Da, very pretty.