Newton's laws of motion in form of Riccati equation

4 Pages·0.11 MB·English

by  M. Nowakowski

Newton’s Laws of Motion in Form of Riccati Equation
Marek Nowakowskiand Haret C. Rosu
Instituto de F´ısica de la Universidad de Guanajuato, Apdo Postal E-143, Le´on, Guanajuato, M´exico
Wediscusstwoapplications ofRiccati equationtoNewton’slaws of motion. Thefirst oneis themotion of aparticle under
2 the influence of a power law central potential V(r) = krǫ. For zero total energy we show that the equation of motion can
0 be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where
0 the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics,
2 where again theRiccati equation appears naturally,are problemsinvolvingquadraticfriction. Weusemethodsreminiscent to

n nonrelativistic supersymmetry to generalize and solve such problems.
a
J PACS number(s): 45.20.-d, 11.30 Pb

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1
I. INTRODUCTION II. THE POWER LAW CENTRAL POTENTIALS

]
h
p It is known that Riccati equations, in general of the After implementation of the angular momentum con-
- type servation law, the equation for the energy conservation
s
s in the case of a central potential V(r) is given by the
a dy =f(x)y2+g(x)y+h(x) , (1) standard expression
l dx
c
. 1 l2
s findsurprisinglymanyapplicationsinphysicsandmath- E = mr˙2+ +V(r) . (4)
c 2 2mr2
ematics. For example, supersymmetric quantum me-
i
s chanics [1], variational calculus [2], nonlinear physics TakingaderivativewithrespecttotimeofEq.(4)results
y
[3],renormalizationgroupequationsforrunningcoupling into a second fundamental equation of the form
h
p constants in quantum field theories [4] and thermody-
l2 dV(r)
[ namics [5] are just a few topics where Riccati equations mr¨− + =0 . (5)
play a key role. The main reason for their ubiquity is mr3 dr
2
v that the change of function Specializing from now on to a power law potential [6]
6
6 1 d g V(r)=krǫ , (6)
y =− (logz)− , (2)
0 f dx 2
h i
0 where k is the coupling constantand the exponent ǫ can
1 turns it into linear second-order differential equations of be either positive or negative, we obtain from (5)
1 the form
0 mr¨r l2
/ d2z d dz g2 1dg d V(r)=− + . (7)
s − logf − − +h− logf z =0 , ǫ ǫmr2
c dx2 (cid:18)dx (cid:19)dx h 4 2dx dx i
i Inserting the last equation in (4) gives
s
(3)
y
h 1 1 1 l mr¨r
mr˙2+ + − −E =0 . (8)
p that stand as basic mathematical background for many 2 (cid:18)2 ǫ(cid:19)mr2 ǫ
: areas of physics.
v
i SincetheRiccatiequationisawidelystudiednonlinear Under the assumption of E = 0, this expression leads
X equation, knowing that the physical system under con- to a Riccati form. Obviously with E = 0, we restrict
r siderationcanbe broughtintoRiccatiformhascertainly ourselves to the case k < 0. To explicitly derive from
a
many advantages. (8) the Riccati equation we pass (as it is costummary in
It is thereforeofinterestto lookfor yetdifferentphys- centralpotentialproblems)toanangleθasafreevariable
icalproblems which aregovernedby this first ordernon- (i.e., we consider r(θ(t)). With
linear equation. This can be a starting point to new av-
enuesininvestigatinganalyticalsolutionsofyetunsolved θ˙= l , r′ ≡ dr (9)
mr2 dθ
problems. In this paper we concentrate mainly on top-
ics from classicalmechanics and show that certain types and introducing
of Newton’s laws of motion are equivalent to the Riccati
′
equation. r
ω = , (10)
r
1
it can be readily shown, after some algebraic manipula- d2y d2q(x) 1 y
q(x) +y(x) = f , (15)
tions, that (8) reduces to dx2 dx2 q2(x) (cid:18)q(cid:19)
ω′ = ǫ+2ω2+ ǫ+2 . (11) which can be solved by the integrals
2 2
d x
This is the Riccati equation for the motion of a particle dx +a= (cid:16)q(cid:17) , (16)
in a central power law potential assuming E = 0. It is Z q2(x) Z
φ x +b
worthnotingthatnoinformationaboutthecouplingcon- r q
(cid:16) (cid:17)
stantkenterstheRiccatiequation(11). Essentiallywhat
where a and b are integration constants and
we have shown is that any solution of (4) will also sat-
isfy (11). The inverse is not necessarily true and should
be examined in detail. Indeed, the coupling constant k φ(z)≡2 f(z)dz . (17)
Z
shouldbeexplicitlycontainedinthesolutionforr(θ)(see
below). Takingp=const=mandsuitablyrescalingthedistance
A special case which deserves to be briefly mentioned r with the mass m, equation (15) is essentially identical
is ǫ = −2. With this exponent, the choice E = 0 is, in to (5). Indeed, in this case the integrals in (17) give
general, only possible if l2 +k <0. Then directly from
(4) we conclude that r′ i2sma constantwhich, ofcourse, is t−t0 = 1 r dr1 , (18)
compatiblewiththeRriccatiequation(11). However,this mZr0 q2mV(r1)− rl21 +b
constant cannot be determined by means of (11). This
feature is also inherent in the general case. which, with a proper identification of b, is the same as
To discuss the case ǫ 6= 2, we first solve the Riccati directly integrating (5). The interplay between the Er-
equation (11). The solution can be easily found to be makov equation and the central potential problems can
be a useful tool of studying both problems. We con-
ǫ+2 β sin[(ǫ+2)θ+β] jecture that certain invariants of the Ermakov equation
ω(θ)=tan θ+ = ,
(cid:18) 2 2(cid:19) cos[(ǫ+2)θ+β]+1 could be also applied to the central potential problems.
(12)
III. COSMOLOGICAL ANALOGY
where β plays a role of the integration constant. Going
backto the definition ofω in(10)we arriveata solution
We want to point out here a beautiful but formal cos-
for r(θ)
mological analogy to the results of the previous section.
R We recall that in deriving the Riccati equation (9) we
r(θ)= , (13)
[1+cos((ǫ+2)θ+β)]1/(ǫ+2) relied on a power law potential (6), a new parameter θ
(the angle givenin (9)), and the assumptionE =0. The
whereRisaconstant. Asinthecaseǫ=−2thisconstant analogy to cosmology is based on these observations. In
can only be determined by inserting (11) into (4). The Friedmann-Robertson-Walker spacetime the set of Ein-
result is stein’s equationswith the cosmologicalconstantΛ setto
zero reduce to differential equations for the scale factor
1
l2 ǫ+2 a(t),whichisafunctionofthecomovingtimet. Together
R= . (14)
(cid:18)m|k|(cid:19) with the conservation of energy-momentum tensor they
are given by
The last two equations represent then the analytical so-
lution of the posed problem. We obtained this solution 3a¨(t)=−4πG(ρ+3p(ρ))a(t) (19)
bytransformingtheoriginalproblemintoaRiccatiequa-
a(t)a¨(t)+2a˙2(t)+2κ=4πG(ρ−p(ρ))a2(t) (20)
tion. Itmightbe thatthe lawsofmotioninRiccatiform
d
areonlyacuriosity. Given,however,the factthatonlya p˙a3(t)= a2(ρ+p(ρ) . (21)
dt
few analytical solutions of the central potential problem (cid:0) (cid:1)
are known, it is certainly a useful curiosity. Further- In the above G is the Newtonian coupling constant, p is
more, it is not excluded that this novel way opens new the pressure, ρ is the density and κ can take the values
moregeneralmethodstosolveproblemsinmechanics. In 0,±1. Choosing the equation of state to be barotropic
this context, we would like to mention here a yet differ-
ent connection of the central potential problem with the p(ρ)=(γ−1)ρ , (22)
Ermakov nonlinear differential equation [7]. We refer to
the following form of the latter equation [8] fixes essentially ρ to obey a power law behaviour of the
form
2
−3γ
a can be then deduced that the following equivalent form
ρ=ρ , (23)
0(cid:18)a (cid:19) of γ can be obtained
0
and the remaining equation for a(t) reduce to a single γ =g′ −(α−λ )v2 . (30)
1 p
equation, viz
This resembles supersymmetric quantum mechanics and
2
a¨(t) a˙(t) κ 3 we might be tempted to compare v to Witten’s super-
+c +c =0 , c≡ c−1 . (24) p
a(t) (cid:18)a(t)(cid:19) a2(t) 2 potential.
To the new force we again add a quadratic function
Introducing the conformal time η by
with a friction coefficient λ such that the new equation
1
of motion becomes
dη 1
= , (25)
dt a(η) v˙ =γ−λ v2 . (31)
1
itcanbeseenthat(24)isequivalenttoaRiccatiequation
in the function u = a′, where the dot means derivation Thishastheadvantagethatperconstructionvp isapar-
a ticular solution of (31). Equipped with this fact, one
with respect to η
can proceed to construct the general solution which is
u′ +cu2+κc=0 . (26) a standard procedure in the general theory of the Ric-
cati equation. Before doing so, it is instructive to dwell
This cosmological Riccati equation has been previously upon the physical meaning of the new forceγ. Imposing
obtainedbyFaraoni[9]andalsodiscussedbyRosu[10]in g′−(α−λ1)αr22 >0,itcanbeseenthatγ >0. Moreover,
thecontextoflatecosmologicalacceleration. Theformal as obvious from (28) and (30), γ goes to a constant for
analogy to the mechanical case is obvious: the condition largetandhasakink-likebehaviour. Wecanthenenvis-
E = 0 corresponds to Λ = 0, the angle θ is replaced ageasituationwhereγisa‘switch-on’functionforaforce
by the conformal time η, and whereas in the mechanical becoming constant at some time. As mentioned above,
example we had a power law behaviour of the potential, by construction the problem (31) is solvable because vp
the barotropic equation of state forces upon ρ to satisfy isaparticularsolutionof(31). Byinvokingthestandard
ρ ∝ a−3γ. As (11) does not contain the coupling con- Bernoulli ansatz for the general solution vg, namely
stant k, the cosmological Riccati equation (26) loses the
1
information about G. v =v + , (32)
g p
V
we arrive at the differential equation (special case of the
IV. QUADRATIC FRICTION
Bernoulli equation) for V
Starting with a constant force g (free fall, constant V˙ =2λ v V +λ . (33)
1 p 1
electric field, etc) and adding a quadratic friction with a
positivefrictioncoefficientν >0,wehave,perexcellence, Writing v as
p
a Riccati equation for the Newton’s law of motion
1 W˙
v˙ =g′ −αv2 , (27) vp =− p , (34)
λ W
1 p
′
withg ≡g/mandα≡ν/m. Thegeneralsolution(which where
for reasons to be seen later in the text we denote by v )
p
involves a free parameter λ and reads [11] W =e−λ1 vpdt , (35)
p
R
′ r ert−λe−rt ′ one is led to the solution for V
v (t;g ,α,λ)= , r≡αg . (28)
p α(cid:18)ert+λe−rt(cid:19)
λ W2dt+C
1 p
In the following we borrow some techniques from super- V = R W2 , (36)
p
symmetric quantum mechanics. However, we do not fol-
low strictly the supersymmetric scheme as the purposes The general solution is then given by
in the quantum case and the mechanical case are quite
different. We define a new time-dependent force by W2
p
v =v + . (37)
γ(t;g′,α,λ,λ )≡v˙ (t;g′,α,λ)+λ v2(t;g′,α,λ) . (29) g p λ1 Wp2dt+C
1 p 1 p R
The initial value problem, v(0) = v , is solved by fixing
withanewparameterλ >0. Weemphasizethat(29)is 0
1 C through
adefinitiongiventhroughthesolution(28). From(27)it
3
r 1−λ 1
[5] H.C. Rosu and F. Aceves de la Cruz, e-print quant-
v − = . (38)
0
α(cid:18)1+λ(cid:19) C ph/0107043, accepted at Physica Scripta(2002).
[6] Power-lawpotentialsatzeroenergyhavebeenalsostud-
Up to integrals, the equation of motion (31) is solved.
ied by J. Daboul and M.M. Nieto, Int. J. Mod. Phys. A
Setting λ=e−2δ, wecanrewrite (28)inthe moreconve- 11(1996)3801;Phys.Rev.E52(1995)4430;Phys.Lett.
nient form A 190 (1994) 9.
r [7] V. Ermakov, Univ.Izv.Kiev, Series III 9, 1 (1880).
vp = tanh(rt+δ) . (39) [8] P. Espinoza, e-print math-ph/0002005, p. 6.
α
[9] V. Faraoni, Am.J. Phys.67, 732 (1999).
Then Wp can be computed explicitly [10] H.C. Rosu, Mod. Phys. Lett.A 15, 979 (2000).
[11] H. Davis, Introduction to nonlinear differential and in-
1
W = . (40) tegral equations, (Dover Publications, Inc., New York,
p [cosh(rt+δ)]λ1/α 1962), pp.60-62.
It suffices to assume λ = nα, n ∈ N leading to inte-
1
grals of the type cosh−n(x)dx, which can be solved in
aclosedanalyticalRformbyrecursionformulae. Ofcourse,
the procedure outlined here can be generalized by start-
ing with morecomplicatedforcesinsteadof the constant
one.
IV. CONCLUSION
Inthispaperwehavepointedouttheusefulnessofthe
Riccati equation in studying certain mechanical prob-
lems. We derived a Riccati equation for a central po-
tential problem of the power law type assuming E = 0.
This led us to an analytical solution of the problem. In
a second step, we generalized the system of a constant
force plus a quadratic friction to a time-dependent force
and friction. We argued that this time-dependent force
servesasa‘switch-on’function. Theproblemturnedout
to be solvable by means of a construction similar to su-
persymmetric quantum mechanics. As indicated in the
text, both applications can be generalized.
ACKNOWLEDGMENT
The first author thanks CONACyT for financial sup-
port through a Catedra Patrimonial fellowship.
[1] For review, see F. Cooper, A. Khare and U. Sukhatme,
Phys.Rep.251, 267 (1995).
[2] M.I. Zelekin, Homogeneous Spaces and Riccati Equation
in Variational Calculus, (Factorial, Moskow, 1998, in
Russian).
[3] V.B.MatveevandM.A.Salle, Darboux Transformations
and Solitons (Springer, Berlin, 1991).
[4] I.L. Buchbinder, S.D. Odintsov, and I.L. Shapiro, Effec-
tive Action in Quantum Gravity, (IOP Publishing Ltd,
1992), p. 282; K. Milton, S.D. Odintsov, and S. Zerbini,
e-printhep-th/0110051.
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