- #1

Hari Seldon

- 5

- 1

- Homework Statement
- I am trying to find the Lagrangian of a system called "Watt Regulator". My kinetic energy doesn't match the kinetic energy on the book. I would like to know if I am calculating the coordinates of the system in the right way.

- Relevant Equations
- Kinetic energy of the whole system: ##T = m v^2 = m (\dot x^2+\dot y^2)##

I understand that it is a system with two degrees of freedom. And I chose as generalized coordinates the two angles shown in the pic I posted. I am having troubles in finding the kinetic energy of this system, cause the book tells me that the kinetic energy is something different then what I calculated.

So very likely I am wrongly calculating the ##x## and the ##y##.

Indeed I calculated the ##x## as

$$x = l \sin{\varphi}+l \sin{\theta}$$

and the ##y## as

$$y = l \cos{\varphi}+l cos{\theta}$$

Then I calculated the derivative of ##x## and ##y##:

$$\dot x=l \dot \varphi \cos{\varphi}+l \dot \theta \cos{\theta}$$

$$\dot y=-l \dot \varphi \sin{\varphi}-l \dot \theta \sin{\theta}$$

So my kinetic energy would be

$$T = m (\dot x^2+\dot y^2)=m l^2 [\dot \theta^2 +\dot \varphi^2+2\dot \theta \dot \varphi (\cos{\varphi} \cos{\theta}-\sin{\varphi} \sin{\theta})]$$

While the kinetic energy that the book calculated is

$$T=m l^2 (\dot \theta^2+\dot \varphi^2 \sin^2{\theta})$$

What am I doing wrong?