July 11, 2018 Wednesday

Bedtime Story 

Functional 'Action' in Lagrangian Mechanics

Last night I had introduced the concept of ‘action’ in Lagrangian mechanics.

In Lagrangian mechanics, it has nothing to do with the general understanding of the word.

In mechanics it is a very good way in defining a mechanical system that is in action or rather, evolving over time.

Evolving in physics signifies a mechanical system that is dynamic and changing over time. 

In the language of mathematics, action can be defined as that which corresponds to a stationary point when a system evolves over time.

The action is usually stated in terms of integral over time, taken along the path of the system between the initial time and the final time of the development of the system:

𝒮 = ∫(t₁ to t₂) L dt,

Here L is the Lagrangian function.

For a system q(t) between times t₁ and t₂, where q represents the generalized coordinates, the action S[q(t)] is defined as the integral of the Lagrangian L for an input evolution between the two times t₁ and t₂.

𝒮[q(t)] = ∫(t₁ to t₂) L[q(t), q(dot)(t),t] dt

The endpoints of the evolution are fixed and defined as

q₁ = q(t₁) and q₂ = q(t₂).

The q with a dot on top as you will recall is the time derivative or dq/dt which is equal to (dq₁/dt, dq₂/dt,… dq_N/dt)

q represents generalized coordinates which is equal to (q₁, q₂,…,q_N).

Once we have understood the action functional, now the principle of least action can be stated mathematically:

 𝛿𝒮= ∫(t₁ to t₂) L(q,q(dot),t,)dt = 0

This integral formula when restated in English language states that the system takes the path for which the action is least.

So let us briefly what concepts that describe the workings of nature have been covered so far in our bedtime stories:

Angular momentum and its conservation

Conservation Law

Manifold in Riemannian geometry

Principle of Least Action (general)

Calculations of Variations

Lagrangian mechanics (at least for the classical mechanics)

Lagrangian function as an integral over time

Action (functional Action)

Principle of Least Action from the formulation of Lagrangian function and functional Action  

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

                              

Good night Mon Ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg