Key principle of the efficient running, swimming, and flying

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by Valery B. Kokshenev

Submitted to EPL 11 June 2009, resubmitted 31 August 2009

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Abstract. Empirical observations indicate striking similarities among
locomotion in terrestrial animals, birds, and fish, but unifying physical
grounds are lacking. When applied to efficient locomotion, the analytical
mechanics principle of minimum action yields two patterns of mechanical
similarity via two explicit spatiotemporal coherent states. In steady
locomotory modes, the slow muscles determining maximal optimum speeds maintain
universal intrinsic muscular pressure. Otherwise, maximal speeds are due to
constant mass-dependent stiffness of fast muscles generating a uniform force
field, exceeding gravitation. Being coherent in displacements, velocities and
forces, the body appendages of animals are tuned to natural propagation
frequency through the state-dependent elastic muscle moduli.

Key words: variational principle of minimum action (04.20.Fy), locomotion
(87.19.ru), biomechanics (87.85.G-).

##  I Introduction

Although evolutionary biologists and comparative zoologists make wonderful
generalizations about the movements of terrestrial animals, birds, and fish of
different size [1-12], the fundamental physical principles underlying striking
similarities in distinct types of movement for organisms remain a challenge
[13]. Within the scope of the simplest pendulum model (stiff-legged
approximation), it has been demonstrated [14] that humans and other animals,
in contrast to human-made engines, accomplish efficient propulsion (maximum
power output at minimum power consumption) by tuning musculoskeletal system to
the resonant propagation frequency. Storing mechanical energy in elastic
oscillations of body parts and in pendulum oscillations of legs or other
appendages, animals thereby reduce the energy consumption [1,3], which is
minimal at the resonance conditions [14]. In this study, instead of searching
for uncovered principles of body mass effects in biology [5], or doing in-
depth analysis of equations of motion in pendulum [14], spring [7,8], or
vortex [15] approximations and other engineer constructive approaches [9], I
address the key principle of mechanics.

In analytical mechanics, the requirement of __ minimum action between two
fixed points of the conceivable trajectory of an arbitrary isolated mechanical
system determines Lagrangian $\mathcal{L}(q,v)$, the function of time-
dependent coordinates $q(t)$ and instant velocities $v(t)=dq/dt$. The most
general property of a freely __ moving system is spatiotemporal homogeneity
implying that the multiplication of $\mathcal{L}$ on an arbitrary constant
does not affect the equations of motion, arising from $\mathcal{L}$. This
property, designated as a _mechanical similarity_ [16], permits one to
establish the major mechanical constraints without consideration of equations
of motion. Indeed, following Landau and Lifshitz [16], let us consider the
uniform transformation of mechanical trajectories due to linear changing of
all coordinates $q\rightarrow aq$ and times $t\rightarrow bt$, and hence
velocities $v\rightarrow(a/b)v$, via arbitrary coefficients $a$ and $b$. Let
the potential energy change consequently through a certain exponent $s$, i.e.,
$\mathcal{U}(aq)=a^{s}\mathcal{U}(q)$. Being a quadratic function of
velocities, the kinetic energy scales as
$\mathcal{K}(av/b)=\left(a/b\right)^{2}\mathcal{K}(v)$. The requirement of
homogeneity of $\mathcal{L}(q,v)=\mathcal{K}(v)-\mathcal{U}(q)$ is self-
consistent when both the energies change similar, i.e., $(a/b)^{2}=a^{s}$ or
$b=a^{1-s/2}$ . Thereby, the frictionless propagation of a classical system
obeys the scaling relationships imposed on all principal mechanical
characteristics: _period_ $T$, overall-system _speed_ $V$, and _force_
_amplitude_ $F$, namely [16]

$T\backsim t\varpropto L^{1-s/2}\text{, }V\backsim v\varpropto L^{s/2}\text{,
and }F\varpropto L^{s-1}\text{.}$ (1)

The seminal case $s=-1$ introduces Newtonian’s intertrajectory coupling force
$F\backsim M^{2}L^{-2}$, where mass $M$ emerges as the dimensional coefficient
of proportionality.

It will be demonstrated how the mechanical principle of minimum action applied
to musculoskeletal system of animals involved in efficient locomotion may
provide basic patterns of biomechanical similarity.

##  II Minimum action in biomechanics

During locomotion, chemical energy released by muscles and mechanical elastic
energy stored in body system is transformed into external and internal work
and partially lost as a heat. In the case of the off resonance human walking
[17], the small velocity-dependent frictional effects were accounted for in
the second order of perturbation theory, thereby generalizing the Lagrangian
formalism over weakly open systems.

During the muscle forced resonance walking and running (or flying) minimizing
energy consumption, the small damping effects restrict only the amplitude of
motion, i.e., _stride length_ $\Delta L$ (or stroke amplitude) and _muscle
length change_ $\Delta L_{m}$, but not the propagation _speed_ $V=\Delta L/T$
and _period_ $T$, constrained geometrically [14]. Likewise [17], frictional
effects can be therefore neglected in the equations of motion [14], on the
first approximation. With the same precision, the principle of mechanical
similarity (1) provides

$\displaystyle T^{-1}$
$\displaystyle=1/T_{ms}\varpropto\sqrt{E_{ms}}L_{m}^{-1}\text{, }V\backsim
V_{ms}\varpropto\sqrt{E_{ms}}\text{,}$ $\displaystyle F$
$\displaystyle\backsim\Delta F\backsim F_{ms}\backsim\Delta
F_{ms}=\varepsilon_{m}A_{m}E_{ms}\text{, with
}E_{ms}\varpropto(L_{m})^{s}\text{ and }g_{ms}\varpropto(L_{m})^{s-1}\text{,}$
(2)

when presented in the linear-displacement body ($\Delta L\backsim L$) and
muscle ($\Delta L_{m}\backsim L_{m}$) approximation. Introducing in eq. (2)
the force change $\Delta F$ for the body _force output_ $F$, driving a given
animal (of _characteristic length_ $L$, _cross-sectional area_ $A$, and _body
mass_ $M$) through the environment, and the effective body rigidness, or
longitudinal _stiffness_ $K=\Delta F/\Delta L$, one also determines the
natural (resonant) _cyclic frequency_ $T^{-1}\backsim\sqrt{K/M}$
[1,7,8,17,18]. Since the animal locomotion is substantially muscular [1,3,18],
the _muscle stiffness_ $K_{m}=E_{m}A_{m}/L_{m}$ (of a muscle of length $L_{m}$
and cross-sectional area $A_{m}$), controlled by the geometry-independent
muscle rigidity or _elastic modulus_ $E_{m}$ (ratio of _stress_ $\sigma_{m}$
to _strain_ $\varepsilon_{m}$, i.e., $(\Delta F_{m}/A_{m})/(\Delta
L_{m}/L_{m})$) [7, 18], is also under our consideration. To improve the
integrative approach to animal locomotion [1-18] via mechanical [19] and
elastic strain [19, 20] similarities, let us determine a muscle-force _field_
$g_{m}\equiv F_{m}/m$, where the _muscle mass_ $m$ (or _motor mass_[6]) is a
source of the _active force output_ $F_{m}$. Furthermore, the scaling
relations for physical quantities (shown in eq. (2) by symbol $\varpropto$)
result from provided relations and constraints imposed by the invariable
_body_ _density_ $\rho$ ($=M/AL$) and _muscle density_ $\rho_{m}$
($m/A_{m}L_{m}$), all common in scaling biomechanics [1,7,9,18].

In this study, the intrinsic muscle modulus $E_{ms}$, substituting $E_{m}$ in
eq. (2), describes a new dynamic degree of freedom characterizing muscle
ability of tuning to the resonance [15] in different locomotory gaits
distinguished by the single _dynamic-state exponent_ $s$.

##  III Results and discussion

The _steady-speed locomotion_ for flight mode was first recognized by Hill:
”the frequencies of hovering birds are in inverse proportionality to the cube
roots of the weights, i.e., to the linear size” [2]. This dynamic regime is
pronounced in eq. (2), taken with $s=0$, by the propagation frequency
$T^{-1}\backsim\sqrt{E_{m0}/\rho}L^{-1}$, contrasting with the rigid-pendulum
estimate $T_{pend}^{-1}\backsim\sqrt{g}L^{-1/2}$ ($g$ is gravitation field)
[7,14]. Broadly speaking, Hill’s observation plays the role similar to
Kepler’s observation of third law for planets $T^{2}\varpropto L^{3}$,
following from eq. (1) with $s=-1$.

Hence, when the animal’s body travels or cruises slowly for long distances [4]
with the constant optimum speed $V_{body}^{(\max)}$
$\backsim\sqrt{E_{m0}^{(\max)}/\rho}$, invariant with body weight and
frequency, or moves throughout the terrestrial, air, or water environment
resisting drag forces, the legs, wings, and tails suggest to maintain constant
elastic modulus $E_{m0}^{(\max)}$ in _slow muscles_ responsible for the steady
locomotion [21]. Consequently, a constant functional intrinsic muscle stress
$\varepsilon_{m}E_{m0}$ is also predicted in eq. (2) with $s=0$, providing in
turn constant safety factor (ratio of muscle strength to peak functional
stress), also expected by Hill [2]. These and other relevant constraints of
steady-speed locomotion are displayed in table 1.

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$s=0$ | Frequency | Length |  Speed |  Force | Mass
---|---|---|---|---|---
$\ \ T^{-1}$ | $T^{-1}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot L^{{\small-1}}$ | $\rho^{{\small-}\frac{1}{4}}E_{{\small 0}}^{\frac{1}{4}}\cdot V^{-\frac{1}{2}}$ | $F^{0}$ | $\rho^{{\small-}\frac{1}{6}}E_{{\small 0}}^{{\small-}\frac{1}{2}}\cdot M^{{\small-}\frac{1}{3}}$
$\Delta L,L$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot T$ | $L$ | $\rho^{{\small-}\frac{1}{4}}E_{{\small 0}}^{\frac{1}{4}}\cdot V^{\frac{1}{2}}$ | $F^{0}$ | $\rho^{{\small-}\frac{1}{3}}\cdot M^{\frac{1}{3}}$
$V^{(\max)}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot T^{0}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot L^{0}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot F^{0}$ | $\rho^{{\small-}\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}\cdot M^{{\small 0}}$
$K_{body}^{(\max)}$ | $\rho^{\frac{1}{2}}E_{{\small 0}}^{\frac{1}{2}}A\cdot T^{-1}$ | $E_{{\small 0}}A\cdot L^{-1}$ | $\rho^{\frac{1}{4}}E_{{\small 0}}^{-\frac{1}{4}}\cdot V^{-\frac{1}{2}}$ | $L^{-1}\cdot F$ | $\rho^{{\small-}\frac{1}{3}}E_{0}\cdot M^{\frac{1}{3}}$
$\sigma_{slow}^{(\max)}$ | $\varepsilon_{m}E_{0}\cdot T^{0}$ | $\varepsilon_{m}E_{0}\cdot L^{0}$ | $\varepsilon_{m}E_{0}\cdot V^{0}$ | $\varepsilon_{m}E_{0}\cdot F^{0}$ | $\varepsilon_{m}E_{0}\cdot m^{{\small 0}}$
$F_{slow}^{(\max)}$ | $\varepsilon_{m}E_{0}A_{m}\cdot T^{0}$ | $\varepsilon_{m}E_{0}A_{m}\cdot L^{0}$ | $\varepsilon_{m}E_{0}A_{m}\cdot V^{0}$ | $\varepsilon_{m}^{(\max)}E_{0}A_{m}$ | $\rho_{{\small m}}^{{\small-}\frac{2}{3}}\varepsilon_{m}E_{0}\cdot m^{\frac{2}{3}}$

Table 1. Mechanical characteristics of body system and slow individual muscles
in the steady-motion dynamic states $s=0$ prescribed by the principle of
minimum muscular action in eq. (2). _Abbreviation_ : $E_{0}=E_{m0}^{(\max)}$.

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The constant maximum propulsive force $F_{body}^{(\max)}\backsim
E_{m0}^{(\max)}A$, equilibrating all drag forces via slow muscles, i.e.,
$F_{drag}^{(\max)}\backsim F_{slow}^{(\max)}$ shown in table 1, was first
documented by Alexander as the peak body force $F_{body}^{(\exp)}\varpropto
M^{2/3}$ [10] exerted on the environment by running, flying, and swimming
animals ranged over nine orders of body mass. More recently, the slow-fiber
force output $F_{slow}^{(\max)}\varpropto m^{2/3}$ (table 1) was revealed [6]
by statistical regression method in both biological and human-made _slow
motors_. The underlying muscle longitudinal field ”caused by intrinsic muscle
quantity (here associated with $E_{m0}$), equally stimulated electrically and
by the nervous system” [2] decreases linearly with the distance $r$:
$g_{slow}^{(\max)}(r)\thickapprox
E_{m0}^{(\max)}\varepsilon_{m}^{(\max)}/\rho_{m}r$, where
$\varepsilon_{m}^{(\max)}$ is nearly isometric strain, as follows from eq. (2)
with $s=0$.

The resonant efficient locomotion broadly prescribes a concerted behavior
synchronized in time and coordinated in displacements and forces of the body’s
appendages. Consequently, the muscle _duty factor_ $\beta_{m}=\Delta t_{m}/T$,
where $\Delta t_{m}$ is timing of the muscle lengthening/shortening $\Delta
L_{m}$, is constant, besides the body-mass invariable _Strouhal number_
$St=\Delta L/VT$, explaining the tail and wing oscillations in swimmers and
flyers [22]. At maximum propulsive efficiency of cruising dolphins, birds, and
bats, it was observed as $St_{cruis}\thickapprox 0.3$ [4].

The steady-speed locomotion state also was remarkably established in hovering
flying motors via the wing frequencies $1/T^{(\exp)}\varpropto M^{-1/3}$ [23],
as predicted in table 1. However, departures from Hill’s findings rationalized
here by the dynamic-state exponent $s=0$ were also debated [24]. For example,
it was claimed [7] that Hill’s maximal optimum speeds are in sharp
disagreement with the peak trot-gallop _crossover speeds_ $V_{cross}^{(\exp)}$
measured in quadrupeds [12]. The same could refer to the bipeds [11]. However,
as can be seen from the proper empirical data $1/T^{(\exp)}\varpropto
M^{-0.178}$ [11] and $1/T^{(\exp)}\varpropto V_{cross}^{-1}$ $\varpropto
M^{-0.145}$ [7,12], the measured stride frequencies indicate observations of
another kind of mechanical similarity attributed to the _non-steady dynamic
state_ $s=1$, prescribed in eq. (2) through the mass-dependent muscle modulus
$E_{m1}\varpropto L_{m}\varpropto M^{1/3}$.

The minimum muscle action of legs in fast running rats, wallaby, dog, goat,
horse, and human was indirectly revealed through the mechanical similarity
derived with the help of _leg spring model_ [8], providing the stride
frequency $T^{-1}\backsim\Delta t_{leg}^{-1}\varpropto M^{-0.19}$, stride
length $\Delta L\varpropto M^{0.30}$, model-body length $L_{leg}\varpropto
M^{0.34}$, body stiffness $K_{leg}^{(\max)}\varpropto M^{0.67}$, and body
force output $F_{leg}^{(\max)}\varpropto M^{0.97}$. Relations between the
quantities underlying these findings are discussed below and summarized in
table 2.

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$s=1$ | Frequency | Length |  Speed |  Force | Mass
---|---|---|---|---|---
$T^{-1}$ | $T^{-1}$ | $g_{1}^{\frac{1}{2}}\cdot L^{-\frac{1}{2}}$ | $g_{1}\cdot V^{-1}$ | $(\rho g_{1}^{2}A)^{{\small-}\frac{1}{2}}\cdot F^{\frac{1}{2}}$ | $\rho^{\frac{1}{6}}g_{1}^{\frac{1}{2}}\cdot M^{{\small-}\frac{1}{6}}$
$\Delta L,L$ | $g_{1}\cdot T^{2}$ | $L$ | $g_{1}^{-1}\cdot V^{2}$ | $(\rho g_{1}A)^{{\small-1}}\cdot F$ | $\ \rho^{{\small-}\frac{1}{3}}\cdot M^{\frac{1}{3}}$
$V_{cross}^{(\max)}$ | $g_{1}\cdot T$ | $g_{1}^{\frac{1}{2}}\cdot L^{\frac{1}{2}}$ | $V$ | $(\rho A)^{{\small-}\frac{1}{2}}\cdot F^{\frac{1}{2}}$ | $\rho^{{\small-}\frac{1}{6}}g_{1}^{\frac{1}{2}}\cdot M^{\frac{1}{6}}$
$K_{body}^{(\max)}$ | $\rho g_{1}A\cdot T^{0}$ | $\rho g_{1}A\cdot L^{0}$ | $\rho g_{1}A\cdot V^{0}$ | $\rho g_{1}A\cdot F^{0}$ | $\rho^{{\small-}\frac{1}{3}}g_{1}\cdot M^{\frac{2}{3}}$
$\sigma_{fast}^{(\max)}$ | $\rho_{m}g_{1}^{2}\cdot T^{2}$ | $\rho_{m}g_{1}\cdot L_{m}$ | $\rho_{m}\cdot V^{2}$ | $A_{m}^{-1}\cdot F_{m}$ | $\rho_{m}^{\frac{2}{3}}g_{1}\cdot m^{\frac{1}{3}}$
$F_{fast}^{(\max)}$ | $\rho_{m}g_{1}^{2}A_{m}\cdot T^{2}$ | $\rho_{m}g_{1}A_{m}\cdot L$ | $\rho_{m}A_{m}\cdot V^{2}$ | $F_{m}$ | $g_{1}\cdot m$

Table 2. Mechanical characterization of body of animals and fast muscles in
physiologically equivalent non-steady states $s=1$ prescribed by eq. (2).
_Abbreviation_ : $g_{1}=g_{m1}$.

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In accord with table 2, the equilibration of the air drag by wings of flapping
birds is manifested by the observed wing frequencies $1/T^{(\exp)}\varpropto
M^{-1/6}$ [23]. Moreover, the mechanical similarity between animals resisting
air, ground, and water friction forces was demonstrated via the energy cost
minimization [9], where the spatiotemporal correlations $V\varpropto L^{1/2}$
($\varpropto M^{1/6}$) were critically explored on ad hoc basis.

When the non-steady locomotion conditions associated with the physiologically
equivalent (or transient-equilibrium [19]) states $s=1$ are applied to
individual fast-twitch-fiber muscles controlling fast gaits [21], the muscle
field is apparently uniform and likely universal [6]. Indeed, the body force
field $F_{body}^{(\max)}/M\thickapprox 3g$ was first observed via the maximum
force output in fast trotting and hopping quadrupeds [8]. Later, mass-specific
force output $g_{m1}^{(\exp)}$ was empirically established [6] for locomotory
individual muscles associated with _fast motors_ in running, flying, and
swimming animals. One therefore infers that the gravitation field $g$ is not
crucial in fast running modes, as proposed in [9]. Moreover, the principle of
minimum muscular action suggests that fast muscles may generate force into the
whole muscle bulk [25] maintaining constant body stiffness (table 2), unlike
the constant pressure characteristic of steady gaits (table 1). In other
words, the fast muscles are not simple passive springs [3,26], attributed to
$s=2$ and having length-independent period, but are complex systems being able
to activate fibres in both parallel and series. Maintaining the uniform muscle
force field $g_{m1}$, the _Froude number_($Fr=V/\sqrt{gL}$ [1]) must be mass-
invariable, for both muscle system ($Fr_{fast}\backsim\sqrt{g_{m1}/g}$) and
body system, apart from the corresponding Strouhal number. For fast running
gaits in mammals, $Fr_{run}^{(\exp)}\thickapprox 1.5$ and
$St_{run}^{(\exp)}\thickapprox 0.4$ [8].

##  IV Conclusion

The main goal of this letter is to demonstrate how the complex biological
phenomenon of mechanical similarity in animal locomotion allows to be
rationalized and formulated as a predictive, quantitative framework. It has
been shown how the fundamental physical principle of minimum action applied to
locomotory muscles via intrinsic elastic moduli quantifies amazing
similarities established empirically between maximal speeds, frequencies,
forces, and other relevant mechanical characteristics of animals locomoting in
a certain gait. Naturally operating the softness of legs, wings, and tails,
the efficient runners, flyers, and swimmers are shown to maintain constant
Strouhal number via the universal constant muscle pressure, when traveling or
cruising at steady speeds. When acting quickly at higher speeds, escaping from
predators, or when hunting, the successful runners, flyers, and swimmers
appear to maintain the universal field in the whole bulk of fast muscles, at
least at crossover speeds. This uniform field eventually results in the
bodyweight depending, fixed muscle stiffness and universal Froude and Strouhal
numbers. The provided from first principles study illuminates and supplements
a wide spectrum of reliable empirical findings in walking and running bipeds
[3,11], trotting and galloping quadrupeds [6-9,12]; hovering and flapping
birds [2-4,10,11], bats, and insects [3,4,9]; undulating and tail-beating fish
[2-4,9,10], dolphins [2,4], sharks [4], and whales [2].

On the other hand, the study of muscle characteristics, including obtained
scaling relations to muscle and body mass, is limited by the linear-
displacement muscle approximation. It can been shown however that the top
speeds attributed to limiting animal performance [19,24] cannot be achieved by
the linear-strain elastic muscle fields. The consequences of application of
the minimum action to specific fast locomotory muscles structurally adapted to
a certain mechanical activity, such as motor, brake, or strut functions [3]
prescribed by non-linear elastic effects [25] will be discussed elsewhere.

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Acknowledgments. Financial support by the national agency CNPq is
acknowledged.

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