A physical demonstration: Two balls on a trampoline

To illustrate that energy transfer between people on a trampoline can happen through purely passive mechanisms, we designed a simple table-top demonstration involving a store-bought mini-trampoline and two balls (see also Materials and Methods).

Figure 2 shows a series of key frames, illustrating the energy transfer between the two (tennis) balls, dropped nearly but not exactly simultaneously. The two balls contact the trampoline at slightly different times, with some overlapping period when they are both in contact with the trampoline. The mass that makes contact with the trampoline second bounces much higher. See also movie S2, which shows this specific example in slow motion, and also other examples illustrating how when the masses make contact with the trampoline approximately simultaneously, they bounce up to about the same height.

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Figure 2. A table-top demonstration.

This figure shows a sequence of frames illustrating two balls dropped almost simultaneously onto a mini-trampoline, bouncing back up to very different heights. We see that the ball that makes contact second, rises much higher, as also seen in the mathematical models. See slow motion video in movie S2.

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We did not perform carefully controlled drops, make detailed measurements of the resulting bounces, or try to make this table-top demonstration a dynamically scaled version of two humans bouncing on a larger trampoline. We intend this only as a demonstration of the phenomenon.

When dropped by human hands, the two balls often land at slightly different times due to human motor variability, resulting in different amounts of overlap between their contact phases with the trampoline. As a consequence, as seen from the mathematical models below, the rise heights of the masses after the bounce have corresponding variability. When there is no contact overlap, as happens often (if the drops are not nearly simultaneous), there is no dramatic energy transfer.

A mathematical model: passive bouncing of two masses

When people bounce on trampolines, they perform positive mechanical work with their legs to counteract any loss of energy (through passive dissipation or active negative leg work). Here, for simplicity, we restrict ourselves to energy-conservative models: no leg work or dissipation. See Materials and Methods for simulation details for these mathematical models.

We idealize the two players as particles with masses and bouncing on a trampoline, modeled as a taut massless string of length , as shown in Figure 3a–b. The two masses are at horizontal distances and respectively from the nearest fixed ends of the trampoline; the distance between the masses is . The trampoline is at a large initial tension , so that the tension in it does not change to first order when deformed (Fig. 3b). The particles do not slip against the trampoline, and we neglect the horizontal forces on the particles, so that the motion of the particles is purely vertical, for all time.

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Figure 3. Simple model of two people or two balls bouncing on a trampoline, as two point-masses on a massless trampoline.

a) The system can be in one of four phases: neither mass in contact with the trampoline (P0), only mass-1 in contact (P1), only mass-2 in contact (P2), and both masses in contact (P12). b) The geometry of the system is shown, along with the forces on the masses when both are in contact with the trampoline.

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The vertical position of the two masses are denoted and , positive upward (Figure 3b). The undeflected trampoline is at . We divide the state space into four phases based on which masses are in contact with the trampoline: P0 (neither in contact with trampoline, both in flight), P1 (only in contact), P2 (only in contact), P12 (both in contact). See Figure 3a. The corresponding equations of motion are:(1)These equations are linear, and assuming small vertical deflections of the trampoline. The total dynamical system, consisting of these four phases patched together, is piecewise linear, and therefore, ultimately, nonlinear and non-smooth. Transition between phases occurs with no discontinuous change in position and velocity. A mass leaves the trampoline when the upward force on it by the trampoline becomes zero while it is moving upward. This take-off event coincides with the trampoline becoming a single straight line to the left and to the right of the mass; that is, the deflection in the trampoline vanishes locally. We assume two take-off or landing events do not happen simultaneously. The trampoline comes immediately to rest when neither mass is in contact; that is, the trampoline has no intrinsic dynamics. These constitutive assumptions are consistent with energy conservation.

The total system energy consists of the kinetic and gravitational potential energies of the two masses, namely and respectively, and the stored energy in the stretched string. The stored energy in the string in the various phases are:(2)(3)(4)(5)In the following discussions, we will sometimes refer to the “energy of a particular mass,” implicitly partitioning the total system energy between the two masses. The partitioning of the total system mechanical energy into the two masses is clearest when both masses are in flight — then, each mass is associated with the sum of its kinetic and gravitational potential energy. When both masses are in contact with the trampoline, there is no objective partitioning of the total energy between the two masses. When exactly one mass is in contact with the trampoline, we use the convention that the mass that is in contact gets credit for the energy stored in the string.

See Materials and Methods for how this non-smooth dynamical system is simulated.

Passive dynamics predicts energy transfer and complex dynamics

For the following simulations of the above model pertaining to bouncing people (as opposed to bouncing tennis balls described later), we use the following parameters: kg, m, , and ms⁻². The vertical stiffness of the trampoline at its midpoint is . We picked tension such that this midpoint stiffness was equal to 5000 N/m, roughly the secant stiffness of the trampoline described in [1].

Before considering a single bounce and the energy transfer in greater detail, we examine simulations of the passive dynamical system for a long time period. Simulating this dynamical system from any generic initial condition (which does not result immediately in the two masses touching the trampoline simultaneously in the first bounce results) in a complex non-periodic bouncing motion of the two masses, as shown in Figure 4a. Also, see video of the animation (movie S3).

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Figure 4. Long-term bouncing dynamics.

a) The motion and starting at rest from initial conditions and . b) The total energy (kinetic+gravitational potential) in mass-1 when both masses are in flight, as a fraction of the total energy. c) The state of the system when the right mass either just takes off (red dots) or when the right mass just lands (blue dots), when the left mass is already in the air.

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The two masses repeatedly exchange energy with each other, sometimes one mass bounces higher and sometimes the other mass bounces higher: Figure 4b shows the fluctuating energy content in mass-1. Thus, over a long enough simulation, if the “game” is stopped at some random moment sufficiently far into the future, there is equal likelihood of either player “winning” i.e., having more energy. The energy transfer between masses occurs when the masses are in simultaneous contact with the trampoline, performing work on each other through the trampoline. Thus, it appears that the passive model is sufficient to explain the energy transfer.

To be clear, while the mechanics of a single bounce interaction of the two masses may be comparable to that of the interaction between humans on a trampoline, the details of the long-time simulation may not be of direct applicability to long-time human bouncing. We discuss this long-time simulation further for its own intrinsic dynamical properties.

In a single long simulation, the state of the system appears to come arbitrarily close to almost every region of the accessible phase space, consistent with energy conservation. Figure 4c gives a scatter-plot of the snap-shots of the state, in a single long simulation lasting about phases, at transitions between phases P0 and P2: when mass-2 lands (, red dots) or takes off (, blue dots), with mass-1 already in the air (). The interior of the disk in Figure 4c is the set of all possible states consistent with constant total energy. We found that the histogram of energies for each mass over time is not a constant function of energy, but has a minimum near symmetric bouncing when each mass has half the energy. Indeed the thin slivers of empty regions in Figure 4c near corresponds to states at which each mass has exactly half the energy – in particular, where is the average height of the masses at initial condition. (We do not know if the system is ergodic [2].)

The complex dynamics observed for this dynamical system is not entirely unanticipated. A well-studied dynamical system is a mass bouncing, elastically or inelastically, on a much more massive paddle oscillating vertically and exactly sinusoidally [3], [4]; this system is known to be chaotic in certain parameter regimes. Note that this mass on an oscillating paddle system can be obtained as a distinguished limit of our dynamical system by making and by ensuring that never has a flight phase, so that it oscillates exactly sinusoidally. More recently, apparently unaware of this earlier work, similar chaotic dynamics were observed and analyzed for water droplets bouncing on a fluid surface, acting as a trampoline [5], [6]. One qualitative difference between these earlier systems and the two mass system consider in this article, is that typically the paddle or the fluid trampoline in these systems is oscillated using external energy input, so that the total system energy need not be constant.

Maximizing energy transfer

In this section (and in the Appendix S1), the passive bouncing model will be used to evaluate the effectiveness of various variables that players could control in order to gain energy from their opponent and win the “seat drop war” game. Players can control their jump timing relative to their opponent, their energy at impact, and the distance between themselves and the opponent. We find that players should aim to contact the trampoline when their opponent has maximally deflected the trampoline (half of the contact time), and should attempt to have more energy than the opponent (either due to larger mass or higher jump). The transfer of energy between players is larger when the players are closer together, than when they are farther apart.

Energy can get transferred only when both masses are in simultaneous contact with the trampoline. Without loss of generality, consider the situation in which mass-2 lands on the trampoline when mass-1 is already in contact, so that the two masses are in simultaneous contact with the trampoline for a while. Now simulate the system forward in time until both masses are in flight again i.e., phase P0 is reached. We examine the energy increase in the two masses when P0 is reached, as a function of the time difference between when mass-1 makes contact and mass-2 makes contact with the trampoline — the contact time-lag (Figures 5a–c). The energy increase in the two balls is normalized by the pre-contact energy of each mass (), and the contact time-lag has been normalized by the contact period of mass-1 in the absence of interference by mass-2. The model parameters are as noted in the previous sub-section, including , except when specifically overridden below. An alternative version of the plot would record the energies at the first moment one of the balls begins flight. In this alternate version, the ball still in contact will get credit for the stored elastic energy in the string. We note that this version of the plot (not shown) looks slightly different from the plots shown, as the ball in flight has the opportunity to re-contact the trampoline before the other ball takes off, providing further opportunity for energy exchange.

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Figure 5. Energy transfer and contact time-lag.

Energy increase in mass-1 (red) and mass-2 (blue) as a function of the impact time-lag. The masses are equal and are placed symmetrically on the trampoline, so that . The separation between the masses . a) The initial energies are equivalent to dropping from rest from a height of 1 m. b) Mass-1's initial energy is the same as for panel-a, but . c) Mass-1's initial energy is the same as for panel-a, except mass-2's energy is .

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The energy increase in Figure 5a–c are discontinuous functions of the contact time-lag. This discontinuity arises because we record the energy transfer only at the first transition to phase P0 after mass-2 lands. The number of phases that the system goes through before reaching phase P0 can depend on the initial conditions. Thus, on one side of a discontinuity, a mass barely takes off, with close to zero velocity. And on the other side of the discontinuity, this same mass comes very close to taking off, but does not have enough energy to do so, resulting in the system passing through more contact phases phases, P1, P2, and P12, before phase P0 can happen.

In Figure 5a, the masses had initially the same energy. So the normalized energy increases in the two masses are mirror images of each other about the x-axis because of energy conservation (), and because they have been normalized by the same quantity ().

Here, the mass that lands second, namely mass-2, always gains energy and mass-1 always loses energy, whatever the contact time-lag (Figure 5a). As would be expected, we see in Figure 5a that the energy transfer is close to zero when mass-2 lands when the mass-1 has just landed (close to symmetric simultaneous bounce), or about to take off (close to no interaction). More significantly, we see that the normalized energy increase of mass-2 reaches unity (), around when the time-lag is about half of mass-1's contact period. That is, if mass-2 makes contact when mass-1 is approximately half-way through its bounce, the energy transfer is essentially 100% in a single bounce.

When such complete energy transfer occurs, mass-2 makes contact when mass-1 just starts to rise or just before it starts to rise (note the two contact time-lags on either side of contact timing = 0.5 for which the energy transfer is perfect). As mass-2 pulls the string down, mass-1 remains in contact for a brief while and then leaves contact with an upward velocity, earlier than it would otherwise have in the absence of mass-2. This mass-1's upward velocity is such that, if left alone, the mass-1 would just about reach y = 0 at roughly the same time that mass-2 leaves contact, so that both masses are in flight with mass-1 with close to zero energy.

The basic energy transfer mechanism can be most simply understood using an ‘instantaneous argument’ as illustrated in Figure 6. Given a state as in Figure 6a, in which mass-1 is rising while in contact with the string, the presence of mass-2 lowers the upwards vertical force on mass-1 (Fig. 6c) compared to when mass-2 is not in contact (Fig. 6b, keeping position of mass-1 fixed). Thus, for any given upward , the corresponding acceleration and the instantaneous positive power on mass-1 by the string would be lower than if mass-2 were not in contact. If this situation persists until mass-1 takes off, it would take off with an upward speed smaller than if mass-2 had not interfered. Note that the situation described in this ‘instantaneous’ argument need not persist in general until phase P0 is reached — the bounce dynamics can be quite complex in certain parameter regimes. This heuristic ‘instantaneous’ argument is likely most directly applicable when mass-2 lands on the string when mass-1 is just about to take off, when it is likely that the situation presented in Fig. 6a would persist until at least mass-1 takes off.

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Figure 6. Basic mechanism of passive energy transfer from mass-1 to mass-2.

When mass-1 is moving up, the presence of mass-2 lowers the work done by the string on mass-1. Thus mass-1 takes off with lesser upward velocity than if mass-2 had not interfered.

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The details of the energy transfer's dependence on the contact time-lag can be complex and dependent on various other system parameters including differences in energy of mass-1 and mass-2 before contact (Figure 5b–c). When mass-2 has twice mass-1's initial energy (Figures 5b), mass-1 gains half of mass-2's energy when both contact simultaneously (contact time-lag = 0). In contrast, when mass-2 has half mass-1's initial energy (Figures 5c), mass-2 gains half of mass-1's initial energy when they contact simultaneously (contact time-lag = 0).

Even though the motion is governed by relatively simple linear differential equations in each phase, an analytical treatment to obtain the dependencies in Figure 5 was found to be cumbersome (although likely feasible) because the differential equations had to be integrated until a certain event happened (namely one or both masses taking off) and patched together, rather than simply integrated until a particular time.

In Appendix S1, we discuss energy transfer between the balls under some simplifying limits: (1) Large energy “collisional” limit, in which the collision consequence becomes independent of gravity and many other quantities; (2) limit of masses very close to each other, in which the mass in contact leaves contact when the other mass makes contact; and (3) limit of one mass much higher than the other, in which case the energy transfer scenario is similar to that of a “freshman physics” demonstration involving dropping a small ball sitting atop a big ball, from rest, onto the ground [38], [39].

Stability of symmetric bouncing

When the masses are equal () and symmetrically positioned (), a symmetric periodic motion is achieved by dropping the two balls from the same height. There is a one-parameter family of such symmetric motions parameterized by the initial height. In numerical simulations, we find that a generic small perturbation of the initial conditions for such symmetric bouncing leads to the two balls making contact not quite simultaneously, and this asymmetry grows with time, eventually leading the motion to be far away from symmetric bouncing.

The stability of symmetric motion is examined by considering the properties of a time-period-based ‘Poincare map’ [3], [7] as described below. (A state-section-based Poincare map has closely related properties.) Say the two balls are dropped from height and bounce periodically and synchronously with period . We define our Poincare map as the mapping of states at time to time . We approximate the Poincare map's Jacobian about the fixed point using a central difference scheme. Even though the map is perhaps not arbitrarily differentiable (because the differential equations are non-smooth), the map appears once continuously differentiable, as multiple finite difference approximations using random state perturbations give identical eigenvalues for the Jacobian, up to numerical errors. See Figure 7 for the eigenvalues as functions of bouncing height . The Jacobian has two unit eigenvalues (equal to +1) for any because of energy conservation. We further noticed that for all heights , the two non-unit eigenvalues, real or complex, were reciprocals of each other ( and ), and therefore, the product of all four eigenvalues equals one (true numerically, up to error ). These properties follow from the map being “symplectic” [8], as the map is derived from a Hamiltonian (energy conservative, holonomic) system [25]. For most initial heights , the Jacobian has two non-unit real eigenvalues, with one real eigenvalue greater than one in absolute value, and the other real eigenvalue less than one. Thus symmetric bouncing displays obvious linear instability for these heights. For a small range of heights, the two non-unit eigenvalues were complex conjugates with absolute value equal to one, within numerical error, about , suggesting “spectral stability.” We did not perform any similar analyses that take the non-smooth dynamics into account more carefully. Nevertheless, we note that for all heights, even for heights where all eigenvalues seem to have unit magnitudes, long-enough numerical simulations eventually take the system far away from symmetric bouncing. For these simulations, we used a high accuracy integrator that preserved energy at a level of over long durations of integration, but not a ‘symplectic’ integrator [9].

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Figure 7. Stability of symmetric passive bouncing.

The four eigenvalues of the Jacobian (Floquet multipliers) corresponding to the mapping of the state over one period of the periodic motion: symmetric bouncing for symmetric masses. The product of the eigenvalues was equal to 1 (with an error of about ). Two eigenvalues are equal to . In the intermediate regime shown, all four eigenvalues, two of which are complex conjugates and reciprocals of each other, have unit absolute values. At other regimes, one eigenvalue has magnitude greater than one, implying linear instability.

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Game-theoretic analysis: bouncing as a zero sum game

We began this article with a game played on the trampoline as motivation, but all the analysis so far has been of passive mechanical models. Consider, for instance, two players dropping from the same height, with legs extended. Say player 1 knows that player 2 will bounce passively. Then, player 1 can bend her knees during flight, just enough so that she will land slightly after player 2, gaining most of player 2's energy and winning the game. But what if player 2 is thinking strategically as well, trying to time her landing just right so that she can gain all of player 1's energy? What strategy should a player adopt knowing that the other player is also thinking strategically? A rational analysis of such strategic interactions is the purview of game theory. In this section, we examine two people jumping on a trampoline from a game theoretic perspective.

For the reader unfamiliar with game theory, we recommend [10] for a non-technical introduction, and [11] for a more advanced mathematical treatment and precise definitions. Game theory is broadly applicable to analysis of any strategic interaction, be it between humans, other animals, computer software – indeed, between any set of agents, such that the consequences of any one agent's action are affected by the actions of other agents. Game theory, pioneered initially by von Neuman [12], has been applied, for instance, to the arms race [13] and to the evolution of species and animal behavior [14].

Model with legs.

Consider two players modeled as point-masses as shown in Figure 8, now with mass-less legs that the players can extend and contract within a range of lengths. The leg lengths are respectively and , and . We assume that the leg lengths are picked before contact, and during a bounce, the legs are completely rigid and perform no mechanical work. After interacting with the trampoline, eventually, both players reach flight phase (phase P0) and the leg lengths go back to zero. The object of the game, then, is for each player to pick their leg length so that their energy is maximum when they reach phase P0, when both are in flight again.

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Figure 8. Bouncing as a zero sum game.

a) Two players drop from the same height and they can pick a rigid leg length to modify their contact times. b) The normalized energy increase in mass-1 (the payoff function) as a function of the leg length choices of the two players, when the two players drop from an initial height of 2 m. The vertical line (blue/red) shows the minimax and maximin strategy that coincide. c) The payoff function when initial height is 0.9 m. For this height, the pure minimax (denoted player 2) and maximin (denoted player 1) strategy do not coincide. The second panel shows the optimal probabilities corresponding to mixed minimax strategies. For these calculations, we used parameter values pertaining to people, as used earlier (kg, etc).

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We analyze the energy transfer over only one bounce; that is, over only one complete interaction through the trampoline. Strategies for maximizing energy transfer over multiple bounces were not considered. Picking a non-zero leg length has two possible effects for a player: (1) It makes the player contact the trampoline earlier, thus altering the contact timing relative to the other player. (2) It reduces the initial effective potential energy of the player.

A zero-sum game.

Player 1 wishes to maximize her energy increase and player 2 wishes to maximize her energy increase . For this game, in the language of game theory, the energy gain after a bounce is the players' payoff and the leg length choice during flight is their strategy or move. This is a continuous game [15] in that the players pick continuous-valued variables and . The energy gain functions and are called payoff functions or payoff surfaces.

Because energy is conserved in this system, we have . Thus, player 1's energy gain is player 2's energy loss and vice versa. Games such as these, in which the total payoff to all players add up to zero, in which one player's gains exactly equal total losses of the other player(s), are called zero-sum games or strictly competitive games [11]. Thus, instead of stating that player 2 wishes to maximize , we could equivalently state that player 2 wishes to minimize .

The payoff functions for our problem (namely, energy transfer ) are shown as surfaces in Figure 9a–b, for two different cases in which the two players start at rest from identical initial heights 0.9 m and 2 m respectively. To construct these payoff surfaces, we discretized the continuous space of strategies, by using a 100×100 grid of pairs, each ranging from 0 to 0.5 m. We computed the energy transfers for each pair on this grid, by performing numerical integrations similar to those that produced Figure 5. This gives a 100×100 payoff matrix, a discrete approximation to payoff function .

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Figure 9. Symmetric games: Dropping from the same height.

a) The normalized energy increase in mass-1 (the payoff function) as a function of the leg length choices of the two players, when the two players drop from an initial height of 2 m. The vertical line (blue/red) shows the minimax and maximin strategy that coincide. c) The payoff function when initial height is 0.9 m. For this height, the pure minimax (denoted player 2) and maximin (denoted player 1) strategy do not coincide. The second panel shows the optimal probabilities corresponding to mixed minimax strategies. For these calculations, we used parameter values pertaining to people, as used earlier (kg, etc).

https://doi.org/10.1371/journal.pone.0078645.g009

Given this zero sum game, how should players pick their strategies? Next, we discuss two kinds of game-theoretic solution strategies, namely, deterministic and probabilistic (mixed).

Strategies and game-theoretic equilibria for bouncing on a trampoline

We compute the game theoretic solutions by first reducing the continuous game to a finite game by discretizing the space of leg length strategies: 50 leg length choices for each player, so that the payoff function is a 50×50 payoff matrix. Then, the computation of the pure minimax/maximin strategies reduce to finding the minimum or maximum of rows and columns (as appropriate) of the payoff matrix. The computation of the mixed minimax or Nash equilibrium can be reduced to a linear programming problem: that is, the minimization of a linear function subject to linear equality and inequality constraints [17]. See Materials and Methods for more methodological details. The unknowns in this linear optimization problem are the probabilities at which the various and values are chosen by the respective players. A mixed Nash equilibrium always exists for a game with finitely many strategies; and this is true of our game, once we have reduced our continuous game to a finite game.

Mixed strategies in asymmetric bouncing games: Rich get richer.

When a zerosum game is not symmetric, there is no longer a theorem that states that the expected value of the payoffs will be zero. The bouncing game is not symmetric (generically) if the players start with different initial heights from rest, different initial velocities, have different masses, or have asymmetric positions along the trampoline (lengths ). We considered all these asymmetries, one at a time, and computed the expected payoff of each player's minimax strategy, also the Nash equilibrium.

For Figure 10a, we considered the two players dropping from rest from slightly different initial heights, holding all other parameters symmetric as before; we have m, and is varied between 1.6 and 2.4 m. For each of these initial heights, we first compute the payoff surface and then obtain the mixed Nash/minimax solution by linear programming. For this range of initial heights, we find that the mixed Nash/minimax solution always had the following property: the player that initially has a higher energy – that is, drops from a higher initial height – had an expected energy gain. That is, the rich get richer. Thus, any small initial energy asymmetry only grows in time, making synchronous bouncing unstable (unstable in the sense of Figure 7 which pertains to time-evolution of the bouncing – rather than unstable in the sense of Nash). Similarly, in Figure 10b, we consider the effect of mass ranging from 50 to 90 kg, while keeping everything else symmetric. Again, we find that the mixed minimax/Nash is such that the player that has the greater mass – and therefore starts with more energy – stands to gain even more energy in a single bounce.

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Figure 10. Asymmetric games.

All parameters are kept at their default symmetric values, except for the one parameter that is varied for each panel. a) Players drop from different initial heights. Player 1 has initial height m; player 2's initial height is varied as shown. All else is symmetric. The player that starts higher, that is, has higher initial energy, gains even more energy in a single bounce, further increasing her energy. b) Players have different masses. Player 1's mass is kept at 70 kg and player 2's mass is varied as shown. Again, the player with greater initial energy gains even more energy. c) Players have different initial speeds. Player 1 starts at rest and player 2's initial speed is changed. When player 2 has a positive or upward initial speed, it lands second and gains energy from player 1. d) Players have asymmetric positions on the trampoline: player 2's position is moved to the right by . We find that the player that is closest to its end of the trampoline loses more energy.

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For Figure 10c, we keep everything symmetric except change the initial speed of player 2, keeping . Thus, the initial energy of player 2 is greater than that of player 1 for all the cases considered here. Nevertheless, one player can gain energy from the other — we see that the player with the greater downward speed – that is, the player that lands first – gains more energy. Note that this trend is consistent with that observed earlier with the passive bouncing – for instance, similar to Figure 5a, which however has the two masses start with equal energy.

Finally, for Figure 10d, we keep everything symmetric except for the horizontal position of player 2 along the trampoline. That is, we keep the position of player 1 fixed, change length by and change length by . Here, the players start with the same energy, and we find that the player closer to the center gains more energy.

Overall, the results for asymmetric games in Figure 10 are similar to those obtained earlier for passive bouncing. This is because even though the panels in Figure 10 were obtained by computing the mixed minimax/Nash strategies, ultimately, the ‘mixed’ strategies obtained were pure. That is, almost all computed strategies for Figure 10 involved using with unit probability. The only exception we found was for especially large and small in Figure 10a, for which the computed mixed strategies actually had a non-trivial probability distribution spread over the different available leg lengths.

Many balls bouncing

As an aside for future work, we generalized our two-ball simulation to the the bouncing of balls on the trampoline (). While we did not examine this system in great detail, we found that even with small numbers of balls, say about 10, the system starts to exhibit properties reminiscent of macroscopic statistical mechanical systems. For instance, Hamiltonian systems with very large number of degrees of freedom, even though energy-conservative, are capable of phenomena analogous to ‘damping’ — conversion of macroscopically observable kinetic energy to internal degrees of freedom. Figure 11 demonstrates this phenomenon in a 25 ball system. Here, initially the macroscopically coherent motion of the masses, moving together as a whole, gets converted into largely incoherent motion of the individual particles. Thus, we see that the oscillation amplitude of the string decays, even through the total energy in the masses is conserved. The mean velocity of the masses (center of mass speed) decreases, while the kinetic energy relative to the system's center of mass increases. See section on Materials and Methods for simulation details. See video animation in movie S4.

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Figure 11. Statistical mechanics of many balls bouncing passively.

a) Twenty five balls are dropped from approximately but not exactly the same height. b) The angle of the string is shown as a function of time. Note that the string oscillates macroscopically initially, as the masses bounce together coherently on it. However, eventually this macroscopic coherent motion of the masses and the coherent oscillatory string motion gets ‘damped out’ with the energy getting transferred to incoherent motion of the masses. c) A ‘macroscopic’ kinetic energy computed as with mean ball velocity decreases with time, as the masses' velocities cancel each other more.

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Non-smooth dynamics simulations

The equations of motion (Eq. 1) describing two balls bouncing were integrated in MATLAB using standard ordinary differential equation solvers (ode45 and ode15s with high accuracy specifications ). Switching from one phase to another phase of the hybrid system is achieved with the in-built ‘event detection’ capabilities of the ODE solvers, so that the solution is formed as a patch-work of solutions to smooth differential equations. See Programs S1 for the complete MATLAB code.

For ball simulations, , we now present the equations of motion in an algorithmic form. Let denote the horizontal and vertical positions of the ball, . If ball- is not in contact with the string, its motion is governed by . At any moment, the set of balls that are in contact with the string are determined as those with . Say there are balls in contact and the set contains the indices of the balls in contact. For instance, if balls 3, 5, and 8 are in contact, and .

The equation for ball in contact () is:in which, in addition to the above definitions, we have , . Also, and are the fixed positions of the left and right ends of the string. See Programs S1 for the complete MATLAB code. For this ODE solution, we simply integrate over the non-smoothness in the right-hand side of the differential equation, using the stiff solver ode15s. While this is not ideal for solution accuracy (as the solvers assume higher-order differentiability), the adaptive step-sizing keeps the error small enough that energy is conserved over reasonable time-scales. This accuracy seems sufficient for obtaining overall qualitative or statistical properties of the dynamics (as verified by comparing with the more accurate two-ball simulation).

Table-top demonstration

For the physical table-top demonstration, we used lacrosse and tennis balls dropped onto Gold's Gym mini trampoline, 36 inches in diameter. We did not tune any of the stiffnesses, and our calculations suggest that the effects will be seen on trampolines of vastly different mechanical properties.

Computing game-theoretic solutions: Pure strategies

As noted, first, we evaluate the payoff function at finitely many and to obtain a payoff matrix. Evaluating the payoff requires solving the differential equations until both players are in flight again. We used 50-element lists of and , namely and , with the lengths ranging from 0 to 0.5 m, to get 50×50 payoff matrices. Let us call the payoff matrix for player 1 and for player 2, to distinguish them from the corresponding continuous functions and . We use the convention that the matrix element , where represents the element of the list of 's.

Given the payoff matrix A, we find the maximin strategy for player 1 by first finding the minimum of each row of ; and then finding the row with the maximum minimum. The and corresponding to those rows and columns give the maximin solution. Equivalently, the minimax strategy for player 2 is found by repeating this same maximin procedure on . Alternatively, the minimax strategy for player 2 is found by first finding the maximum for every column of and then finding the column with the minimum maximum.

Computing game-theoretic solutions: Mixed strategies

The mixed equilibria, as noted earlier, are solved using linear programming, briefly described as follows. Consider the perspective of player 1, who wishes to maximize . Again, we discretize the continuous space of strategies, so that we can examine finitely many strategies and chosen by the two players (, say). The corresponding payoff matrix is , with elements . Player 1 chooses strategy with probability (). These probabilities are our unknowns and are found by solving the following maximin problem:(8)(9)(10)As outlined in [17], [36], in his maximin problem can be converted into a simple optimization problem, with linear objective function and linear constraints – a linear programming problem – by introducing an additional variable as follows:such that(11)(12)(13)(14)where the solution for variable equation gives the expected payoff for player 1. This linear programming problem has unknowns, namely, the probabilities and u. The optimization problem can be solved reliably with provable guarantees using standard software (linprog in MATLAB). See [37] for an introduction to linear programming.

To obtain the probabilities with which player 2 should choose leg length , we repeat the above procedure after replacing every occurrence of replaced by . We verified that the expected payoffs for the two players, obtained by solving the problem independently from each player's perspective, are identical up to numerical tolerances.

The MATLAB code used to solve these problems are part of Programs S1.