Double pendulum
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Double Pendulum. 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 t  t ( g / l ) 1/2. Find conjugate momenta as angular momenta.

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Double Pendulum

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Double pendulum

Double Pendulum


Double pendulum1

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


Dimensionless form

Make substitutions:

Divide by mgl

tt(g/l)1/2

Find conjugate momenta as angular momenta.

Dimensionless Form


Hamilton s equations

Make substitutions:

Divide by mgl

tt(g/l)1/2

Find conjugate momenta as angular momenta.

Hamilton’s Equations


Small angle approximation

Small Angle Approximation

  • 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.


Phase space

Phase Space

  • 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


Boundaries

Boundaries

  • 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


Fixed points

Fixed Points

  • For small angle deflections there should be two fixed points.

    • Correspond to normal modes

Jf

2

1

f


Invariant tori

Invariant Tori

  • 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

next


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