# The priciple of least action

1. Sep 8, 2006

### chieutim

have just started the first lessons on theoretical physics, and have some quesions on the principle of least action.I would be very graceful to you for help me understand this.
As the variable under the intergral operator is time, so it is supposed to be more special than other space quatity(qi).But this make time is more "important", it is some thing not appreciate to the special theory of relativity.Although , in theory of relativity, the variable is s, but what happen to the system of two paticles, and general systems.And more over ,the priciple in electric-magnetic field use the 4-dimention variable, can we provide the princile an general form so that can apply it to all cases?
About the principle in classic theory, I have tried to consider time as other coordinate, so a mechanical system of degree s consider to be a point of a (s+1) maniford (this provides an interesting definition of mass-point , and the principle is not dependent on the concept of space and time ).But to evaluate time under the integeral operator, I have find the expresion of ds repect to dqi,unfortunately, although in a simple case this expresion is not beautiful (like the expresion of metric on maniford).For instant , we consider the 2-diemention motion in the field U(x), we gain the expresion ds=-U(x)dt+1/2mdx/dt dx , this mean that ds is not a differential form becase the matric (-U(x) 1/2dx/dt) depend on the direct on the Maniford, this is unusual.
So I want to know the gerneral form of the principle.Please help me if you know, thank you for your word.
(Im sorry that my language is not good, so grammar and word may not not exactly)