Question on Lagrangian and Action

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In summary, the author discusses minimizing an action and finds the extrema by computing the delta of the action.
  • #1
RelativeQuanta
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I'm reading my textbook and trying to follow the math on how to minimize the action for an arbitrary Lagrangian. The author states that the action is:
[tex]
S[x(t)] = \int^{t_B}_{t_A} dt L( \dot x(t),x(t))
[/tex]

Then the author goes on to talk about finding the extrema for the action by computing [itex] \delta S[x(t)] [/itex]. The author says to compute this by substituing [itex] x(t) + \delta x(t) [/itex] into the definition for the action, expanding to 1st order and integrate by parts. The text then shows this:
[tex]
\delta S[x(t)] = \int^{t_B}_{t_A} dt [\frac {\partial L}{\partial \dot x(t)} \delta \dot x(t) + \frac {\partial L}{\partial x(t)} \delta x(t)] = [ \frac {\partial L}{\partial \dot x(t)} \delta x(t) ]^{t_B}_{t_A} + \int^{t_B}_{t_A} dt [ - \frac {d}{dt} \frac {\partial L}{\partial \dot x(t)} + \frac {\partial L}{\partial x(t)} ] \delta x(t)
[/tex]

What I don't understand is how the author got this. If all he did was substitute and expand like he said to do, what happened to the 0th order term from the expansion? I'm also unsure how he got the right most form of the equation using integration by parts.
 
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  • #2
Given a function [itex]X(t)=x(t)+\delta{x}(t)[/itex], (where x(t) is the assumed solution of the resulting diff. eq.) he is looking at the first order term of:
[tex]\bigtriangleup{S}\equiv{S}(X(t))-S(x(t))[/tex]
That first order expression is proportional to [itex]\delta{x}(t)[/itex], and is called [itex]\delta{S}(x(t))[/itex]
 
  • #3
Ah, ok. That makes sense, I just wish the author had made that clear.

Thanks
 
  • #4
One other question that's sort of related to the first. In this case, the author talked about a value [itex] \delta S[x(t)] [/itex]. I notice that if I simpily take the first order derivative, I get something that looks exactly the same except for all the little deltas changing to d's. Is there any real difference? Or is the delta notation just to keep you from being confused as to what is being integrated?
 
  • #5
Well, the variation can be regarded as a differential on A FUNCTION SPACE, rather than, say, on the real line of numbers.

It is the type of space we're working with that is different; the basic "idea" is the same.
 

1. What is the Lagrangian and Action in physics?

The Lagrangian and Action are two fundamental concepts in classical mechanics that are used to describe the motion of a physical system. The Lagrangian is a function that summarizes the energy of the system in terms of its position and velocity, while the Action is the integral of the Lagrangian over time.

2. How is the Lagrangian different from other equations in physics?

The Lagrangian is unique because it takes into account the entire history of a system's motion, rather than just its current state. This allows for a more comprehensive understanding of the system's behavior and can be used to derive the equations of motion for the system.

3. What is the significance of the Action in classical mechanics?

The Action is a fundamental quantity in the principle of least action, which states that the actual path taken by a physical system is the one that minimizes the Action. This principle is a fundamental principle in classical mechanics and is used to derive the equations of motion for a system.

4. How is the Lagrangian and Action used in quantum mechanics?

In quantum mechanics, the Lagrangian and Action are used to describe the behavior of particles at the microscopic level. The Lagrangian is used to calculate the probability amplitude for a particle to move between two points, while the Action is used to calculate the path integral, which is used to determine the probability of a particle's path.

5. Are there any real-world applications of the Lagrangian and Action?

Yes, the Lagrangian and Action have many real-world applications, from predicting the motion of planets in our solar system to understanding the behavior of particles in particle accelerators. They are also used in engineering applications, such as designing efficient systems for controlling the motion of robots or spacecraft.

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