Generalized potential for apparent forces: the Coriolis effect
Elmo Benedetto, Ivana Bochicchio, Christian Corda, Fabiano Feleppa, Ettore Laserra
aa r X i v : . [ phy s i c s . g e n - ph ] M a r Generalized potential for apparentforces: the Coriolis effect
Elmo Benedetto a , Ivana Bochicchio b , Christian Corda c , FabianoFeleppa d , Ettore Laserra b a Department of Computer Science, University of Salerno, Via Giovanni PaoloII, 132, 84084 Fisciano (Sa), Italy b Department of Civil Engineering, University of Salerno, Via Giovanni PaoloII, 132, 84084 Fisciano (Sa), Italy c International Institute for Applicable Mathematics and Information Sciences,B. M. Birla Science Centre, Adarshnagar, Hyderabad 500063, India, andDepartment of Physics, Faculty of Science, Istanbul University, Istanbul,34134, Turkey d Department of Physics, University of Trieste, via Valerio 2, 34127 Trieste,Italy
Abstract
It is well known, from Newtonian physics, that apparent forces appearwhen the motion of masses is described by using a non-inertial frame ofreference. The generalized potential of such forces is rigorously analyzedfocusing on their mathematical aspects.
In the framework of classical mechanics, the configuration P = ( P , .., P N ) ofa system with n degree of freedom can described through the parametrization P ( t, q ), where t is the time and q = ( q , ..., q n ) is the Lagrangian parameter.The well known Eulero–Lagrange equation of motion is written as ddt (cid:18) ∂T∂ ˙ q i (cid:19) − ∂T∂q i = Q i , (1)where T = T ( t, q, ˙ q ) is the kinetic energy, ˙ q i = dq i dt are the generalized velocitiesand Q i the generalized components of the forces acting on the system definedas Q i = N X j =1 F j · ∂P j ∂q i , i = 1 , ..., n. (2)1ow, for conservative system, it is possible to introduce the potential U = U ( q )such that Q i = N X j =1 F j · ∂P j ∂q i = ∂U∂q i ( q ) , i = 1 , ..., n, (3)where F j ( t, P, ˙ P ) are the total conservative forces applied to each point P i .Hence, one can introduce the Lagrangian L = T + U and consider the equationof motion in the equivalent form ddt (cid:18) ∂L∂ ˙ q i (cid:19) − ∂L∂q i = 0 i = 1 , ..., n. (4)Suppose now we have a generalized force that can be written in terms of avelocity-dependent potential U ( q, ˙ q, t ) as Q i = ∂U∂q i − ddt (cid:18) ∂U∂ ˙ q i (cid:19) . (5)If this is the case, substituting (5) into (1), we can conclude that the Euler-Lagrange equation still holds in the form (4) for a Lagrangian function L thatcan defined once more as L = T + U . The potential U may be called a “gener-alized potential” or “velocity-dependent potential”. It is not a potential in theconventional sense because it depends on more than just the particle positionand it cannot be calculated from a line integral of the generalized force. Despitethis fact, the interesting aspect of such a formalism lies in the fact that it per-mits once more the use of a Lagrangian and the Euler-Lagrange equation andhence to re-obtain, in a generalized context, all the consequent properties. Inaddition, from a velocity-dependent potential can be derived interesting forces,such as the Lorentz force of a magnetic field B acting on a moving chargedparticle F = e ( E + v × B ) , (6)where e is the particle charge, v the particle velocity, E the electric field. Inparticular, we will show how forces deriving from a generalized potential canbe written in the form of (6), giving the suitable definition of the two vectorfields E and B . Following this line, the paper is so organized: In Section 2 werecall the expression of apparent forces in non-inertial frames. The generalizedpotential is analyzed in Section 3 and evaluated in Section 4 in case of apparentforces.Finally, we (wish to) point out that the level of the paper is educational one, andhence, we restrict ourselves to examples from classical mechanics. Let’s remarkthat this methods could be extended to the effects of relativistic mechanics Consider a free material point (
P, m ), not subjected to effective forces, in ab-solute motion with respect to an inertial frame of reference T Ω ≡ Ω ξηζ and2n relative motion with respect to a non-inertial reference frame T O ≡ Oxyz ,moving in any rigid translational motion with respect to T Ω . If so, the principleof relatives motions is valid and we have v ( a ) = v + v τ , (7)where v ( a ) , v and v τ are respectively absolute, relative and translational veloc-ity. In particular, if ω τ is the angular velocity vector, the translational velocityis given by the well-known fundamental formula of rigid kinematics v τ = v O + ω τ × ( P − O ) , (8)where v O is the velocity of the origin of the non-inertial reference frame T O . Inits relative motion, the material point is subjected to the apparent forces F = − m a τ − m a c , (9)where a τ = a O + ˙ ω τ × ( P − O ) − ω τ × ( P − O ) × ω τ (10)is the translational acceleration, and a c = 2 ω τ × v (11)is the Coriolis acceleration. Therefore F τ = − m a τ = − m [ a O + ˙ ω τ × ( P − O ) − ω τ × ( P − O ) × ω τ ] (12)is the translational force (sometimes known as ‘dragging force’) and F c = − m a c = − m ω τ × v (13)is the Coriolis force. The sum of F τ and F c provides the most general inertialforce acting in the non-inertial reference frame. Let us derive a generalizedpotential for such a force.The absolute motion of the point with respect to theinertial frame rests on Lagrange’s three scalar equations, that, in this case,since we use the three Cartesian coordinates as Lagrangian coordinates, may besummarized in the following vector equation: ddt ∂ L ( a ) ∂ v ( a ) − ∂ L ( a ) ∂P = 0 , (14)where, by definition, we have ∂∂ v ( a ) = i ∂∂ ˙ ξ + j ∂∂ ˙ η + k ∂∂ ˙ ζ , ∂∂P = i ∂∂ξ + j ∂∂η + k ∂∂ζ . (15)Being the material point, in its absolute motion, not subjected to effective forces,the absolute Lagrangian L ( a ) coincides with the absolute kinetic energy, so L ( a ) = m v a ) = T ( a ) . (16)3o pass to the Lagrangian L in relative motion, we only have to substitute tothe absolute velocity the expression given by (7), so we obtain L = m v + v τ ) = m v + m (cid:20) v · v τ + 12 v τ (cid:21) , (17)where T = m v is the apparent kinetic energy. Afterwards we will prove thatthe terms in square brackets as the second side of (17) represent the generalizedpotential of apparent forces, not considering the sign and the mass m . Remark 2.1
Following (2) we can obtain the generalized components Q D and Q Cor of the dragging and Coriolis forces, respectively Q Dh = − m [ a O + ˙ ω τ × ( P − O ) + ω τ × ( P − O ) × ω τ ] · ∂P∂q h (18) Q Corh = − m ω τ × v · ∂P∂q h (19) In the physical space, the most general force resulting from a generalized po-tential can be expressed by the following classical theorem [3]:
Theorem 3.1
If a force is of the type F = E + v × B , (20) where v is the velocity of the material point on which the force is exerted, thanthe two vectorial fields (eventually depending on time) E and B must satisfy ∇ · B = 0 , ∇ × E + ∂ B ∂t = 0 . (21) Moreover, a corresponding generalized potential is U ( P, v , t ) = φ ( P, t ) + A ( P, t ) · v , (22) where φ ( P, t ) is a so-called scalar potential and A ( P, t ) is a so-called vectorpotential. In addition, the connection between force and generalized potential isprovided by the two equations B = ∇ × A , E = ∇ φ − ∂ A ∂t . (23)4his is the case, for example, of the Lorentz electromagnetic force acting on afree point particle of charge e , position P and velocity v in the presence of anelectric field E ( t, P ) and a magnetic induction B ( t, P ): F = e E + e v × B . (24)This is a force of type (20) and the corresponding equations (21) represent thefirst couple of the fundamental Maxwell equations. These equations are deducedas necessary conditions for the existence of a potential, without other physicalconsiderations. This section is devoted to explicit the connection given in Equation (23) betweenthe force of type (20) and generalized potential. Precisely, starting from a forceof type (20) and following Theorem 3.1, we known that it admits a generalizedpotential of type expressed in (22), i.e. U ( P, v , t ) = φ ( P, t ) + A ( P, t ) · v . (25)Now, imposing A ( P, t ) = m v τ = m [ v O + ω τ × ( P − O )] (26)as the vector potential and φ ( P, t ) = A m v τ E and B . Remark 4.1
It could be directly done by making explicit the terms appearingin the Lagrangian equations, but we prefer to use a different strategy based ontheorem appearing in [1, p. 64]. Precisely, we observe that the two arguments P and t of the vector field A ( P, t ) must be considered as independent from eachother. Furthermore, once the motion of the non-inertial reference frame T O with respect to the fixed (inertial) one T Ω , namely the translational motion, isassigned, the origin O , the velocity v O and the angular velocity ω τ , become threeknown functions of time, O = O ( t ) , v O = v O ( t ) , ω τ = ω τ ( t ) (see [3, p. 70]). Let’s consider (23) , that gives back (see for example [4, Eq. (2.3.36) p. 293]) B = ∇ × A = m ∇ × v τ = 2 m ω τ , (28)so, by using also (11), we can write F = E + v × B = E − m ω τ × v = E − m a c . (29)5o calculate the field E we use (23) , that gives E = ∇ φ ( P, t ) − ∂ A ∂t = 12 m ∇ A − ∂ A ∂t . (30)We start to calculate the first term of the right hand side,12 m ∇ A = m ∇ v τ = m ∇ ( v O + 2 v O · ω τ × ( P − O ) + [ ω τ × ( P − O )] ) . (31)Observing that ∇ v O = , we can write ∇ [2 v O · ω τ × ( P − O )] = 2 ∇ [ v O × ω τ · ( P − O )] , (32)and so 12 ∇ A = 2 m ∇ [ v O × ω τ · ( P − O )] + m ∇ [ ω τ × ( P − O )] . (33)Recalling that ( P − O ) is a potential field hence ∇ × ( P − O ) = and using awell-known formula from the vector analysis (see for example [2, p. 230]) ∇ ( A · B ) = ( B · ∇ ) A + ( A · ∇ ) B + B × ∇ × A + A × ∇ × B , (34)we obtain ∇ [ v O × ω τ · ( P − O )] = v O × ω τ . (35)By a substitution in (33) we can write12 ∇ A = m ∇ [ ω τ × ( P − O )] + m v O × ω τ . (36)If P ∗ is the projection of P on the instantaneous rotation axis related to theorigin O , that is the axis passing through O and parallel to ω τ , we get[ ω τ × ( P − O )] = ω τ ( P − P ∗ ) . (37)If, at each fixed instant t , we introduce a system of cylindrical coordinates r (= | P − P ∗ | ) , θ, z , having as its axis z the instantaneous axis of rotation relatedto O , and define { ˆ e = vers ( P − P ∗ ) , ˆ e , ˆ e } the associated basis (orthonormalin this case), the following expression results: ω τ r = ω τ ( P − P ∗ ) , (38)and, by recalling the gradient in cylindrical coordinates ∇ ( r,θ,z ) = ∂∂r ˆ e r + 1 r ∂∂θ ˆ e θ + ∂∂z ˆ e z , (39)we have ∇ [ ω τ × ( P − O )] = ω τ ∂r ∂r ˆ e r = 2 ω τ r ˆ e r = 2 ω τ ( P − P ∗ ) , (40)6hat can be rewritten as12 ∇ [ ω τ × ( P − O )] = ω τ × ( P − O ) × ω τ . (41)Thus 12 ∇ A = m ω τ × ( P − O ) × ω τ + m v O × ω τ . (42)Finally, following Remark 4.1, ∂ A ∂t = m ∂∂t [ v O + ω τ × ( P − O )] = m [ a O + ˙ ω τ × ( P − O ) − ω τ × v O ] . (43)By substituting (42) and (43) into (30) and using (10), we obtain E = − m [ a O + ˙ ω τ × ( P − O ) − ω τ × ( P − O ) × ω τ ] = m a τ . (44)Finally by (29) and (44), we conclude that F = E + v × B = m a τ − m a c , (45)that is exactly the translation forces (9). Remark 4.2
Collecting (8) , (25) , (26) and (27) , we can write the explicit ex-pression of the generalized potential. Moreover, since m v O is certainly inde-pendent of P and ˙ P , it can be neglected and hence U ( P, v , t ) can be regarded asa sum of three contributions: U ( v , t ) = m v O · v ; U ( P, v , t ) = m ω τ × ( P − O ) · v U ( P, t ) = m v O · ω τ × ( P − O ) + m [ ω τ × ( P − O )] (46) Recalling (5) , it is possible to consider: ∂U ∂q j − ddt (cid:18) ∂U ∂ ˙ q j (cid:19) = − m a · ∂P∂q j (47) ∂U ∂q j − ddt (cid:18) ∂U ∂ ˙ q j (cid:19) = − m ω τ × v · ∂P∂q j − m ˙ ω τ × ( P − O ) · ∂P∂q j − m v O × ω τ · ∂P∂q j (48) ∂U ∂q j − ddt (cid:18) ∂U ∂ ˙ q j (cid:19) = + m v O × ω τ · ∂P∂q j − m ω τ × [ ω τ × ( P − O )] · ∂P∂q j (49) from which we obviously obtain (18) and (19) . From all these results we re-mark, as in [6], that, a part of the case of uniformly rotating frames, we can’tseparate the contributions to the only time-dependent Coriolis force and to theonly dragging force, respectively. In other words, the generalized potential is afeature of the whole system of the general inertial forces and not separately tothe the generalized components of each force. Conclusion remarks
In this paper we have reviewed the analogies between the electromagnetic forceand the inertial ones. The generalized potential of Lorentz force is often studiedin standard textbooks while the analogous potential of the Coriolis field is gen-erally overlooked. We have highlighted this aspect describing the mathematicalformalism of the inertial fields emphasizing the role of the Coriolis and Draggingforces.
The Authors thank the referees for useful comments.