Action in Lagrangian Mechanics

In summary, Lagrangian mechanics is built upon calculus of variation and to solve a problem, we need to define the action and find the solution to the Euler - Lagrange equation.
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Dario56
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Lagrangian mechanics is built upon calculus of variation. This means that we want to find out function which is a stationary point of particular function (functional) which in Lagrangian mechanics is called the action.

To know what this function is, action needs to be defined first. Action is defined via integral.

In problems which use calculus of variation such as brachistochrone problem, caternary problem or finding path of least distance between two points, appropriate integral is the integral between two points of question of appropriate variable (time, potential energy, distance etc.), that is the integral of variable which is usually needed to be minimized in the problem (can be maximized as well).

When integral is defined, function is known and with Euler - Lagrange equation we get the solution to the problem. For example that can be function which defines path of least time, distance or shape of the rope as a solution of caternary problem.

What I don't understand is how to define this integral in context of Lagrangian mechanics or in context of action? In another words, how did Lagrange found out that difference in kinetic and potential energy of the system (commonly known as Lagrangian) gives correct equations of motion when plugged in Euler - Lagrange equation?
 
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Dario56 said:
What I don't understand is how to define this integral in context of Lagrangian mechanics or in context of action? In another words, how did Lagrange found out that difference in kinetic and potential energy of the system (commonly known as Lagrangian) gives correct equations of motion when plugged in Euler - Lagrange equation?
Wikipedia has a page on the history and development of the principle of least action:

https://en.wikipedia.org/wiki/Stationary-action_principle
 
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Probably similar to this

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You might try looking at Lanczos
https://www.amazon.com/dp/0486650677/?tag=pfamazon01-20
Rojo and Bloch
https://www.amazon.com/dp/0521869021/?tag=pfamazon01-20
Coopersmith
https://www.amazon.com/dp/0198743041/?tag=pfamazon01-20
 
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You don't need lagrange, you need Newton+hamilton+idea of test particle. We, as physicists, use test particles to learn about the behavior and motion about the world around us.

Newton's first law is basically "test particles, when left alone, will move in a straight line at a constant speed".

Then a guy called Hamilton sets up a premise called "principle of least action" which uses Newton's ideal+geometry, in which you must notice that a straight line between two points can be thought of as "the shortest distance" or "least time" between two points. If you agree with this, then bam, you formulate the following integral formulation:
$$S =-mc \int^{\tau_{final}}_{\tau_{initial}} d \tau$$

Minimizing this for some $\tau$ will give you the trajectory of a free particle. There is more to this story, and I hand waved a little, but I believe your question isn't really with lagrange, but more so with hamilton, Newton, and the ideal of a test particle, and at the core, what IS a principle of least action?!

Now, I've never TRULY gotten classical mechanics because it never sat right with me, but if the above is what you're questioning, I'm sure Feynman can do a better (and more through) job than I can, and you can find more information here:
https://www.feynmanlectures.caltech.edu/II_19.html
 

FAQ: Action in Lagrangian Mechanics

1. What is the Lagrangian in Lagrangian Mechanics?

The Lagrangian in Lagrangian Mechanics is a mathematical function that describes the dynamics of a physical system. It is typically denoted as L and is defined as the difference between the kinetic energy and potential energy of a system.

2. How is the Lagrangian used in Lagrangian Mechanics?

The Lagrangian is used to derive the equations of motion for a system in Lagrangian Mechanics. By using the principle of least action, which states that the actual motion of a system is the one that minimizes the action integral (the integral of the Lagrangian over time), the equations of motion can be obtained.

3. What is the advantage of using Lagrangian Mechanics over Newtonian Mechanics?

Lagrangian Mechanics offers a more elegant and efficient approach to solving problems in classical mechanics compared to Newtonian Mechanics. It allows for the use of generalized coordinates, which can simplify the equations of motion for complex systems. It also takes into account the constraints of a system, making it more suitable for studying systems with non-conservative forces.

4. Can Lagrangian Mechanics be applied to systems with multiple degrees of freedom?

Yes, Lagrangian Mechanics can be applied to systems with multiple degrees of freedom. In fact, it is often used to study systems with many degrees of freedom, such as complex mechanical systems or systems in quantum mechanics.

5. What are some real-world applications of Lagrangian Mechanics?

Lagrangian Mechanics has a wide range of applications in various fields of physics and engineering. Some examples include the study of celestial mechanics, the analysis of fluid dynamics, the design of control systems, and the development of new technologies such as robotics and aerospace engineering.

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