Hi. Please show all steps and explanation thank you! The Lagrangian for the double pendulum is given by L=T-U,where1T = ; (m1 + m2)l¡¢i +maB3 + m2hhg162 cos($1 – 02),U = -(m1 + m2)gl1 cos ¢1 – m2gl2 cos $2.(a) Find the generalized momenta for this pendulum:aLPø2 =Pø1=(b) Find the energy of this pendulum:ƏL02– L,-ớ1 +E =-and show that E =T+U.

L =T-U=12m1+m2l12ϕ12˙+12m2l12ϕ22˙+m2l1l2ϕ1˙ϕ2˙cosϕ1-ϕ2+m1+m2gl1cosϕ1+m2gl2cosϕ2

Answered: The Lagrangian for the double pendulum…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Hi. Please show all steps and explanation thank you!

The Lagrangian for the double pendulum is given by L=T-U,
where
1
T = ; (m1 + m2)l¡¢i +maB3 + m2hhg162 cos($1 – 02),
U = -(m1 + m2)gl1 cos ¢1 – m2gl2 cos $2.
(a) Find the generalized momenta for this pendulum:
aL
Pø2 =
Pø1
=
(b) Find the energy of this pendulum:
ƏL
02– L,
-ớ1 +
E =
-
and show that E =T+U.

Expert Solution

Step 1

$\begin{array}{rcl} L & = & T - U \\ = & \frac{1}{2} (m_{1} + m_{2}) l_{1}^{2} \dot{ϕ_{1}^{2}} + \frac{1}{2} m_{2} l_{1}^{2} \dot{ϕ_{2}^{2}} + m_{2} l_{1} l_{2} \dot{ϕ_{1}} \dot{ϕ_{2}} \cos (ϕ_{1} - ϕ_{2}) + (m_{1} + m_{2}) g l_{1} \cos (ϕ_{1}) + m_{2} g l_{2} \cos (ϕ_{2}) \end{array}$

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