Swinging Atwood's machine

The swinging Atwood's machine is a system with two degrees of freedom. One may derive its equations of motion using either Hamiltonian mechanics or Lagrangian mechanics. Let the swinging mass be m and the non-swinging mass be M. The kinetic energy of the system, T, is: : \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} where r is the distance of the swinging mass to its pivot, and \theta is the angle of the swinging mass relative to pointing straight downwards. The potential energy U is solely due to the acceleration due to gravity: : \begin{align} U &= Mgr - mgr \cos{\theta} \end{align} We may then write down the Lagrangian, \mathcal{L}, and the Hamiltonian, \mathcal{H} of the system: : \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} We can then express the Hamiltonian in terms of the canonical momenta, p_r, p_\theta: : \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} Lagrange analysis can be applied to obtain two second-order coupled ordinary differential equations in r and \theta. First, the \theta equation: : \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} And the r equation: : \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} We simplify the equations by defining the mass ratio \mu = \frac{M}{m}. The above then becomes: :(\mu+1)\ddot{r} - r\dot{\theta}^2 + g(\mu - \cos{\theta}) = 0 Hamiltonian analysis may also be applied to determine four first order ODEs in terms of r, \theta and their corresponding canonical momenta p_r and p_\theta: : \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} Notice that in both of these derivations, if one sets \theta and angular velocity \dot{\theta} to zero, the resulting special case is the regular non-swinging Atwood machine: :\ddot{r} = g \frac{1-\mu}{1+\mu}=g\frac{m-M}{m+M} The swinging Atwood's machine has a four-dimensional phase space defined by r, \theta and their corresponding canonical momenta p_r and p_\theta. 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 moment of inertia I and radius R, the Hamiltonian of the SAM is then: For many other values of the mass ratio (and initial conditions) SAM displays chaotic motion. Numerical studies indicate that when the orbit is singular (initial conditions: r=0, \dot{r}=v, \theta=\theta_0, \dot{\theta}=0), the pendulum executes a single symmetrical loop and returns to the origin, regardless of the value of \theta_0. When \theta_0 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. ==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 complex harmonic motion. The orbit is called nonsingular if the swinging mass does not touch the pulley. Periodic orbits 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 pendulum, and various loops. In general a periodic orbit exists when the following is satisfied: :r(t+\tau) = r(t),\, \theta(t+\tau) = \theta(t) The simplest case of periodic orbits is the "smile" orbit, which Tufillaro termed Type A orbits in his 1984 paper. Singular orbits The motion is singular if at some point, the swinging mass passes through the origin. Since the system is invariant under time reversal and translation, it is equivalent to say that the pendulum starts at the origin and is fired outwards: :r(0) = 0 The region close to the pivot is singular, since r is close to zero and the equations of motion require dividing by r. As such, special techniques must be used to rigorously analyze these cases. The following are plots of arbitrarily selected singular orbits. Collision orbits 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): :r(\tau) = r(0) = 0, \, \tau > 0 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. They were also referred to as teardrop, heart, or rabbit-ear orbits because of their appearance. When the swinging mass returns to the origin, the counterweight mass, M must instantaneously change direction, causing an infinite tension in the connecting string. Thus we may consider the motion to terminate at this time. ==Boundedness==

For any initial position, it can be shown that the swinging mass is bounded by a curve that is a conic section. The pivot is always a 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 m is released from rest at r=r_0 and \theta=\theta_0. The total energy of the system is therefore: : 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} However, notice that in the boundary case, the velocity of the swinging mass is zero. Hence we have: : Mgr - mgr \cos{\theta}=Mgr_0 - mgr_0 \cos{\theta_0} To see that it is the equation of a conic section, we isolate for r: : \begin{align} r&=\frac{h}{1-\frac{\cos{\theta}}{\mu}}\\ h&=r_0\left(1-\frac{\cos{\theta_0}}{\mu}\right) \end{align} 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 h can also be calculated for nonzero initial velocity, and the equation still holds in all cases. The eccentricity of the conic section is \frac{1}{\mu}. For \mu>1, this is an ellipse, and the system is bounded and the swinging mass always stays within the ellipse. For \mu=1, it is a parabola and for \mu it is a hyperbola; in either of these cases, it is not bounded. As \mu gets arbitrarily large, the bounding curve approaches a circle. The region enclosed by the curve is known as the Hill's region. ==Recent three dimensional extension==

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