Effective potential of a spinning heavy symmetric top when magnitudes of conserved angular momenta are not equal
EEffective potential of a spinning heavysymmetric top when magnitudes of conservedangular momenta are not equal
V. Tanrıverdi [email protected]
Abstract
Effective potential for a spinning heavy symmetric top is studiedwhen magnitudes of conserved angular momenta are not equal to eachother. The dependence of effective potential on conserved angularmomenta is analyzed. This study shows that the minimum of effectivepotential goes to a constant derived from conserved angular momentawhen one of the conserved angular momenta is greater than the otherone, and it goes to infinity when the other one is greater. It alsoshows that the usage of strong or weak top separation does not workadequately in all cases.
Motion of a symmetric top can be studied by using either a cubic functionor effective potential. The cubic function is mostly used in works that utilizegeometric techniques [1, 2, 3, 4, 5, 6], and effective potential is mostly used inworks considering physical parameters [7, 8, 9, 10, 11]. In some other works,both the cubic function and effective potential are used [12, 13, 14, 15, 16,17, 18].Effective potential shows different characteristics when one of the con-served angular momenta greater than the other one or equal to. One canfind different aspects of effective potential in the literature when magnitudesof the conserved angular momenta are equal to each other [7, 19]. However,it is not studied when magnitudes of the conserved angular momenta arenot equal to each other except in Greiner’s work, and his study does not1 a r X i v : . [ phy s i c s . c l a ss - ph ] J a n over different possibilities related to the conserved angular momenta andthe minimum of effective potential [17]. Studying this topic helps under-stand the motion of a spinning heavy symmetric top, and in this study, wewill study this case together with the relation between the minimum of ef-fective potential and a constant derived from parameters of gyroscope andconserved angular momenta.In section 2, we will give a quick overview of constants of motion andeffective potential. In section 3, we will study effective potential when mag-nitudes of the conserved angular momenta are not equal to each other. Then,we will give a conclusion. In the appendix, we will compare the cubic functionwith effective potential. For a spinning heavy symmetric top, Lagrangian is [12] L = T − U = I x θ + ˙ φ sin θ ) + I z ψ + ˙ φ cos θ ) − M gl cos θ, (1)where M is the mass of the symmetric top, l is the distance from the centerof mass to the fixed point, I x = I y and I z are moments of inertia, g is thegravitational acceleration, θ is the angle between the stationary z (cid:48) -axis andthe body z -axis, ˙ ψ is the spin angular velocity, ˙ φ is the precession angularvelocity and ˙ θ is the nutation angular velocity. The domain of θ is [0 , π ]. Fora spinning symmetric top on the ground θ should be smaller than π/
2, andif θ > π/
2, then the spinning top is suspended from the fixed point.There are two conserved angular momenta which can be obtained fromLagrangian, and one can define two constants a and b by using these con-served angular momenta as [12] a = I z I x ( ˙ ψ + ˙ φ cos θ ) , (2) b = ˙ φ sin θ + a cos θ, (3)where a = L z /I x and b = L z (cid:48) /I x . Here, L z and L z (cid:48) are conserved angularmomenta in the body z direction and stationary z (cid:48) direction, respectively.One can define a constant from energy as E (cid:48) = I x θ + I x φ sin θ + M gl cos θ, (4)2nd its relation with the energy is E (cid:48) = E − I x a / (2 I z ).By using change of variable u = cos θ , one can obtain the cubic functionfrom (4) as[12] f ( u ) = ( α − βu )(1 − u ) − ( b − au ) (5)which is equal to ˙ u , where α = 2 E (cid:48) /I x and β = 2 M gl/I x . This cubicfunction can be used to find turning angles.From E (cid:48) = I x ˙ θ / U eff [9], it is possible to define an effective potential U eff ( θ ) = I x b − a cos θ ) sin θ + M gl cos θ. (6)By using the derivative of U eff with respect to θdU eff ( θ ) dθ = I x sin θ (cid:20) ( b − a cos θ )( a − b cos θ ) − M glI x sin θ (cid:21) , (7)it is possible to find the minimum of U eff . The factor sin θ is equal tozero when θ is equal to 0 or π , and effective potential goes to infinity atthese angles. The root of equation (7) is between 0 and π , and it will bedesignated by θ r giving the minimum of effective potential, and it can befound numerically. Then, the form of effective potential is like a well. Thegeneral structure of U eff together with E (cid:48) can be seen in figure 1.By using equation (7), one can write [12]˙ φ cos θ − ˙ φa + M glI x = 0 . (8)The root of this equation can also be used to obtain the minimum of U eff .By using the discriminant of this equation, one can define a parameter ˜ a = (cid:112) M gl/I x to make a disrimination between ”strong top” (or fast top) where a > ˜ a and ”weak top” (or slow top) where a < ˜ a [20, 21].The position of the minimum and the shape of U eff can be helpful inunderstanding the motion. If E (cid:48) is equal to the minimum of U eff then theregular precession is observed. If E (cid:48) is greater than the minimum of U eff ,like figure 1, the intersection points of E (cid:48) and U eff give turning angles. And,symmetric top nutates between these two angles periodically. There can bedifferent types of motion, and some of these motions can be determined byusing relations between E (cid:48) & M glb/a and a & b when | a | (cid:54) = | b | [21].3 eff min E’ θ min θ r θ max U e ff θ Figure 1: General structure of U eff ( θ ) and E (cid:48) . θ min and θ max show turningangles, and θ r represents the angle where minimum of U eff occurs. Curve(red) shows U eff , dashed (blue) line shows E (cid:48) and horizontal continious(black) line shows the minimum of U eff . The relation between a and b can affect effective potential. There are threepossible relation between a and b : | a | > | b | , | a | < | b | and | a | = | b | . We willconsider two different possibilities, | a | > | b | and | a | < | b | , to study effectivepotential since the third one is studied previously, i.e. | a | = | b | [7, 19]. Wewill give examples to studied cases, and for examples, the following constantswill be used: M gl = 0 . J , I x = 0 . kg m and I z = 0 . kg m . | a | > | b | In this section, we will study the case when | a | > | b | . After factoring equation(7), it can be written as dU eff ( θ ) dθ = a I x sin θ (cid:20) ( ba − cos θ )(1 − ba cos θ ) − M glI x a sin θ (cid:21) . (9)The angle, making the terms in the parentheses zero, gives the minimum ofeffective potential. If | a | > | b | , the second term in the parentheses is alwaysnegative, and then b/a − cos θ should also be positive for the root. Therefore,the inclination angle should satisfy π > θ > arccos b/a . In the limit where a goes to infinity, θ r goes to arccos b/a . In a goes to zero limit, b should also go4 a) U eff π /2 π U e ff θ (b) θ r π /2 π
0 50 100 150 θ r a (c) U eff min -0.07 0 0.07 0 50 100 150 U e ff m i n a Figure 2: U eff , change of θ r with respect to a and change of U eff min withrespect to a . a) Three different effective potential: a = 10 rad s − (greendashed-dotted curve), a = 30 rad s − (blue dashed curve) and a = 60 rad s − (red continious curve), and all of them satisfy b/a = 0 .
5. Black line shows
M glb/a . b) Change of θ r with respect to a for constant b/a = 0 . b/a ) = 1 .
05. Vertical dotted line showsposition of ˜ a . c) Change of U eff min with respect to a for constant b/a = 0 . M glb/a . Vertical dotted line showsposition of ˜ a .to zero since | a | > | b | , then the first term goes to zero (see equation (7)) andthe second term should also go to zero for the root which is possible when θ r goes to π . If both a and b are negative or positive, θ r is between π/ π when | a | is close to zero, and it is between 0 and π/ | a | and | b | aregreat enough. If only one of them is negative, then θ r is always greater than π/ b = 0, in | a | goes to infinity limit θ r goes to π/
2, and a goes tozero limit does not change and remains as π .These shows that θ r ∈ (arccos b/a, π ). If b/a goes to 1, then arccos b/a goes to 0. Therefore, θ r can take values between 0 and π depending on signsof a and b , the ratio b/a and greatness of a and b .Now, we will consider the change of U eff min when | a | > | b | . We have seenthat as | a | goes to zero, θ r goes to π . Then, it can be seen from equation(6) that U eff min goes to − M gl as | a | goes to zero. As | a | goes to infinity θ r goes to arccos b/a , then U eff min goes to M glb/a from below. Then,
M glb/a is always grater than U eff min when | a | > | b | .As an example, we will consider that there is a constant ratio between a and b : b/a = 0 .
5. In figure 2(a), three different effective potentials for threedifferent a values are shown together with M glb/a . In this figure, it can be5een that the form and magnitude of the minimum of U eff are changing as a changes, and it can also be seen that θ r is also changing. In figure 2(b),it can be seen that θ r takes very close values to π for very small values of a and goes to arccos 0 . . rad as a increases. In figure 2(c), it can beseen that the minimum of U eff takes very close values to − M gl when a issmall, and it goes to M glb/a as a goes to infinity. These are consistent withprevious considerations.It can be considered that there is a shift in the behaviour of θ r and U eff min near a = ˜ a . But this shift is not sudden, and one can say that the usage ˜ a gives an approximate separation when | a | > | b | .In some cases, M gl can be negative and there are some differences ineffective potential in these cases. When
M gl is negative, the second term inequation (9) becomes positive, and then arccos b/a > θ > a goes to infinity, again θ r goes to arccos b/a . In a goes to zerolimit, θ r goes to 0. These show that the interval for the minimum of effectivepotential changed from (arccos b/a, π ) to (0 , arccos b/a ) when M gl changedsign from positive to negative. If both a and b are negative or positive, θ r isbetween 0 and π/
2. If only one of them is negative, then θ r can be greaterthan π/ | a | is great enough. The minimum of U eff goes to −| M gl | when a goes to 0, and it goes to −| M gl | b/a when a goes to infinity when M gl is negative. | b | > | a | In this section, we will study the case when | b | > | a | . After factoring equation(7) in another way, it can be written as dU eff ( θ ) dθ = b I x sin θ (cid:20) (1 − ab cos θ )( ab − cos θ ) − M glI x b sin θ (cid:21) . (10)Similar to the previous case, the first term should be positive, and a/b − cos θ should be positive when | b | > | a | for the root, and then π > θ > arccos a/b . In b goes to infinity limit, the second term in the parentheses goes to zero. Then,as | b | goes to infinity, θ r should go to arccos a/b . In b goes to zero limit, θ r goes to π which can be seen from equation (7) similar to the previous section.Then, θ r goes to π when b goes to zero, and it goes to arccos a/b when | b | goes to infinity.When a and b are both positive or negative, as | b | increases from zeroto infinity, θ r decreases from π to arccos a/b < π/
2. If only one of them is6ositive, then θ r is always greater than π/ a = 0, as | b | goes to infinity θ r goes to π/ π as | b | goes to 0.Similar to the previous case, θ r can take values between 0 and π dependingon signs of a and b , the ratio a/b and greatness of a and b .The magnitude of the minimum of U eff changes with respect to b . In b goes to zero limit, U eff min goes to − M gl since θ r goes to π . In b goes toinfinity limit, θ r goes to arccos a/b , and then the minimum of U eff goes toinfinity with I x b (1 − ( a/b ) ) / (a) U eff π /2 π U e ff θ (b) θ r π /2 π
0 50 100 150 θ r b (c) U eff min U e ff m i n b Figure 3: U eff , change of θ r with respect to b and change of U eff min withrespect to b . a) Three different effective potential: b = 10 rad s − (greendashed-dotted curve), b = 30 rad s − (blue dashed curve) and b = 60 rad s − (red continious curve) with a/b = 0 .
5. Black line shows
M glb/a . b) Changeof θ r with respect to b for constant a/b = 0 . a/b ) = 1 . rad . Vertical dotted line shows the position of b = 2˜ a . c) Change of U eff min with respect to b for constant a/b = 0 . M glb/a . Vertical dotted line shows position of b = 2˜ a . Dotted curve shows I x b (1 − ( a/b ) ) / a and b is considered: This time a/b = 0 .
5. In figure 3(a), three different effec-tive potentials for three different b values are shown similar to the previoussection. In this figure, there are some similarities and differences from figure2(a). One can see that θ r is also different for different b values similar tothe previous section. It can be seen that as b takes different values, the formand magnitude of the minimum of U eff becomes different similar to previouscase, and it can be greater than M glb/a , unlike the previous case. In figure7(b), it can be seen that for very small values of b , θ r is close to π and it goesto arccos 0 . . rad as b increases. In figure 3(c), it can be seen that theminimum of U eff is close to − M gl if b is small, and it goes to infinity with I x b (1 − ( a/b ) ) / b goes to infinity. These are the expected results fromthe explanations given above.By considering these results, it can be said that M glb/a is not importantdifferently from | a | > | b | case. From figures 3(b) and 3(c), one can say thatthe shift in the behaviour of θ r and U eff min does not take place around a = ˜ a ,and the usage of ˜ a for seperation is not suitable when | b | > | a | .When M gl is negative, the second term in equation (9) becomes posi-tive, and then in this case, a/b − cos θ should be negative which is possiblewhen arccos a/b > θ >
0. In the limit where b goes to infinity, again θ r goesto arccos a/b . In b goes to zero limit, θ r goes to 0. Similar to the previ-ous case, the interval for the minimum of effective potential changed from(arccos b/a, π ) to (0 , arccos b/a ). If both a and b are negative or positive, θ r is between 0 and π/
2. If only one of them is negative, then θ r can be greaterthan π/ | b | goes to infinity, and θ r goes to 0 as b goes to zero. When a = 0, in | b | goes to infinity limit θ r goes to π/
2, and | b | goes to zero limitdoes not change and remains as 0. If M gl is negative, the minimum of U eff goes to −| M gl | when b goes to 0, and it goes to infinity as | b | goes to infinity. Effective potential can be helpful in understanding the motion of a symmetrictop in different ways. E (cid:48) should be equal to or greater than the minimumof U eff for physical motions. By using the limits given in section 3, onecan say that the regular precession takes place at greater angles when a and b are small, and as a and b increase, it takes place at smaller angles. Toobserve regular precession smaller than π/ a and b should have the samesign and have greater magnitudes. The limiting angle when | a | or | b | goes toinfinity can be found by using inverse cosine of b/a and a/b when | a | > | b | and | b | > | a | , respectively. If E (cid:48) is greater than the minimum of U eff , thendifferent types of motions can be seen [21]. These motion will take place closerangles to θ r when E (cid:48) is close to the minimum of U eff , and by consideringsigns and magnitudes of a and b one can have an opinion on the angles wherethe motion takes place.If a and/or b are small, then there can be a high asymmetry in the form of8 eff . From the definitions of U eff and E (cid:48) , one can say that ˙ θ is propotionalto the difference E (cid:48) − U eff ( θ ) for a specific θ value. Therefore, one can saythat as θ increases from θ min to θ r , the change in ˙ θ is gradual, and as θ increases from θ r to θ max , the change in ˙ θ is more rapid when a and/or b aresmall. As θ changes from θ max to θ min , this change in ˙ θ is firstly rapid andthen gradual.If a and b are great enough and the difference E (cid:48) − U eff min is small enough,then the asymmetry in U eff can be ignored. In these cases, one can makean approximation and find an exact solution for this approximation [12, 13].This approximation works better when the asymmetry in U eff is least.We have seen that comparison of | a | with ˜ a can be used when | a | > | b | for an approximate seperation, and it is not suitable when | b | > | a | . Butcomparison between | b | and ˜ a can be used when | b | > | a | , and if it is used,one should use a naming other than ”strong top” or ”weak top”. We shouldnote that comparison of | a | with ˜ a is very useful when | a | = | b | [19].Another thing that should be taken into account is the relation between M glb/a and E (cid:48) [21]. This study has shown that the minimum of U eff isalways smaller than M glb/a when | a | > | b | , which shows that one can alwaysobserve all possible motions when | a | > | b | . On the other hand, M glb/a canbe greater than or smaller than the minimum of U eff when | b | > | a | .These results show that effective potential has different advantages overthe cubic function in understanding the motion of a spinning heavy symmet-ric top. However, the cubic function is still important since it is better forproofs. There is an alternative to effective potential: the cubic function given inequation (5).Here, we will compare the cubic function with effective potential. Thecubic function is equal to ˙ u , and its roots give the points where ˙ u = 0. ˙ θ is equal to zero at two of these three points, and the third root is irrelevantto turning angles. Then, one can use the cubic function to obtain turningangles. If these two roots are the same, i.e. double root, then one can alsosay that this case gives regular precession. These turning angles can also beobtained from effective potential by using E (cid:48) = U eff ( θ ). And, if E (cid:48) = U eff min then the regular precession is observed as explained above.9n the other hand, there is not any correspondence between the minimumof U eff and the maximum of f ( u ). The reason for this is the multiplicationwith 1 − u during the change of variable. Then, f ( u ) can not be used tomake further analyses similar to U eff , given above.We will consider a case satisfying α = 575 . s − , a = 10 rad s − , b =2 rad s − as an example. For the symmetric top with previously given pa-rameters, β becomes 596 . s − . U eff and f ( u ) can be seen in figure 4. Onecan see that θ min = 1 . rad and θ max = 2 . rad can be obtained fromarccos( u
2) = 1 . rad and arccos( u
1) = 2 . rad , respectively. On the otherhand, θ r = 2 . rad can not be obtained from arccos( u m ) = 2 . (a) U eff -0.04 -0.02 0 0.021.2 1.6 2.0 2.4 2.8 U e ff θ U eff min θ min θ r θ max (b) f ( u ) -400-200 0 200 -1 0 1 f ( u ) u f max u u m u u Figure 4: U eff and f ( u ) when α = 575 . s − , β = 596 . s − , a = 10 rad s − and b = 2 rad s − . a) U eff continious (red) curve, E (cid:48) = − . J dashed(blue) line, θ min = 1 . rad , θ max = 2 . rad , θ r = 2 . rad and U eff min = − . J . b) f ( u ) continious (red) curve, u = − . u = − . u =1 . u m = − .
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Eur. J. Phys. https://doi.org/10.1088/1361-6404/ab6415 [20] Klein F and Sommerfeld A 2010 The theory of the Top, Volume II (NewYork: Birkhauser)[21] Tanrıverdi V 2020 Motion of the heavy symmetric top when magnitudesof conserved angular momenta are different https://arxiv.org/abs/2011.09348https://arxiv.org/abs/2011.09348