A clear physical description of a particle\u27s motion in terms of the components of its acceleration is obtained if the trajectory is described in terms of the Frenet-Serret frame. Within this frame, centripetal acceleration emerges as a natural consequence of a particle\u27s motion.Ako se putanja čestice opisuje u Frenet–Serretovom sustavu, postiže se jasan fizički opis njenog gibanja preko komponenata njenog ubrzanja. U tom se sustavu centripetalno ubrzanje javlja kao prirodna posljedica gibanja čestice

Bevelacqua, Joseph John

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ISSN1330–0008CODENFIZAE4EMERGENCE OF CENTRIPETAL ACCELERATION WITHIN THEFRENET–SERRET FRAMEJOSEPH JOHN BEVELACQUABevelacqua Resources, Suite 100, 343 Adair Drive, Richland, WA 99352 U.S.A.E-mail address: bevelresou@aol.comReceived 6 October 2003; Accepted 15 December 2003Online 19 April 2004A clear physical description of a particle’s motion in terms of the components of itsacceleration is obtained if the trajectory is described in terms of the Frenet–Serretframe. Within this frame, centripetal acceleration emerges as a natural consequenceof a particle’s motion.PACS numbers: 01.30.Rr, 01.40.-d, 02.40.Hw UDC 531.122Keywords: particle’s motion, Frenet–Serret frame, centripetal acceleration1. IntroductionClassical mechanics textbooks universally address particle motion and associ-ated quantities such as acceleration. The concept of acceleration is developed as therate of change of velocity with special cases (e.g., centripetal acceleration) developedto describe particular types of motion (e.g., circular motion). In most textbooks,the concept of centripetal acceleration appears as a separate concept that is nottied to the general acceleration concept, and is often presented as an artifact ofcircular motion rather than a consequence of the generalized motion of a particle.One possible reason is that the mathematical treatment in classical mechanics textsrelies on relatively simple arguments involving Cartesian or polar coordinates thatare not naturally adapted to the curve describing the general motion of a particle.The author does not intend to slight these excellent pedagogical efforts, but feelsan improved treatment is possible using a junior physics major’s knowledge of basiccalculus.An improved physical description of the acceleration of a particle is obtainedif it is viewed in terms of the Frenet–Serret frame [1–6]. This frame is defined inFIZIKA A 12 (2003) 3, 143–150 143bevelacqua: emergence of centripetal acceleration within . . .terms of tangent and normal vectors that are physically tied to the particle’s motionand the underlying space-time geometry along the path of motion of the particle.Herein, we will demonstrate that centripetal acceleration is a natural consequenceof the motion when viewed within the Frenet–Serret frame.2. Frenet–Serret frameThe movement of a particle may be described in terms of its motion along somecurve β(s). For any given point P on this curve, a set of orthonormal unit vectorsmay be defined:T = unit tangent vector at each point PN = unit normal vector at each point PB = unit binormal vector at each point PThis set of orthogonal unit vectors, known as the Frenet–Serret frame, has thefollowing propertiesT · T = N · N = B · B = 1, (1)T · N = T · B = N · B = 0, (2)B = T × N , (3)T = N × B, (4)N = B × T . (5)Given this set of coordinates, let β(s) be a curve parameterized by the arc length(s) and let T (s) be the vectorT (s) = β ′(s). (6)where the prime indicates differentiation with respect to s. While there might beother canonical parameterizations, only a parameterization by the arc length leadsto a normalized vector T (s).The vector T (s) is tangential to the curve and it has unit length. We will referto T as the unit tangent vector. Eq. (1)(T · T = 1)may be differentiated withrespect to s to yield a relationship between T and T ′,2T (s) · T ′(s) = 0, (7)that suggests T and T ′ are orthogonal. If we let N be a unit vector orthogonal toT , we can writeT ′(s) = κN(s), (8)144 FIZIKA A 12 (2003) 3, 143–150bevelacqua: emergence of centripetal acceleration within . . .where κ is a scalar called the curvature and N is called the unit normal to thecurve. If the length of both sides of Eq. (8) is taken, then we obtainκ =∣∣T ′(s)∣∣. (9)In Eq. (9), the curvature measures the rate of change of the tangent vector, whichis a measure of how much the curve is curving.The third vector B is the binormal vector. The properties of B may be gleanedby differentiating Eq. (1)(B · B = 1)with respect to s2B(s) · B ′(s) = 0. (10)Therefore B ′ is orthogonal to B. We can also differentiate Eq. (2)(T · B = 0)B ′(s) · T (s) + B(s) · T ′(s) = 0. (11)Using Eq. (8), this equation may be rewritten asB ′(s) · T (s) = −B(s) · T ′(s) = −κN(s) · B(s). (12)Since T and T ′ are orthogonal, and since B is perpendicular to T according toEq. (3), Eq. (12) suggests that B ′ is also orthogonal to T . This can only occur ifB ′ is orthogonal to the T -B plane. Therefore, B ′ is proportional to NB ′(s) = −τ N(s). (13)for a quantity τ which we will call the torsion. The scalar τ is chosen such thatthe binormal vector is a unit vector. Torsion is similar to the curvature in that itmeasures the rate of change of a quantity. Torsion is an indication of the rate ofchange of the binormal.The advantage of the Frenet–Serret frame is that it propagates with the particle.The tangent vector points in the direction of motion. The normal and binormalvectors point towards the directions in which the trajectory is tending to curve.The properties of the vector N may be established by differentiating Eq. (1)(N · N = 1)2N(s) · N ′(s) = 0. (14)Therefore, N ′ and N are orthogonal. This suggests N ′ lies in the T -B plane, andtherefore can be expressed as a linear combination of T and BN ′ = aT + bB, (15)where a and b are scalars. Taking the dot product of Eq. (15) with T and B,respectively yields the following results for a and ba = N ′ · T , (16)b = N ′ · B. (17)FIZIKA A 12 (2003) 3, 143–150 145bevelacqua: emergence of centripetal acceleration within . . .When the equations N ·T = 0 and N ·B = 0 are differentiated and combined withEqs. (8) and (13), we obtainN ′ · T = −N · T ′ = −N ·(κN)= −κ, (18)N ′ · B = −N · B ′ = −N ·(− τ N)= τ. (19)Eqs. (16)–(19) suggest that a = −κ and b = τ . With this result, Eq. (15) becomesN ′ = −κT + τ B, (20)which completes the specification of T , N , and B.The Frenet–Serret frame equations may be simplified by writing them in matrixform[T ′N ′B ′]=[0 κ 0−κ 0 τ0 −τ 0][TNB]. (21)3. Motion within the Frenet–Serret frameThe motion associated with a curve in the Frenet–Serret frame may be used tocompute the velocity and acceleration along that curve. We assume the arc lengths is a function of time (t) and consider the motion along the curve β(s).Using Eq. (6), the velocity v may be written asv =ddtβ(s(t))=dβ(s)dsdsdt= β ′(s)v = v T , (22)wherev =dsdt. (23)The acceleration a is obtained in a similar mannera =d2dt2β(s(t))=dvdtT + vdTdsdsdt, (24)a = aT + κv2 N , (25)where Eqs. (8) and (24) have been used to simplify Eq. (25) anda =dvdt. (26)Eq. (25) demonstrates the logical emergence of the centripetal acceleration.The equation states that a particle moving along a curve in space–time feels a146 FIZIKA A 12 (2003) 3, 143–150bevelacqua: emergence of centripetal acceleration within . . .component of acceleration along the direction of motion whenever there is a changein speed. The second component in Eq. (25) is the centripetal acceleration in thedirection of the normal.In order to more fully demonstrate the relationship between the second term inEq. (25) and the centripetal acceleration, we will evaluate the relationship betweenκ and τ and the curve β(s). We let β(s) be a curve with curvature κ and torsionτ . Using Eqs. (6) and (8), we can writeT ′(s) = β ′′(s) = κN(s). (27)The relationship between κ and β ′′(s) may be obtained by taking the dot productof β ′′(s) with itselfβ ′′(s) · β ′′(s) = κN(s) · κN(s) = κ2, (28)orκ =∣∣β ′′(s)∣∣. (29)In a similar manner, τ may be obtained from β ′′′(s) and Eq. (27)β ′′′(s) = κ′ N(s) + κN ′(s). (30)Using Eq. (20), we obtainβ ′′′(s) = κ′ N(s) + κ(− κT (s) + τ B(s)), (31)β ′′′(s) = κ′ N(s) − κ2 T (s) + κτ B(s). (32)The desired expression for τ is obtained by computing the quantity β ′(s) ·[β ′′(s)×β ′′′(s)]. Using Eqs. (3), (4), (6), (27) and (32), we can writeβ ′(s) ·[β ′′(s) × β ′′′(s)] = T ·[κN ×(κ′ N − κ2 T + κτ B)], (33)β ′(s) ·[β ′′(s) × β ′′′(s)] = T ·[κ3 B + κ2τ T], (34)β ′(s) ·[β ′′(s) × β ′′′(s)] = κ2τ. (35)Solving for τ we obtainτ =β ′(s) ·[β ′′(s) × β ′′′(s)]κ2. (36)Using Eq. (28), Eq. (36) becomesτ =β ′(s) ·[β ′′(s) × β ′′′(s)]β ′′(s) · β ′′(s). (37)FIZIKA A 12 (2003) 3, 143–150 147bevelacqua: emergence of centripetal acceleration within . . .The assertions regarding the centripetal acceleration may be verified by consid-ering the specific case of circular motion. For circular motionβ(s) =(r cos θ, r sin θ, 0), (38)where r is the radius of the circle and θ is the polar angle, which is a function ofboth s and rs = r θ. (39)Using Eq. (39), Eq. (38) may be written in terms of sβ(s) =(r cossr, r sinsr, 0). (40)The derivatives β ′ and β ′′ follow from Eq. (40)β ′(s) =(− sinsr, cossr, 0), (41)β ′′(s) =(−1rcossr,−1rsinsr, 0). (42)Determining a value for κ will complete the specification of the acceleration definedby Eq. (25). Using Eq. (29) κ is determinedκ =∣∣β ′′(s)∣∣ =√(−1rcossr)2+(−1rsinsr)2+ 0, (43)κ =√(1r)2(sin2sr+ cos2sr)=1r. (44)Using Eq. (44), Eq. (25) defined the total acceleration a,a = aT +v2rN . (45)The first term is the acceleration associated with the change in velocity in thedirection of the tangent to the curve describing the particle’s motion, and thesecond term is the acceleration normal to the curve or the centripetal accelerationhaving the expected v2/r form for circular motion. Eq. (45) can be derived in amore compact and elegant manner using differential geometry through the use ofChristoffel symbols [6,7]. However, that approach is beyond the knowledge levelof most students in a junior level physics course. Such an approach would be awelcome addition to advanced undergraduate or graduate students as an initialapplication illustrating the usefulness of affine connection coefficients.148 FIZIKA A 12 (2003) 3, 143–150bevelacqua: emergence of centripetal acceleration within . . .4. ConclusionsThe motion of a particle within the Frenet–Serret frame leads to the develop-ment of the components of acceleration that logically lead to the introduction ofcentripetal acceleration. This approach presents the concept of acceleration in alogical and consistent manner, and provides a junior physics major with a unifieddescription of acceleration and also provides the more advanced student with anatural tie to differential geometry.References[1] L. Godeaux, Bull. Soc. Math. Belg. 23 (1971) 179.[2] J. L. Synge, Relativity: The General Theory, North Holland, Amsterdam (1960) p. 86.[3] E. Hönig, E. L. Schücking and C. W. Vishveshwara, J. Math Phys. 15 (1974) 744.[4] B. R. Iyer and C. V. Vishveshwara, Phys. Rev. D 48 (1993) 5706.[5] D. Bini, R. T. Jantzen and A. Merloni, Class. Quantum Grav. 16 (1999) 1333.[6] J. H. Heinbockel, Introduction to Tensor Calculus and Continuum Mechanics, TraffordPublishing, Victoria, BC (2001) p. 108.[7] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, Freeman, San Francisco(1973) p. 210.FIZIKA A 12 (2003) 3, 143–150 149bevelacqua: emergence of centripetal acceleration within . . .NASTANAK CENTRIPETALNOG UBRZANJA U FRENET–SERRETOVOMSUSTAVUAko se putanja čestice opisuje u Frenet–Serretovom sustavu, postiže se jasan fizičkiopis njenog gibanja preko komponenata njenog ubrzanja. U tom se sustavu cen-tripetalno ubrzanje javlja kao prirodna posljedica gibanja čestice.150 FIZIKA A 12 (2003) 3, 143–150

Nastanak centripetalnog ubrzanja u Frenet–Serretovom sustavu

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