The priciple of least action

[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 principle 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)

 

[Jhero]

First of all, congratulations on starting your journey into theoretical physics! It is a fascinating and challenging field that requires a lot of dedication and hard work. As for your questions about the principle of least action, I will try my best to provide some insights and clarification.

The principle of least action is a fundamental principle in classical mechanics that states that the path taken by a system between two points in time is the one that minimizes the action, which is a mathematical quantity defined as the integral of the Lagrangian over time. The Lagrangian is a function that describes the energy of a system in terms of its generalized coordinates, which can include spatial coordinates (qi) and time (t). So, in this sense, time is not considered more special than other coordinates, but it is an integral part of the Lagrangian and plays a crucial role in determining the path of a system.

Now, in the theory of relativity, the concept of time is indeed treated differently than in classical mechanics. This is because in relativity, time is considered as a fourth dimension in addition to the three spatial dimensions. However, the principle of least action still holds in relativity, but the action and Lagrangian take different forms to account for the effects of time dilation and the curvature of spacetime.

In terms of applying the principle of least action to different systems, it is indeed possible to generalize it for systems with multiple particles or interacting fields. In these cases, the Lagrangian becomes a function of multiple coordinates and their derivatives, and the action is still minimized by taking the path that satisfies the Euler-Lagrange equations (equations that describe the motion of a system in terms of its Lagrangian). So, in short, the principle can be applied to all cases, but the specific form of the Lagrangian and action will vary depending on the system.

Regarding your idea of considering time as another coordinate, it is an interesting approach, but it may not be entirely accurate. Time is not just another coordinate like the spatial coordinates, but it is a fundamental aspect of the fabric of the universe. This is why it is treated differently in relativity and plays a crucial role in quantum mechanics as well.

In conclusion, the principle of least action is a powerful tool in theoretical physics that helps us understand the motion and behavior of systems. It can be applied to a wide range of systems, but the specific form of the Lagrangian and action will vary