Question

Why is this expression called action
\[ \int_0^t { (T - V) } dt\]
where T is kinetic energy and V is potential energy.

Answer 1

T-V = L (Lagrangian)

Answer 2

http://en.wikipedia.org/wiki/Action_%28physics%29

Answer 3

\[[\text{energy}]\cdot[\text{time}]\]

Answer 4

what exactly is action? how does [energy][time] give action?

Answer 5

It's just a name, don't trouble yourself with the semantics of it.

Answer 6

what is the significance of it?

Answer 7

the Lagrange equation of motion is based on minimizing this value ...

Answer 8

Yes. And when that's done, it reproduces newton's equations in cartesian coordinates.

Answer 9

looks like i need to study calculus of variation before that.

Answer 10

The calculus of variations can be used to derive the Lagrangian equations of motion, but is not necessary for actually using it.

Answer 11

a body will always take the part of least action

Answer 12

I want to understand it ...
must be something like Fermat's principle .. of least action

Answer 13

path*

Answer 14

What exactly do you want to understand?

Answer 15

that why \int T-V dt gives the trajectory

Answer 16

If you minimize that function, L = T- V, then you get the following equation from the calculus of variations:

\[ \frac{\partial L}{\partial x} = \frac{d}{dt} \frac{\partial L}{\partial \dot{x}}\]
where \[\dot{x} = v_x \]

If we assume T is independent of position (i.e. T = 1/2 mv^2) then the left hand side becomes
\[\frac{-\partial U}{\partial x} \]
And if U is independent of velocity (as is often the case) we can associate the right hand side with the momentum, since
\[\frac{\partial T}{\partial \dot{x}} = m\dot{x} = p\]
So we get
\[\frac{d}{dt} p = - \frac{\partial U}{\partial x} = F\]

which is just Newton's 2nd Law

Answer 17

The Lagrangian formulation and the Newtonian formulation are equivalent statements, in the sense that you can derive one from the other, but the Lagrangian often has many advantages over the Newtonian picture. You need to start somewhere, and you can either start by saying "F = ma" or you can say that the Lagrangian is minimized and go from there.

Answer 18

well ... i understand the application part. Just need to understand what the creators thought before formulating ... can we discuss this tomorrow? I need some time to work out.
I got something on Google
http://physics.stackexchange.com/questions/3912/does-action-in-classical-mechanics-have-a-interpretation

Answer 19

also here
http://physics.stackexchange.com/questions/9/hamiltons-principle

Answer 20

They just wanted to reproduce newton's equations from a scalar that easily incorporated constraints, that's all.