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Double Pendulum

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Double Pendulum

- The double pendulum is a conservative system.
- Two degrees of freedom

- The exact Lagrangian can be written without approximation.

l

q

m

l

f

m

Make substitutions:

Divide by mgl

tt(g/l)1/2

Find conjugate momenta as angular momenta.

Make substitutions:

Divide by mgl

tt(g/l)1/2

Find conjugate momenta as angular momenta.

- For small angles the Lagrangian simplifies.
- The energy is E = -3.

- The mode frequencies can be found from the matrix form.
- The winding number W is irrational.

- The cotangent manifold T*Q is 4-dimensional.
- Q is a torus T2.
- Energy conservation constrains T*Q to an n-torus

- Take a Poincare section.
- Hyperplane q= 0
- Select dq/dt > 0

q

f

1

2

Jf

- The greatest motion in f-space occurs when there is no energy in the q-dimension
- Points must lie within a boundary curve.

Jf

2

1

f

- For small angle deflections there should be two fixed points.
- Correspond to normal modes

Jf

2

1

f

- An orbit on the Poincare section corresponds to a torus.
- The motion does not leave the torus.
- Motion is “invariant”

- Orbits correspond to different energies.
- Mixture of normal modes

Jf

f

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