Double Pendulum
– A Simple Analysis of a Dynamic System
Presenters:
Brian Okimoto
Yifan Shen
Jonah Taylor
Introduction
• Euler Lagrange System
• Hamiltonian System
• Linearization and Equilibrium Points
• Modeling the Trajectory
• Chaos
• Visualization of This 4-D Dynamic System
Deriving equations of motion: Da math
• Choose ∅1 and ∅2 as coordinates
• 𝑥1 = 𝑙1 sin(∅1 )
• 𝑦1 = 𝑙1 cos(∅1 )
• 𝑥2 = 𝑙1 sin ∅1 + 𝑙2 sin(∅2 )
• 𝑦2 = −𝑙1 cos ∅1 − 𝑙2 cos(∅2 )
Deriving Equations of motion: Lagrangian
• Lagrangian, L = T – U
•
•
𝑡2
𝐴𝑐𝑡𝑖𝑜𝑛 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑆 = 𝑡 𝐿𝑑𝑡
1
𝑑 𝜕𝐿
Euler-Lagrange equation:
𝑑𝑡 𝜕𝑞 ′ 𝑖
=
𝜕𝐿
𝜕𝑞𝑖
• Takes Lagrangian function as input and outputs the equations of
motion of our coordinates for which action is minimized
Deriving equations of motion: Da math part 2
•
1
T= 𝑚1 (𝑥1′2
2
+ 𝑦1′2 )
+
1
𝑚2 (𝑥2′2
2
+ 𝑦2′2 )
• U=𝑚1 𝑔𝑦1 + 𝑚2 𝑔𝑦2
• Using our values from before and simplifying we get
• T=
1
(𝑚1 +𝑚2 )𝑙12 ∅1′2
2
1
+ 𝑚2 𝑙22 ∅′2
2
2
+ 𝑚2 𝑙1 𝑙2 ∅1′ ∅′2 cos ∅1 − ∅2
• 𝑈 = − 𝑚1 + 𝑚2 𝑔𝑙1 cos ∅1 − 𝑚2 𝑔𝑙2 cos(∅2 ).
•
1
L= (𝑚1 +𝑚2 )𝑙12 ∅1′2
2
+
1
𝑚2 𝑙22 ∅′2
2
2
+ 𝑚2 𝑙1 𝑙2 ∅1′ ∅′2 cos ∅1 − ∅2 +(𝑚1 +
Deriving equations of motion: Da math part 3
• Our two equations to solve are
𝑑 𝜕𝐿
𝜕𝐿
=
𝑑𝑡 𝜕∅1 ′ 𝜕∅1
and
𝑑 𝜕𝐿
𝑑𝑡 𝜕∅2 ′
=
𝜕𝐿
𝜕∅2
• Plugging in our value for the lagrangian and taking appropriate
derivatives we find that;
′2
• 𝑚1 + 𝑚2 𝑙1 ∅1′′ = −𝑚2 𝑙2 ∅′′
cos
∅
−
∅
−
𝑚
𝑙
𝑙
∅
1
2
2 1 2 2 sin(∅1 −
2
Hamiltonian System
• The Hamiltonian function is a quantity that is conserved for a system,
𝜕𝐻
that is
=0
𝜕𝑡
• For the double pendulum this is the total energy H=T+U as the system
is undamped and un-driven
• In general the Hamiltonian can be written as H= 𝑝𝑖 𝑞𝑖′ − 𝐿
• Our two momentums are 𝑝∅1 =
𝜕𝐿
𝜕∅′1
and 𝑝∅2 =
𝜕𝐿
𝜕∅′2
• Solving now for H in terms of position and momentum we get
Hamiltonian System Part 2
•
•
𝑑𝑥
𝑑𝑦
For a given system = 𝑓 𝑥, 𝑦 𝑎𝑛𝑑 = 𝑔 𝑥, 𝑦
𝑑𝑡
𝑑𝑡
𝑑𝑥
𝜕𝐻
𝑑𝑦
𝜕𝐻
𝜕𝑓
Hamiltonian if =
and = − , exists if
𝑑𝑡
𝜕𝑦
𝑑𝑡
𝜕𝑥
𝜕𝑥
=
𝜕𝑔
−
𝜕𝑦
• Solving these equations using our coordinates 𝑝∅1 , 𝑝∅2 , ∅1 𝑎𝑛𝑑 ∅2 we find;
Our Hamiltonian exists
𝜕∅′1
𝜕∅1
=−
𝜕𝑝∅′ 1
𝜕𝑝∅1
and
𝜕∅′2
𝜕∅2
=−
𝜕𝑝∅′ 2
𝜕𝑝∅2
Rewrite into First Order System
Linearization
• Equilibrium Points {ø1, ø2, ø’1, ø’2}
•
•
•
•
(0,0,0,0) Stable
(Pi,0,0,0) Unstable
(0,Pi,0,0) Unstable
(Pi,Pi,0,0) Unstable
(0,0,0,0)
Imaginary Eigenvalues in 4D space.
(Pi,0,0,0),(0,Pi,0,0),(Pi,Pi,0,0)
Same Jacobian and Eigenvalues
Parametric Plot
• Pendulum Lengths (l1 =
5, l2 = 10)
• Gravity (9.81)
• Donut Effect
• https://www.youtube.c
om/watch?v=QXf95_EK
S6E
Chaos
• Sensitivity to the initial conditions
• Dense Orbit
• Topological mixing
Sensitivity of the Initial Conditions
• Small change in initial conditions will result in a great change in longterm.
θ1 = 0
θ1 = 0.00000000000001
θ1 = 0.0000000000000000001
θ1 = 0.0000000000000000005
Sensitivity of the Initial Conditions
• Lyapunov Exponents
• Sensitivity
• classify the type of attractor
Dense Orbit
• Basically, solutions that are arbitrarily close will eventually behave
very differently
Topological Mixing
• Basically, solutions that are arbitrarily far will eventually come
together – Strange Attractor
• 4-D System, 4-D Object
• 3-D “cross-sectionals”