Nusc
- 752
- 2
Homework Statement
Rather than solve the double pendulum problem with two masses in the usual way.
Instead express the coordinates of the second mass, in terms of the coordinates of the mass above it.
[tex] $ x2=x_1+\xi = L_1Sin[\theta]Cos[\phi]+L_2Sin[\alpha]Cos[\beta]$\\<br /> $ y2=y_1+ \eta = L_1Sin[\theta]Sin[\phi]+L_2Sin[\alpha]Sin[\beta]$\\<br /> $ z2=z_1-\xi = L_1-L_1Cos[\theta]-L_2Sin[\alpha]Cos[\beta]$[/tex]Wouldn't you suspect that the Lagrangian remain invariant? Is there a way to reparameterize these equations?
Homework Equations
The Attempt at a Solution
Attachments
![Mathematica input and output cells showing long trigonometric expressions in Cos, Sin, OverDot[theta], OverDot[phi], OverDot[alpha] and OverDot[beta], followed by FullSimplify applied to a similar expression.](/data/attachments/18/18077-a0ecad9476807318ee4abcd164a7d869.jpg?hash=oOytlHaAcx)
![Handwritten notes on lined paper showing potential energy V = V1 + V2 = (M+m)gL(1 - cos θ) - mgL cos α, kinetic energy T2 = (m/2)[L^2(θ̇ + sin^2 θ φ̇^2) + l^2(α̇^2 + sin^2 α β̇^2)], and a circled term +mLl[simplify] labelled interaction term (IT).](/data/attachments/18/18192-a50fed89c6e93c439cd9bdd6948c91d7.jpg?hash=pQ_ticbpPE)
