NationMaster - Encyclopedia: Double pendulum

In horology, a double pendulum is a system of two simple pendulums on a common mounting which move in anti-phase. Image File history File links Double-Pendulum. ... Image File history File links Double-Pendulum. ... Image File history File links Size of this preview: 409 Ã— 600 pixelsFull resolution (1971 Ã— 2891 pixel, file size: 552 KB, MIME type: image/jpeg) This is a double pendulum with a shutter length of 1 second L1 ~ L2 and m1 > m2 I, the copyright holder of this work, hereby grant... Image File history File links Size of this preview: 409 Ã— 600 pixelsFull resolution (1971 Ã— 2891 pixel, file size: 552 KB, MIME type: image/jpeg) This is a double pendulum with a shutter length of 1 second L1 ~ L2 and m1 > m2 I, the copyright holder of this work, hereby grant... Horology is the study of the science and art of timekeeping devices. ... Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ...

In mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior. The motion of a double pendulum is governed by a set of coupled ordinary differential equations. Above a certain energy its motion is chaotic. Also see pendulum (mathematics). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ... Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ... A physical system is a system that is comprised of matter and energy. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... For other uses, see Chaos Theory (disambiguation). ... The mathematics of pendulums can be quite complex, but some formula and proofs are given below. ...

The double pendulum consists of two thin rods (moment of inertia, $I=frac{1}{12} M ell^2$ ) connected by a pivot and the end of one rod suspended from a pivot. It is natural to define the coordinates to be the angle between each rod and the vertical. These are denoted by θ₁ and θ₂. The position of the centre of mass of the two rods may be written in terms of these coordinates. If the origin of the Cartesian coordinate system is assumed to be at the point of contact of the wall and the first pendulum, then the centre of mass is located at: Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2), is the rotational analog of mass. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... Fig. ...

$x_1 = frac{ell}{2} sin theta_1,$

$x_2 = ell left ( sin theta_1 + frac{1}{2} sin theta_2 right ),$

$y_1 = -frac{ell}{2} cos theta_1$

and

$y_2 = -ell left ( cos theta_1 + frac{1}{2} cos theta_2 right ).$

This is enough information to write out the Lagrangian.

1 Lagrangian
2 Chaotic motion
3 References
4 External links

Lagrangian

The Lagrangian is given by A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

$L = frac{1}{2} m left ( v_1^2 + v_2^2 right ) + frac{1}{2} I left ( {dot theta_1}^2 + {dot theta_2}^2 right ) - m g left ( y_1 + y_2 right )$

where the first term is the kinetic energy of the bodies, the second term is the kinetic energy of the center of mass of each rod, and the last term is the potential energy of the bodies in a uniform gravitational field. The kinetic energy of an object is the extra energy which it possesses due to its motion. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... Potential energy is energy stored within a Physical system. ...

Substituting the coordinates above and rearranging the equation gives

$L = frac{1}{6} m ell^2 left [ {dot theta_2}^2 + 4 {dot theta_1}^2 + 3 {dot theta_1} {dot theta_2} cos (theta_1-theta_2) right ] + frac{1}{2} m g ell left ( 3 cos theta_1 + cos theta_2 right ).$

There is only one conserved quantity (the energy), and no conserved momenta. The two momenta may be written as In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ...

$p_{theta_1} = frac{partial L}{partial {dot theta_1}} = frac{1}{6} m ell^2 left [ 8 {dot theta_1} + 3 {dot theta_2} cos (theta_1-theta_2) right ]$

and

$p_{theta_2} = frac{partial L}{partial {dot theta_2}} = frac{1}{6} m ell^2 left [ 2 {dot theta_2} + 3 {dot theta_1} cos (theta_1-theta_2) right ].$

These expressions may be inverted to get

${dot theta_1} = frac{6}{mell^2} frac{ 2 p_{theta_1} - 3 cos(theta_1-theta_2) p_{theta_2}}{16 - 9 cos^2(theta_1-theta_2)}$

and

${dot theta_2} = frac{6}{mell^2} frac{ 8 p_{theta_2} - 3 cos(theta_1-theta_2) p_{theta_1}}{16 - 9 cos^2(theta_1-theta_2)}.$

The remaining equations of motion are written as

${dot p_{theta_1}} = frac{partial L}{partial theta_1} = -frac{1}{2} m ell^2 left [ {dot theta_1} {dot theta_2} sin (theta_1-theta_2) + 3 frac{g}{ell} sin theta_1 right ]$

and

${dot p_{theta_2}} = frac{partial L}{partial theta_2} = -frac{1}{2} m ell^2 left [ -{dot theta_1} {dot theta_2} sin (theta_1-theta_2) + frac{g}{ell} sin theta_2 right ].$

Chaotic motion

Graph of the time for the pendulum to flip over as a function of initial conditions

The double pendulum undergoes chaotic motion, and shows a sensitive dependence on initial conditions. The image to the right shows the amount of elapsed time before the pendulum "flips over", as a function of initial conditions. Here, the initial value of θ₁ ranges along the x-direction, from −3 to 3. The initial value θ₂ ranges along the y-direction, from −3 to 3. The colour of each pixel indicates whether either pendulum flips within $10sqrt{g/ell }$ (green), within 100 (red), 1000 (purple) or 10000 (blue). Initial conditions that don't lead to a flip within $10000sqrt{g/ell }$ are plotted white. Image File history File links All. ... Image File history File links All. ... A plot of the trajectory Lorenz system for values r = 28, Ïƒ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

The boundary of the central white region is defined in part by energy conservation with the following curve:

$3 cos theta_1 + cos theta_2 = 2. ,$

Within the region defined by this curve, that is if

$3 cos theta_1 + cos theta_2 > 2, ,$

then it is energetically impossible for either pendulum to flip. Outside this region, the pendulum can flip, but it is a complex question to determine when it will flip.

References

Eric W. Weisstein, Double pendulum (2005), ScienceWorld. (Contains details of the complicated equations involved.)
Peter Lynch, Double Pendulum, (2001). (Java applet simulation.)
Northwestern University, Double Pendulum, (Java applet simulation.)
Theoretical High-Energy Astrophysics Group at UBC, Double pendulum, (2005).
Meirovitch, Leonard (1986). Elements of Vibration Analysis, 2nd edition, McGraw-Hill Science/Engineering/Math. ISBN 0-07-041342-8.

External links

Animations and explanations of a double pendulum and a physical double pendulum (two square plates) by Mike Wheatland (Univ. Sydney)

Double pendulum physics simulation from www.myphysicslab.com

Simple explanation of chaos theory and double pendulums at chaoticpendulums.com

Categories: Pendulums | Chaotic maps

Results from FactBites:

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Chaotic motion

References

External links

Lagrangian