View source for Physics:Swinging Atwood's machine

{{Short description|Variation of Atwood's machine incorporating a pendulum}}
[[File:Swinging Atwoods Machine.svg|thumb|The Swinging Atwood's machine. The smaller mass, labelled m, is allowed to swing freely whereas the larger mass, M, can only move up and down. Assume the pivots to be points.]]

The '''swinging Atwood's machine''' (SAM) is a mechanism that resembles a simple [[Physics:Atwood machine|Atwood's machine]] except that one of the masses is allowed to [[Oscillation|swing]] in a two-dimensional plane, producing a [[Dynamical system|dynamical system]] that is [[Chaos theory|chaotic]] for some system parameters and initial conditions.

Specifically, it comprises two masses (the [[Physics:Pendulum|pendulum]], mass {{mvar|m}} and counterweight, mass {{mvar|M}})  connected by an [[Physics:Kinematics#Inextensible cord|inextensible]], massless string suspended on two frictionless pulleys of zero radius such that the pendulum can swing freely around its pulley without colliding with the counterweight.<ref name="Tufillaro1984"/>

The conventional Atwood's machine allows only "runaway" solutions (''i.e.'' either the pendulum or counterweight eventually collides with its pulley), except for <math>M=m</math>. However, the swinging Atwood's machine with <math>M>m</math> has a large [[Parameter space|parameter space]] of conditions that lead to a variety of motions that can be classified as terminating or non-terminating, [[Periodic function|periodic]], quasiperiodic or chaotic, bounded or unbounded, singular or non-singular<ref name="Tufillaro1984"/><ref name="bound"/> due to the pendulum's [[Physics:Reactive centrifugal force|reactive centrifugal force]] counteracting the counterweight's weight.<ref name="Tufillaro1984"/> Research on the SAM started as part of a 1982 senior thesis entitled ''Smiles and Teardrops'' (referring to the shape of some trajectories of the system) by Nicholas Tufillaro at [[Organization:Reed College|Reed College]], directed by [[Biography:David J. Griffiths|David J. Griffiths]].<ref name="smiles and teardrops"/>

==Equations of motion==
[[Image:Swinging-atwood's-machine-4.5.gif|thumb|Motion of Swinging Atwood's Machine for M/m = 4.5]]
The swinging Atwood's machine is a system with two degrees of freedom. We may derive its equations of motion using either [[Physics:Hamiltonian mechanics|Hamiltonian mechanics]] or [[Lagrangian mechanics]]. Let the swinging mass be <math>m</math> and the non-swinging mass be <math>M</math>. The kinetic energy of the system, <math>T</math>, is:

:<math>
\begin{align}
T &= \frac{1}{2} M v^2_M + \frac{1}{2} mv^2_m \\
&= \frac{1}{2}M \dot{r}^2+\frac{1}{2} m \left(\dot{r}^2+r^2\dot{\theta}^2\right)
\end{align}
</math>

where <math>r</math> is the distance of the swinging mass to its pivot, and <math>\theta</math> is the angle of the swinging mass relative to pointing straight downwards. The potential energy <math>U</math> is solely due to the [[Physics:Standard gravity|acceleration due to gravity]]:

:<math>
\begin{align}
U &= Mgr - mgr \cos{\theta}
\end{align}
</math>

We may then write down the Lagrangian, <math>\mathcal{L}</math>, and the Hamiltonian, <math>\mathcal{H}</math> of the system:

:<math>
\begin{align}
\mathcal{L} &= T-U\\
&= \frac{1}{2}M \dot{r}^2+\frac{1}{2} m \left(\dot{r}^2+r^2\dot{\theta}^2\right) - Mgr + mgr \cos{\theta}\\
\mathcal{H} &= T+U\\
&= \frac{1}{2}M \dot{r}^2+\frac{1}{2} m \left(\dot{r}^2+r^2\dot{\theta}^2\right) + Mgr - mgr \cos{\theta}
\end{align}
</math>

We can then express the Hamiltonian in terms of the canonical momenta, <math>p_r</math>, <math>p_\theta</math>:

:<math>
\begin{align}
p_r &= \frac{\partial{\mathcal{L}}}{\partial \dot{r}} = \frac{\partial T}{\partial \dot{r}} = (M+m)\dot{r}\\
p_\theta &= \frac{\partial {\mathcal{L}}}{\partial \dot{\theta}} = \frac{\partial T}{\partial \dot{\theta}} = mr^2 \dot{\theta}\\
\therefore \mathcal{H} &= \frac{p_r^2}{2(M+m)} + \frac{p_\theta^2}{2mr^2} + Mgr - mgr \cos{\theta}
\end{align}
</math>

Lagrange analysis can be applied to obtain two second-order coupled ordinary differential equations in <math>r</math> and <math>\theta</math>. First, the <math>\theta</math> equation:

:<math>
\begin{align}
\frac{\partial {\mathcal{L}}}{\partial \theta} &= \frac{d}{dt} \left(\frac{\partial {\mathcal{L}}}{\partial \dot{\theta}}\right)\\
-mgr \sin{\theta} &= 2mr \dot{r}\dot{\theta} + mr^2 \ddot{\theta}\\
r\ddot{\theta} + 2\dot{r}\dot{\theta} + g\sin{\theta} &= 0
\end{align}
</math>

And the <math>r</math> equation:

:<math>
\begin{align}
\frac{\partial {\mathcal{L}}}{\partial r} &= \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{r}}\right)\\
mr\dot{\theta}^2 - Mg + mg\cos{\theta} &= (M+m) \ddot{r}
\end{align}
</math>

We simplify the equations by defining the mass ratio <math>\mu = \frac{M}{m}</math>. The above then becomes:

:<math>(\mu+1)\ddot{r} - r\dot{\theta}^2 + g(\mu - \cos{\theta}) = 0</math>

Hamiltonian analysis may also be applied to determine four first order ODEs in terms of <math>r</math>, <math>\theta</math> and their corresponding canonical momenta <math>p_r</math> and <math>p_\theta</math>:

:<math>
\begin{align}
\dot{r}&=\frac {\partial{\mathcal{H}}} {\partial{p_r}} = \frac {p_r}{M+m} \\
\dot{p_r} &= - \frac {\partial{\mathcal{H}}} {\partial{r}} = \frac {p_\theta ^2}  {mr^3}  - Mg + mg\cos{\theta} \\
\dot{\theta}&=\frac {\partial{\mathcal{H}}} {\partial{p_\theta}} = \frac {p_\theta} {mr^2} \\
\dot{p_\theta} &= - \frac {\partial{\mathcal{H}}} {\partial{\theta}} = -mgr\sin{\theta}
\end{align}
</math>

Notice that in both of these derivations, if one sets <math>\theta</math> and angular velocity <math>\dot{\theta}</math> to zero, the resulting special case is the regular non-swinging [[Physics:Atwood machine|Atwood machine]]:

:<math>\ddot{r} = g \frac{1-\mu}{1+\mu}=g\frac{m-M}{m+M}</math>

The swinging Atwood's machine has a four-dimensional [[Physics:Phase space|phase space]] defined by <math>r</math>, <math>\theta</math> and their corresponding canonical momenta <math>p_r</math> and <math>p_\theta</math>. However, due to energy conservation, the phase space is constrained to three dimensions.

===System with massive pulleys===
If the pulleys in the system are taken to have [[Physics:Moment of inertia|moment of inertia]] <math>I</math> and radius <math>R</math>, the Hamiltonian of the SAM is then:<ref name="Pujol2010"/>

:<math>\mathcal{H}\left(r, \theta, \dot{r}, \dot{\theta} \right) =
    \underbrace{ \frac{1}{2} M_t \left( R \dot{\theta} - \dot{r} \right) ^2
        + \frac{1}{2} m r^2 \dot{\theta}^2 }_{T}
    + \underbrace{ gr \left(M - m \cos{\theta} \right)
        + gR \left( m \sin{\theta} - M \theta \right)}_{U},
</math>

Where {{var|M}}<sub>t</sub> is the effective total mass of the system,
:<math>M_t = M + m + \frac{I}{R^2}</math>

This reduces to the version above when <math>R</math> and <math>I</math> become zero. The equations of motion are now:<ref name="Pujol2010"/>

:<math>\begin{align}
 \mu_t ( \ddot{r} - R \ddot{\theta}) & = r \dot{\theta}^2 + g (\cos {\theta} - \mu ) \\
 r \ddot{\theta}    & = - 2 \dot{r} \dot{\theta} + R \dot{\theta}^2 - g \sin {\theta} \\
\end{align}
</math>

where <math>\mu_t = M_t / m</math>.

===Integrability===
[[Physics:Hamiltonian system|Hamiltonian system]]s can be classified as [[Physics:Integrable system|integrable]] and nonintegrable. SAM is integrable when the mass ratio <math>\mu = M/m = 3</math>.<ref name="Integrable"/> The system also looks pretty regular for <math>\mu = 4 n^2 - 1 = 3, 15, 35, ...</math>, but the <math>\mu = 3</math> case is the only known integrable mass ratio. It has been shown that the system is not integrable for <math>\mu \in (0,1) \cup (3,\infty)</math>.<ref>{{Cite journal|last=Casasayas|first=J.|last2=Nunes|first2=A.|last3=Tufillaro|first3=N.|date=1990|title=Swinging Atwood's Machine : integrability and dynamics|url=http://www.edpsciences.org/10.1051/jphys:0199000510160169300|journal=Journal de Physique|volume=51|issue=16|pages=1693–1702|doi=10.1051/jphys:0199000510160169300|issn=0302-0738}}</ref> For many other values of the mass ratio (and initial conditions) SAM displays [[Chaos theory|chaotic motion]].

Numerical studies indicate that when the orbit is singular (initial conditions: <math>r=0, \dot{r}=v, \theta=\theta_0, \dot{\theta}=0</math>), the pendulum executes a single symmetrical loop and returns to the origin, regardless of the value of <math>\theta_0</math>. When <math>\theta_0</math> is small (near vertical), the trajectory describes a "teardrop", when it is large, it describes a "heart". These trajectories can be exactly solved algebraically, which is unusual for a system with a non-linear Hamiltonian.<ref name="smiles and hearts"/>

==Trajectories==
The swinging mass of the swinging Atwood's machine undergoes interesting trajectories or orbits when subject to different initial conditions, and for different mass ratios. These include periodic orbits and collision orbits.

===Nonsingular orbits===
For certain conditions, system exhibits [[Physics:Complex harmonic motion|complex harmonic motion]].<ref name="Tufillaro1984"/> The orbit is called nonsingular if the swinging mass does not touch the pulley.
{{gallery|align=center
| title = Selection of nonsingular orbits
| Sam mu 2.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=2</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 3.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=3</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 5.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=5</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 6.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=6</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 16.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=16</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 19.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=19</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 21.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=21</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 24.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=24</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
}}

====Periodic orbits====
[[File:Sam type A.svg|thumb|right|Type A orbits for <math>\theta_0</math> ranging from 0.1 to 3.1.]]
When the different harmonic components in the system are in phase, the resulting trajectory is simple and periodic, such as the "smile" trajectory, which resembles that of an ordinary [[Physics:Pendulum|pendulum]], and various loops.<ref name="smiles and teardrops" /><ref name="motions"/> In general a periodic orbit exists when the following is satisfied:<ref name="Tufillaro1984"/>

:<math>r(t+\tau) = r(t),\, \theta(t+\tau) = \theta(t)</math>

The simplest case of periodic orbits is the "smile" orbit, which Tufillaro termed '''Type A''' orbits in his 1984 paper.<ref name="Tufillaro1984" />

{{gallery|align=center
| title = Selection of periodic orbits
| Sam mu 1.665000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|A "smile" orbit of the swinging Atwood's machine for <math>\mu=1.665</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 2.394000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=2.394</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 1.172700 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=1.1727</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 1.555000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=1.555</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
}}

===Singular orbits===
The motion is singular if at some point, the swinging mass passes through the origin. Since the system is [[Physics:Invariant|invariant]] under time reversal and translation, it is equivalent to say that the pendulum starts at the origin and is fired outwards:<ref name=Tufillaro1984/>

:<math>r(0) = 0</math>

The region close to the pivot is singular, since <math>r</math> is close to zero and the equations of motion require dividing by <math>r</math>. As such, special techniques must be used to rigorously analyze these cases.<ref name="collision"/>

The following are plots of arbitrarily selected singular orbits.
{{gallery|align=center
| title = Selection of singular orbits
| Sam mu 10.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=10</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
| Sam mu 25.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|An orbit of the swinging Atwood's machine for <math>\mu=25</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.
}}

====Collision orbits====
[[File:Sam type B.svg|thumb|right|Type B orbits for <math>\theta_0</math> ranging from 0.1 to 3.1.]]

Collision (or terminating singular) orbits are subset of singular orbits formed when the swinging mass is ejected from its pivot with an initial velocity, such that it returns to the pivot (i.e. it collides with the pivot):

:<math>r(\tau) = r(0) = 0, \, \tau > 0</math>

The simplest case of collision orbits are the ones with a mass ratio of 3, which will always return symmetrically to the origin after being ejected from the origin, and were termed '''Type B''' orbits in Tufillaro's initial paper.<ref name="Tufillaro1984" /> They were also referred to as teardrop, heart, or rabbit-ear orbits because of their appearance.<ref name="smiles and teardrops" /><ref name="smiles and hearts"/><ref name="motions"/><ref name="collision"/>

When the swinging mass returns to the origin, the counterweight mass, <math>M</math> must instantaneously change direction, causing an infinite tension in the connecting string. Thus we may consider the motion to terminate at this time.<ref name="Tufillaro1984"/>

==Boundedness==
For any initial position, it can be shown that the swinging mass is bounded by a curve that is a [[Conic section|conic section]].<ref name="bound"/> The pivot is always a [[Focus (geometry)|focus]] of this bounding curve. The equation for this curve can be derived by analyzing the energy of the system, and using conservation of energy. Let us suppose that <math>m</math> is released from rest at <math>r=r_0</math> and <math>\theta=\theta_0</math>. The total energy of the system is therefore:

:<math>
E = \frac{1}{2}M \dot{r}^2+\frac{1}{2} m \left(\dot{r}^2+r^2\dot{\theta}^2\right) + Mgr - mgr \cos{\theta} = Mgr_0 - mgr_0 \cos{\theta_0}
</math>

However, notice that in the boundary case, the velocity of the swinging mass is zero.<ref name="bound"/> Hence we have:

:<math>
Mgr - mgr \cos{\theta}=Mgr_0 - mgr_0 \cos{\theta_0}
</math>

To see that it is the equation of a conic section, we isolate for <math>r</math>:

:<math>
\begin{align}
r&=\frac{h}{1-\frac{\cos{\theta}}{\mu}}\\
h&=r_0\left(1-\frac{\cos{\theta_0}}{\mu}\right)
\end{align}
</math>

Note that the numerator is a constant dependent only on the initial position in this case, as we have assumed the initial condition to be at rest. However, the energy constant <math>h</math> can also be calculated for nonzero initial velocity, and the equation still holds in all cases.<ref name="bound"/> The [[Eccentricity (mathematics)|eccentricity]] of the conic section is <math>\frac{1}{\mu}</math>. For <math>\mu>1</math>, this is an ellipse, and the system is bounded and the swinging mass always stays within the ellipse. For <math>\mu=1</math>, it is a parabola and for <math>\mu<1</math> it is a hyperbola; in either of these cases, it is not bounded. As <math>\mu</math> gets arbitrarily large, the bounding curve approaches a circle. The region enclosed by the curve is known as the Hill's region.<ref name="bound"/>

==Recent three dimensional extension==

A new integrable case for the problem of three dimensional Swinging Atwood Machine (3D-SAM) was announced in 2016.<ref name="3D-SAM"/> Like the 2D version, the problem is integrable when <math>M = 3m</math>.

==References==
<references>
<ref name="Integrable">
{{cite journal
 | title = Integrable motion of a swinging Atwood's machine
 | year = 1986
 | journal = [[Physics:American Journal of Physics|American Journal of Physics]]
 | pages = 142
 | volume = 54
 | issue = 2
 | doi   = 10.1119/1.14710
 | last1 = Tufillaro	 | first1 =  Nicholas B.
|bibcode = 1986AmJPh..54..142T }}</ref>
<ref name="smiles and teardrops">{{cite thesis
 | title   = Smiles and Teardrops
 | last    = Tufillaro 
 | first   = Nicholas B.
 | year    = 1982
 | publisher = [[Organization:Reed College|Reed College]]
}}</ref>
<ref name="smiles and hearts">{{cite journal
 | last1 = Tufillaro	 | first1 =  Nicholas B.
 | title = Teardrop and heart orbits of a swinging Atwoods machine,
 | year  = 1994
 | journal = [[Physics:American Journal of Physics|American Journal of Physics]]
 | volume = 62
 | issue  = 3
 | pages  = 231–233
 | doi    = 10.1119/1.17602
|bibcode = 1994AmJPh..62..231T | arxiv= chao-dyn/9302006}}</ref>
<ref name="Tufillaro1984">{{cite journal
 | title = Swinging Atwood's Machine
 | year = 1984
 | journal = [[Physics:American Journal of Physics|American Journal of Physics]]
 | volume = 52
 | issue = 10 
 | pages = 895–903
 | last1 = Tufillaro	 | first1 =  Nicholas B.
 | last2 =  Abbott	 | first2 =  Tyler A.
 | last3 =  Griffiths	 | first3 =  David J.
 | doi   = 10.1119/1.13791
|bibcode = 1984AmJPh..52..895T }}</ref>
<ref name="Pujol2010">{{cite journal
 | title   = Swinging Atwood's Machine: Experimental and numerical results, and a theoretical study
 | year    = 2010
 | journal = [[Physics:Physica D|Physica D]]
 | volume  = 239
 | issue   = 12
 | pages   = 1067–1081
 | last1   = Pujol     | first1 =  Olivier
 | last2   = Perez     | first2 = J.P.
 | last3   = Simo      | first3 = C.
 | last4   = Simon     | first4 = S.
 | last5   = Weil      | first5 = J.A.
 | doi     = 10.1016/j.physd.2010.02.017
|bibcode = 2010PhyD..239.1067P | arxiv= 0912.5168}}</ref>
<ref name="bound">{{cite journal
 | title = Unbounded orbits of a swinging Atwood's machine
 | year = 1988
 | journal = [[Physics:American Journal of Physics|American Journal of Physics]]
 | pages = 1117
 | volume = 56
 | last1 = Tufillaro	 | first1 =  Nicholas B.
 | last2 =  Nunes	 | first2 =  A.
 | last3 =  Casasayas	 | first3 =  J.
 | doi        = 10.1119/1.15774
|bibcode = 1988AmJPh..56.1117T }}</ref>
<ref name="motions">{{cite journal
 | title   = Motions of a swinging Atwood's machine
 | last1   = Tufillaro	 | first1 =  Nicholas B.
 | journal = Journal de Physique
 | volume  = 46
 | issue   = 9
 | year    = 1985
 | pages   = 1495–1500
 | doi     = 10.1051/jphys:019850046090149500
}}</ref>
<ref name="collision">{{cite journal
 | last1   = Tufillaro	 | first1 =  Nicholas B.
 | year    = 1985
 | title   = Collision orbits of a swinging Atwood's machine
 | journal = Journal de Physique
 | volume  = 46
 | pages   = 2053–2056
 | doi     = 10.1051/jphys:0198500460120205300
| url= http://hal.archives-ouvertes.fr/docs/00/21/01/53/PDF/ajp-jphys_1985_46_12_2053_0.pdf
 }}</ref>
<ref name="3D-SAM">{{cite journal
 | title   = On the integrability of the motion of 3D-Swinging Atwood machine and related problems
 | year    = 2016
 | journal = Physics Letters A
 | pages   = 989
 | volume  = 380
 | last1   = Elmandouh	 | first1 =  A.A.
 | doi     = 10.1016/j.physleta.2016.01.021
|bibcode = 2016PhLA..380..989E }}</ref>
</references>

===Further reading===
*Almeida, M.A., Moreira, I.C. and Santos, F.C. (1998) "On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems", ''[[Physics:Brazilian Journal of Physics|Brazilian Journal of Physics]]'' Vol.28 n.4  São Paulo Dec.
*Barrera, Emmanuel Jan (2003) ''Dynamics of a Double-Swinging Atwood's machine'', B.S. Thesis,  National Institute of Physics, University of the Philippines.
*Babelon, O, M. Talon, MC Peyranere (2010), "Kowalevski's analysis of a swinging Atwood's machine," Journal of Physics A: Mathematical and Theoretical Vol. 43 (8).
*Bruhn, B. (1987) "Chaos and order in weakly coupled systems of nonlinear oscillators," ''[[Physics:Physica Scripta|Physica Scripta]]'' Vol.35(1).
*Casasayas, J., N. B. Tufillaro, and A. Nunes (1989) "Infinity manifold of a swinging Atwood's machine," ''[[Physics:European Journal of Physics|European Journal of Physics]]'' Vol.10(10), p173.
*Casasayas, J, A. Nunes, and N. B. Tufillaro (1990) "Swinging Atwood's machine: integrability and dynamics," ''Journal de Physique'' Vol.51, p1693.
*Chowdhury, A. Roy and M. Debnath (1988) "Swinging Atwood Machine. Far- and near-resonance region", ''[[Physics:International Journal of Theoretical Physics|International Journal of Theoretical Physics]]'', Vol. 27(11), p1405-1410.
*Griffiths D. J. and T. A. Abbott (1992) "Comment on ""A surprising mechanics demonstration,"" ''[[Physics:American Journal of Physics|American Journal of Physics]]'' Vol.60(10), p951-953.
*Moreira, I.C. and M.A. Almeida (1991) "Noether symmetries and the Swinging Atwood Machine", ''Journal of Physics'' II France 1, p711-715.
*Nunes, A.,  J. Casasayas, and N. B. Tufillaro (1995) "Periodic orbits of the integrable swinging Atwood's machine," ''[[Physics:American Journal of Physics|American Journal of Physics]]'' Vol.63(2), p121-126.
*Ouazzani-T.H., A. and Ouzzani-Jamil, M., (1995) "Bifurcations of Liouville tori of an integrable case of swinging Atwood's machine," ''Il Nuovo Cimento B'' Vol. 110 (9).
*Olivier, Pujol, JP Perez, JP Ramis, C. Simo, S. Simon, JA Weil (2010), "Swinging Atwood's Machine: Experimental and numerical results, and a theoretical study," [[Physics:Physica D|Physica D]] 239, pp.&nbsp;1067–1081.
*Sears, R. (1995) "Comment on "A surprising mechanics demonstration," ''[[Physics:American Journal of Physics|American Journal of Physics]]'', Vol. 63(9), p854-855.
*Yehia, H.M., (2006) "On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine", ''Mechanics Research Communications'' Vol. 33 (5), p711–716.

==External links==
*[https://web.archive.org/web/20051223175337/http://metric.ma.ic.ac.uk/articles/samos98/Swinging.pdf#search=%22swinging%20atwood's%20machine%22 Imperial College Course]
*[http://www.sc.ehu.es/sbweb/fisica/oscilaciones/atwood/atwood.htm Oscilaciones en la máquina de Atwood]
*[https://aquahue.net/aquahue/papers/sam_00.pdf "Smiles and Teardrops" (1982)]
*[https://web.archive.org/web/20110721023414/http://www-loa.univ-lille1.fr/~pujol/ 2010 Videos of an experimental Swinging Atwood's Machine]
*[http://www.compadre.org/osp/items/detail.cfm?ID=11247 Open source Java code for running the Swinging Atwood's Machine]

{{chaos theory}}

[[Category:Hamiltonian mechanics]]

{{Sourceattribution|Swinging Atwood's machine}}