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European Journal of Applied Sciences – Vol. 12, No. 4

Publication Date: August 25, 2024

DOI:10.14738/aivp.124.17314.

Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied

Sciences, Vol - 12(4). 164-182.

Services for Science and Education – United Kingdom

Harnessing Active Force: The Pumping Mechanism of Child’s

Swing Motion

Rong Li

Research Center for Industries of the Future, Westlake University, Hangzhou,

Zhejiang 310030, China and Key Laboratory of Coastal Environment and

Resources of Zhejiang Province, School of Engineering, Westlake University,

Hangzhou, Zhejiang 310030, China

Weicheng Cui

Research Center for Industries of the Future, Westlake University, Hangzhou,

Zhejiang 310030, China and Key Laboratory of Coastal Environment and

Resources of Zhejiang Province, School of Engineering, Westlake University,

Hangzhou, Zhejiang 310030, China

ABSTRACT

Life mechanics, an emerging field, focuses on the self-organizing motions

manipulated by the mind within living systems. This study introduces the concept

of 'active force’, generated by mind-body-environment interactions, as a

fundamental driver underlying these self-organizing movements. As an example,

we propose a new set of control equations to model the self-pumping swing

motion by incorporating the active force into Newton's second law. With this new

mechanical framework, we inversely derived the total (i.e., responsive) active

force due to the body-environment interaction from the child’s swing motions

with rapid standing and squatting movements. It revealed a pulse-like pattern of

the total active force along the swing length, driving changes in the radial speed

and swing length. This force counteracts the resistance and propels the swing,

which is not attainable by the stone. Consequently, the active force serves as the

foundational principle for self-organization in living systems, offering a novel

mechanical approach for understanding and predicting extraordinary movements

(e.g., sports and rehabilitation) regulated by the mind (e.g., nervous system) in

biological systems.

Keywords: Mind-body interaction, Active force, Newton's Second Law, Swing, Pulse

INTRODUCTION

Dyson, a physicist, remarked that the twenty-first century may be the century of biology [1].

Life, the most intricate of complex systems, is usually defined as a system that exhibits many

nontrivial movements, including responsiveness, energy transformation, metabolism, growth,

reproduction, and evolution [2]. Understanding these living movements poses the greatest

challenge in modern science [3, 4]. Accepting the axiom that force is the only reason for the

change in body movement, Schrödinger's seminal question "What is life?" [5] can be reframed

in the context of Newtonian mechanics: can we construct a mechanical model that describes

nontrivial movements inherent in life?

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Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -

12(4). 164-182.

URL: http://dx.doi.org/10.14738/aivp.124.17314

Traditional Newtonian mechanics, which views the human body as the mechanical sum of its

parts, overlooks the complexity of mind-manipulating interactions and emergent behaviors.

From the late 19th century, it became clear that viewing life merely as a machine was

insufficient for understanding phenomena, such as embryonic cell development. For instance,

Driesch's experiments suggest that cells have an inherent ability to adapt to changing

environments [6, 7]. Further studies led to the birth of the modern system theory in the 1930s

[8]. The “vitality” and “entelechy” postulated by early vitalists [9] found modern

interpretation in the concept of “self-organization” in complex system science [10, 11].

However, despite the development of numerous phenomenological differential equations and

theories for complexities and lives [11-15], mechanical descriptions of the dynamics of self- determined (i.e., by mind) movement in life remain rare. This void in physicists’

understanding signals the need for the development of life mechanics [16].

In response to this need, we propose the concept of "active force”, an internal force and its

direct response arising from mind-body-environment interactions, as an integral part of life

mechanics. Recently, studying the autonomous motions in living and engineering systems has

led to a conceptual innovation related to “active” mechanics, for instance, “active matter” in

physics [17] and “active cell mechanics” in biomechanics [18], reminiscent the “action

potential” in electrophysiology [19]. Therefore, the active force is a straightforward

conceptual development along this concept series to define the mechanical deriver of

autonomous motions, which ranges from cells, fish, birds, and people. Indeed, the concept of

active force has been mentioned in a minority of literature, for instance, in experimental

analyses of certain skeletal and muscular motions [20, 21] as well as in active cell mechanics

[18], where it acts as the (stochastic) force driving self-organization behaviors. However, its

mechanical study is still in its infancy, possibly due to its vague mechanical definition,

calculation, and measuring complexity.

Therefore, there remains a substantial gap in understanding the generation mechanisms,

temporal patterns, and physiological significance of the active force. To fill this gap, our new

general system theory (NGST) [22-25] begins a preliminary study into its generation

mechanisms, classification, and mechanical representation in Newton’s laws. It presents

unified mechanics incorporating this active force from mind-body-environment interactions

and the passive forces arising from external interactions with other objects or the inanimate

matter-matter interactions independent of the mind. In other words, whether the force is

active or passive only depends on whether it is generated by mind-body interaction. This

classification doesn't violate Newton's laws; thus, the concept of active force is not a violation

but rather an extension of classical mechanics to incorporate the dynamics of living objects

under the living state. Our previous work on NGST has also shown that the active force must

compensate for the energy dissipated by resistance in a changing environment through doing

work [26, 27].

In this study, we examined the dynamics of a simple pendulum system to illustrate the

necessity of an active force to explain the observed phenomena in swing motions. Specifically,

by comparing the motion patterns of a child and a stone of equivalent weight, we demonstrate

that an active force that extends beyond the conventional forces of gravity and friction is

introduced by a child to generate motion patterns that diverge from those of a lifeless swing.

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This difference originates from the mind manipulating internal interactions and relative

motions between elements (e.g., leg muscles in swing motion or walk) of the body, which

induce a subsequent extra body-environment interaction, propelling the center of mass

motions of the swing. Our findings underscore the importance of incorporating an active force

into Newton's second law and the mechanical analysis of the mind-body-environment

interaction as a fundamental paradigm in active life mechanics.

GENERAL FRAMEWORK OF NEWTON'S SECOND LAW WITH ACTIVE FORCE

Consider the mechanics of a particle within a multiparticle system situated in an Earth-fixed,

non-inertial coordinate system (as depicted in Fig.1). This model is grounded in the

perception that our planet is in motion, an understanding that dates back to the era of Galileo

Galilei. In this context, the governing equation for each particle is derived from Newton's

second law:

d

2

dt

2 miri = Fi

P + Fi

A + Fi

D

, (1)

where mi and ri are the mass and displacement vectors of the ith particle, respectively, Fi

P

, Fi

A

and Fi

D

are the passive-driven force, active force, and dissipative force, respectively. Note that

the vectors are indicated by Roman letters.

Fig. 1: A schematic representation of a N-body system in an earth-fixed coordinate system. It

means that we only consider the motions of the particle systems relative to the observer on the

earth.

Eq. (1) embodies Newton's axiom that force is the agent of the motion change. The dynamics

of nonliving objects can be adequately described by the passive-driven forces Fi

P

and

dissipation forces Fi

D

. These passive-driven forces were generated from other objects,

including particles and the earth, in the system we studied. The most common passive-driven

forces in the macroscopic world are gravity (Fi

G = GmM/r

2

) and the static electromagnetic

force, that is, Fi

M = qi(E + vi × B), where E is the electronic field, B is the magnetic field

strength, qi and vi = ṙ

i are the charge and velocity of the particle, respectively. These forces

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Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -

12(4). 164-182.

URL: http://dx.doi.org/10.14738/aivp.124.17314

can be expressed as a derivative of the generalized potential: Fi

P = −∇U + d(∂U/∂ṙ

i

)/dt,

where the generalized potential is U = VG + qφ − qA ∙ v. Here, VG is the gravity potential and

φ and A are the scalar and vector potentials of the electromagnetic field, respectively. This

analysis employs an Earth-fixed coordinate system and implicitly assumes the validity of

Newton's second law in a non-inertial coordinate system. According to the NGST ontology [26,

27], we must abandon the assumption of inertial coordinate systems because they do not exist

for human observers. However, the origin and expression of Fi

D

are typically complex. The

classical linear friction (Fi

D = −kivi

) and its corresponding Rayleigh dissipation function D =

∑ kivi

2

i /2 [28] is only a particular case (i.e., n = 1) of the general formula, that is, Fi

D = −kivi

n

.

Although complex, friction is always defined along the inverse direction of the velocity.

In addition, a new type of force, referred to as the "active force,” was introduced to explain the

initiating movement changes in living entities [22]. As introduced in the Introduction section,

the active force is the internal force directly generated by the mind-body interaction in a

living system. In contrast, the passive forces arise from external interactions with other

objects or the internal but inanimate matter-matter interactions independent of the mind.

Therefore, the critical distinction between active and passive forces is whether directly

generated by the mind-body interaction. Thus, any entity possessing a mind can exert an

active force; the separation of the mind from the body signifies the death of a living organism.

After death, a living object no longer exhibits an active force.

Thus, two immediate questions arise. How can the internal active force generate the center of

mass motions, and how can this active force be calculated? From our daily experiences,

humans exhibit active forces during their self-determined movements. For example, humans

and other animals can walk, run, and swing by using both the internal active force and the

body environment interactions. To generate these self-determined movements, one generally

utilizes the mechanism of mind-body-environment interaction: the mind issuing biosignal

(e.g., Nerve impulse) to modulate the interactions between elements of the body and their

relative motions. Subsequently, the body exerts extra force changes to the environment,

which in turn applies a reactive force that propels the body into the center of mass motion.

This tripartite interaction is essential to two kinds of active forces for the self-determined

movements of the living system. Compared to the baseline state of no motion or external

stimulation, we define the force change of the interactions between body elements as the

“internal active force” and the force change of the body-environment interaction as the

“responsive active force”. Thus, for each part of the body, we can obtain a decomposition of

the active force

Fi

A = Fi

I + Fi

R

, (2)

where Fi

I

is the internal active force and Fi

R

is the responsive active force. The mind-body

interaction mainly determines the former, while the environment constrains the latter.

Therefore, active force is interpreted as a mind-body-environment interaction. Without the

support of environment, the active force could not be generated.

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However, calculating these active forces is a significant challenge, as it is governed by the

unpredictable free will of the mind, as well as the complex mind-body-environment

interaction. We suggest initially adopting a model that explains this phenomenon qualitatively

and subsequently developing methods to quantify the active force. This study presented an

early effort and some initial results. In general, everyday human movements such as walking,

running, and swinging are inherently complex, necessitating rigorous modeling of the

interactions and movements of various body parts [29-31]. However, this paper aims not to

explore modeling too many complex motions but to clarify the differences between active

forces in living systems and those in inanimate bodies. Therefore, our primary focus in this

study was on the motion of the center of mass for its simplicity of degree of freedom and the

more straightforward measurement of the responsive active force compared to the internal

active force. In this case, the internal active forces of different body parts cancel with each

other (i.e., ∑ Fi

I

i = 0 following Newton’s third law), while the summation of the responsive

active force from the environment to the body is nonzero. Thus, in line with the definition of

the center of mass coordinates (R = ∑i miri / ∑i mi

), Newton’s second law can be obtained

from the summation of Eq. (1) as,

M

d

2

dt

2

R = Fp + Fa + f, (3)

where M = ∑i mi

is the total mass, Fp = ∑ Fi

P

i

is total passive (environment-to-body) force at

the baseline state of the body, f = ∑ Fi

D

i

is the total passive-driven force, total active-driven

force, and the total dissipation force, respectively. Fa = ∑ Fi

A

i = ∑ Fi

R

i

is the total active force

of the body relative to the baseline state, equaling the total responsive active force since

∑ Fi

I

i = 0. In this paper, we employ Eq. (2) to extract the total (responsive) active force

driving the swing's motion, deferring the elucidation of internal active forces to future

investigations.

THE RESPONSIVE ACTIVE FORCE FOR PUMPING A SWING BY A LIVING SYSTEM

This section explores how an active force propels a swing, mainly by comparing the dynamic

differences (displacement, velocity, and force) between the swing motions of a child and a

stone. This comparison enables us to quantitatively derive the temporal pattern of the

responsive active force, revealing the essential role of active force in explaining the motions of

living systems.

The Governing Equation for Pumping A Swing

Applying Eq. (3) elucidates the differences between the swing motions of a child and a stone,

considering only simple pendulum motions within a vertical two-dimensional plane. Fig. 2

illustrates the simple pendulum system: (a) represents a classical case with a non-living stone,

whereas (b) substitutes the stone with a child of equivalent weight. In the stone case,

L denotes the length of the rigid massless rod, m denotes the mass of the stone, and θ

represents the angle of the rod along the vertical axis. Furthermore, three forces acted at the

center of the mass of the stone. First, gravity, Fg = mg acts in the downward vertical direction.

Second, a passive tension force, Tp acts along the rod owing to the balance of gravity and

centrifugal force toward the frictionless pivot. Third, a friction force f resists the motion of

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Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -

12(4). 164-182.

URL: http://dx.doi.org/10.14738/aivp.124.17314

the stone through the medium, which is assumed to be proportional to the velocity of the

stone with a coefficient of friction b in the present work for simplicity. Hence, Newton’s

second law is expressed as follows:

m

dv

dt = Fg + Tp − bv. (4)

Because the passive tension force Tp is balanced by gravity and centrifugal force, the

pendulum length is constant, that is, L = L0. Therefore, it is straightforward to derive the

equation for circumferential motion v⊥ = Lθ̇ from Eq. (4) as

mLθ̈(t) = −mg sin θ(t) − bLθ̇(t). (5)

If we assume that the initial angle is very small, a general solution can be obtained as

θ(t) = e

−

bt

2m(c+e

iωt + c−e

−iωt), (6)

where ω = √g/L0 − b

2/4m2 is the frequency of the damped pendulum, and c+ and c− are two

coefficients determined by the initial conditions. Here, the time-dependent coefficient e

−bt/2m

reveals that the pendulum motion is a damped oscillation that decays with time and finally

ceases. That is, it is a passive motion that depends on the external pumping force.

Fig. 2: (a) A force analysis of a simple pendulum of a lifeless stone. (b) A force

analysis of a living pendulum, i.e., a child named Bob. Tp = mgcos θ(t) + mv⊥

2

/L—

represents the swing rope tension owing to the passive balance of gravity and

centrifugal force. f is the external friction force. Fa is the responsive active force. A

child’s active motion on the swing originates from the mind issuing signals that

modulate the interactions between elements (e.g., leg muscle) of the body, which is

termed as the internal active forces. The terms Fi

I and −Fi

I

represent a pair of

internal active forces acting in opposite directions, indicated in the dashed circle.

Subsequently, the body exerts extra force changes to the environment, which in

turn applies a reactive force (the responsive active force Fa) that propels the body

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37]. It can be considered as a coupled oscillator system composed of a swing and a human.

Typically, there are two pumping strategies: pumping from a standing position (as depicted in

Fig. 3) and a seated position. In the former instance, the person stands at the lowest point and

crouches at the highest point during the swing motion. Each stand-crouch cycle enhances the

swing amplitude. The analysis demonstrated that each crouch-stand cycle provides a swing

with an energy boost from the rider. In the latter scenario, the person abruptly rotates their

body around the end of the swing chain. The amplitude of the swing increases as these

rotations elevate the rider slightly above the highest level.

FIG. 3: Strategy for pumping a swing while standing, adapted from Ref. [38]. The child stands up

near the lowest point and crouches down near the highest point during the swing motion.

In the child’s swing motion, the mind determines the pumping strategies, while the active

force derives the body to motion, and environmental constraints provide the control

conditions for this driver; all these three factors of mind, active force and environmental

constraints are paramount. However, prior research on swing mechanics has primarily

concentrated on effective pumping strategies and environmental constraints, such as the

pumping mode (standing or seated [34, 38]), the modulation of frequency, and the initial

phase [37] under the swing constraints. On the other hand, the force underlying swing

pumping, particularly the active force that drives these movements, has been less explored.

This indicates that the essential force mechanism generated by the mind-body-environment

interaction— has not been adequately addressed. Our study aims to fill this gap by focusing

on the (internal or responsive) active force, which directly results from the mind-body- environment interaction and serves as the driver of the swing's active motion. This work

mainly focuses on the responsive active force, representing the body-environment

interaction. Understanding the temporal evolution pattern of this responsive active force is

crucial, as it not only reveals the direct capacity of individuals to drive the center of mass

movements but also how the environment responds to the body. Given that the force pattern

corresponding to the stand-crouch motion is simpler, we mainly concentrate on pumping

from the standing position in this study.

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mgL0

(1 − cosθn−1

) ≈ mg(L0 − k)(1 − cosθn

). (17)

From this relation, we can obtain the nth maximum angle, height, and velocity as:

θn ≈ arccos [

cosθn−1 − k

′

1 − k

′

], (18)

where k

′ = k/L0. Eq. (18) reveals that the swing amplitude increases every half-cycle

according to a function related to 1/(1 − k

′

). This reveals that the swing accumulates the net

energy from the child’s active motions in each pumping cycle. This net energy stems from the

work done by the child’s active force when standing up, which increases the height by k, over

the energy spent when squatting down, which decreases the height by kcosθn. It is worth

mentioning that Eq. (18) is obtained based on the neglect of the impact of standing squatting

motions. This contribution is −2l(̇ t)θ̇(t)/L(t) for θ̈(t) in Eq. (13), revealing that the standing

results in an increase for |θ̇(t)|. Therefore, the realistic increase of θn (shown in Fig. 4) should

be greater the prediction θn = arccos[1.11(cosθn−1 − 0.1)] by substituting parameters in the

numerical simulations into Eq. (18).

The Responsive Active Force for Pumping A Swing with Linear Friction

Next, we explore a more realistic scenario in which both active and frictional forces are at

play. A particularly noteworthy situation occurs when the responsive active force

counterbalances the frictional force, leading to stable oscillation with a constant amplitude

that neither decays nor expands. Such a state represents a stable equilibrium that is

sometimes observed in swing sports. An intriguing question is whether our active-force

model (Eq. (12) and (13)) can accurately simulate this condition.

Fig. 5: Active-motion simulations using Eqns. (12-16) showcasing a child pumping a swing in

the presence of finite friction, as shown by the solid red lines. To serve as comparisons, the

solid blue lines (indicating finite friction) and dashed black lines (representing no friction)