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Richard Geary (Honda Inc.) helped with an early version of the tabletop demonstration. This acknowledged help does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials as well as authorship, as detailed online in PLOS ONE's guide for authors.

Conceived and designed the experiments: MS AS. Performed the experiments: MS AS. Analyzed the data: MS AS YW. Contributed reagents/materials/analysis tools: MS YW AS. Wrote the paper: MS AS YW. Wrote the MATLAB simulations and performed mathematical analysis: MS YW.

Jumping on trampolines is a popular backyard recreation. In some trampoline games (e.g., “seat drop war”), when two people land on the trampoline with only a small time-lag, one person bounces much higher than the other, as if energy has been transferred from one to the other. First, we illustrate this energy-transfer in a table-top demonstration, consisting of two balls dropped onto a mini-trampoline, landing almost simultaneously, sometimes resulting in one ball bouncing much higher than the other. Next, using a simple mathematical model of two masses bouncing passively on a massless trampoline with no dissipation, we show that with specific landing conditions, it is possible to transfer all the kinetic energy of one mass to the other through the trampoline – in a single bounce. For human-like parameters, starting with equal energy, the energy transfer is maximal when one person lands approximately when the other is at the bottom of her bounce. The energy transfer persists even for very stiff surfaces. The energy-conservative mathematical model exhibits complex non-periodic long-term motions. To complement this passive bouncing model, we also performed a game-theoretic analysis, appropriate when both players are acting strategically to steal the other player's energy. We consider a zero-sum game in which each player's goal is to gain the other player's kinetic energy during a single bounce, by extending her leg during flight. For high initial energy and a symmetric situation, the best strategy for both subjects (minimax strategy and Nash equilibrium) is to use the shortest available leg length and not extend their legs. On the other hand, an asymmetry in initial heights allows the player with more energy to gain even more energy in the next bounce. Thus synchronous bouncing unstable is unstable both for passive bouncing and when leg lengths are controlled as in game-theoretic equilibria.

Bouncing on a trampoline has evolved from a backyard activity for children to an Olympic Sport. While Olympic trampolining only has one person bouncing on a trampoline, in its recreational form, it is quite common for more than one person to bounce on the trampoline simultaneously. In particular, children play a two-person game on trampolines called “seat drop war.” In this game, each player bounces alternatively with her feet and her ‘seat’ (being in an L-shaped body configuration), as shown in

Only one player shown. Each player alternatively bounces with her feet and her ‘seat’. The sequence of body configurations for one player is shown schematically. The other player goes through a similar sequence of configurations, but possibly with a phase difference.

Here, we show that the dramatic energy transfer is observed even in the passive bouncing of inanimate masses. We first describe a simple physical demonstration of the energy transfer: dropping two balls simultaneously onto a small trampoline sometimes results in one ball bouncing much higher than the other. Then, we construct a simple energy-conservative mathematical model, with the two people modeled as masses bouncing passively on a trampoline. This model also exhibits the dramatic energy transfer observed in seat drop war. We call the energy transfer ‘dramatic’ because essentially all the energy of one person/ball gets transferred to the other in a single brief interaction. We show that there is typically an optimal difference between the landing times of the two masses (hereafter called the ‘contact time-lag’) that maximizes energy transfer. The mathematical model, in absence of dissipation or sideways movement of masses, displays complex non-periodic motion, with repeated transfer of energy between the two masses.

Finally, we make a first step at analyzing the game, not as a simple passive dynamics problem involving two balls bouncing, but as a strategic competitive game between two players from a game theoretic perspective, obtaining the optimal strategies for the two players for the zero-sum game.

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

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

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.

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

We idealize the two players as particles with masses

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.

The vertical position of the two masses are denoted

The total system energy consists of the kinetic and gravitational potential energies of the two masses, namely

See

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: ^{−2}. The vertical stiffness of the trampoline at its midpoint

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

a) The motion

The two masses repeatedly exchange energy with each other, sometimes one mass bounces higher and sometimes the other mass bounces higher:

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.

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

In this section (and in the

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 (

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 energy increase in

In

Here, the mass that lands second, namely mass-2, always gains energy and mass-1 always loses energy, whatever the contact time-lag (

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

The basic energy transfer mechanism can be most simply understood using an ‘instantaneous argument’ as illustrated in

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.

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 (

Even though the motion is governed by relatively simple linear differential equations in each phase, an analytical treatment to obtain the dependencies in

In

When the masses are equal (

The stability of symmetric motion is examined by considering the properties of a time-period-based ‘Poincare map’

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

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

For the reader unfamiliar with game theory, we recommend

Consider two players modeled as point-masses as shown in

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

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.

Player 1 wishes to maximize her energy increase

Because energy is conserved in this system, we have

The payoff functions for our problem (namely, energy transfer

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

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).

If the player 1 chooses her leg-length

When the two players have to pick their strategies (leg lengths) simultaneously, another solution concept called the

When the two players have to pick their strategies simultaneously, without knowledge of the other player's strategy, it is appropriate to not just consider the deterministic pure strategies as above, but expand the space of strategies to allow probabilistic strategies – called

The mixed Nash equilibrium is the ordered pair of mixed strategies such that a player cannot improve the expected value of her payoff by changing her mixed strategy unilaterally, while the other player keeps her strategy fixed. Given the probability distributions, the expected value of the payoffs

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

For our bouncing game, the pure maximin and minimax strategies are shown by vertical blue and red lines, overlaid on the payoff functions in

When the initial heights are equal and all other parameters are symmetric, we have a ‘symmetric’ zero-sum game

When the initial heights both equal 2 m, the optimal mixed strategy coincides with the pure strategies, namely

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

For

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

For

Finally, for

Overall, the results for asymmetric games in

As an aside for future work, we generalized our two-ball simulation to the the bouncing of

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

In this article, we have examined the mechanics of energy transfer between two masses bouncing on a trampoline and various aspects of their corresponding dynamics. First, we demonstrated this energy transfer with a table-top set-up, consisting of two balls dropped onto a mini-trampoline landing almost simultaneously. We find that sometimes, when the timing between the two balls landing is just right, one ball bounces much higher than the other. Next, we devised a simple mathematical model of two masses bouncing passively on a massless trampoline with no dissipation. With this mathematical model, we showed that with specific landing conditions, it is possible to transfer all the kinetic energy of one mass to the other through the trampoline, explaining the dramatic energy transfer observed in actual bouncing with humans as well as our table-top demonstration. To our knowledge, we do not know of a prior mathematically based explanation or documentation of this energy transfer. For human-like parameters, starting with equal energy, the energy transfer is maximal when one person lands approximately when the other is at the bottom of her bounce. The passive energy conservative model also has the interesting property that over a long time, the energy shuttles back and forth between the two masses, sometimes almost entirely in one mass and sometimes in the other.

Thus, while the passive model tells us what a player should do (land a little later) to steal the other person's energy in an otherwise passive bounce, it does not tell us what each player would or should do when she knows that the other player is also thinking strategically about the game. We address this strategic interaction question through a game theoretic analysis. To the passive model, we added legs that can change length in flight so as to affect when the player lands and the energy at landing. Starting from both players in flight, we computed the energetic payoff of each player choosing a range of leg lengths before landing. We then computed Nash equilibria and minimax-maximin strategies for the players, which tell us what each player should do given the other player is also thinking strategically. Not surprisingly, we found that if the two players start from the same height, the Nash equilibrium strategies involve neither player gaining or losing energy after a single bounce. However, under asymmetric conditions, one player can gain another player's energy by appropriate choice of leg lengths. For instance, when the two players started from slightly different initial heights, the player that had the greater initial height could pick a leg length that will increase her energy further after a single bounce, for whatever other player does.

Thus, we find that symmetric bouncing is unstable, both for the passive mathematical model and for the game-theoretic version in which leg lengths are chosen strategically by the subjects. The game theoretic model is also able to predict – how body parameters and trampoline location might or not give one or another player a strategic advantage, given the state of the game.

In our game theoretic analysis, we assumed a simplification of the game that preserved total energy conservation, making the game zero-sum. Allowing for active leg work or passive dissipation during contact with the trampoline will make the game non-zero sum. Also, allowing for positive or negative leg work will require appropriately discretizing the space of leg actuation strategies, perhaps similar to that used in other optimization studies of human movement

There have been a few specific applications of game theory to physical games played by humans, such as soccer

Further, while we considered a simple adversarial situation in which each player is trying to steal energy from the other, we could also consider other scenarios that involve a cooperative flavor, in which the subjects are trying to maximize their combined energy or decide how they should cooperate to avoid a non-optimal solution to each, analogous to the

We found that our mathematical model of the passive bouncing dynamics has complex dynamics, which might be of interest to dynamicists. This system is energy-conservative and Hamiltonian

The phenomenon of dramatic energy transfer discussed here has relevance to the mechanics of collisions. The energy transfer here persists even when the trampoline's stiffness is very large and the contact duration very short – our table-top demonstration is already near this ‘collisional’ limit. In this limit, the dramatic energy transfer between the two masses through the trampoline is reminiscent of certain results in the literature on simultaneous ‘rigid-body collisions.’ In particular, simultaneous collisions of nominally rigid-bodies – collisions in which there are multiple points that are making contact at the same time – are known to be often ill-posed. That is, when these simultaneous collisions are ‘regularized’ either by making the contacts happen in sequence or by making the collisions last a non-infinitesimal period with varying time-overlaps between the various contacts, it is found that the details assumed (either the collision sequence or the overlap details) substantially affect the collision consequence

Finally, we comment on possible implications to the bouncing games that children and adults play on trampolines. The dramatic energy transfer between players bouncing on a trampoline may have an effect of trampoline injuries. Some epidemiological studies

The equations of motion (

For

The equation for

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.

As noted, first, we evaluate the payoff function

Given the payoff matrix A, we find the maximin strategy for player 1 by first finding the minimum of each row of

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

To obtain the probabilities with which player 2 should choose leg length

The MATLAB code used to solve these problems are part of

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We are indebted to Danny Finn (Trinity College, Dublin) and the Dublin University Trampoline Club (