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Hagedorn's theorem in some special cases of rheonomic systems
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 32 (2005) 265–280 www.elsevier.com/locate/mechrescom
Hagedorn’s theorem in some special cases of rheonomic systems c V.M. Covi
a,*
, M.M. Veskovic
b
a b
Faculty of Mechanical Engineering, University of Belgrade, 27 Marta 80, 11000 Belgrade, Serbia and Montenegro Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, 36000 Kraljevo, Serbia and Montenegro Available online 25 February 2004
Abstract Hagedorn’s theorem on instability [Arch. Rational Mech. Anal. 58 (1976) 1], deduced from Jacobi’s form of Hamilton’s principle, refers to scleronomic mechanical systems. In this paper we shall prove that Hagedorn’s methodology can be generalized to a class of rheonomic mechanical systems with differential equations of motion which allow the existence of Painleve’s integral of energy. The application of this methodology to the case of rheonomic systems which allow, together with Painleve’s integral, cyclic integrals, as well as to the mechanical systems having resultant motion, with prescribed transport motion, and, finally, to the systems having Mayer’s rheonomic potential, are also considered. Obtained results are illustrated by an example. 2004 Elsevier Ltd. All rights reserved. Keywords: Rheonomic systems; Stability of motion; Variational method; Hagedorn’s theorem
1. Introduction In order to apply Hagedorn’s variational methodology, used analysing the instability of motion of a scleronomic system (see (Hagedorn, 1976)), in this paper are identified the classes of rheonomic systems for which the differential equations of motion can be deduced from a variational principle of Hamilton’s type, with the condition in the form of the Painleve integral. This problem is solved by finding special classes of rheonomic mechanical systems (in Lagrangian function of the system time appears explicitly either through the rheonomic holonomic constraints, or by means of the given equations of motion with respect to the part of generalized coordinates of the system, or, finally, thanks to the rheonomic field of potential forces and forces having rheonomic Mayer’s potential)
*
Corresponding author. Fax: +381-11-3246-382. c). E-mail address: [email protected] (V.M. Covi
0093-6413/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2004.02.009
266
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for which we establish the conditions providing the existence of the Painleve integral of energy. By analogy with the case of the Jacobi integral, this integral (which, in contrast to the scleronomic case, contains also the potential of rheonomic constraints) transforms Hamilton’s principle to the form which is equivalent to Jacobi’s principle. By this the connection with Hagedorn’s variational methodology is established. Section 2 of the paper are exposed to the conditions of existence of the Painleve integral of energy in the case of the motion of a mechanical system in a potential rheonomic field forces, this motion being subject to ideal holonomic rheonomic constraints. It is shown that in such a case the Painleve integral does not depend explicitly on time and that, as the consequence of the existence of that integral, the differential equations of motion can be deduced from the Hamilton principle in Jacobi’s form, which corresponds to a Lagrangian function which is guage invariant with respect to the original one. In this new Lagrangian function the potential of rheonomic constraints is added to the potential forces. The fact that, in the case considered, Hamilton’s principle can be represented in Jacobi’s form, made it possible to formulate the theorem on instability (Theorem 1) of the trivial solution of differential equations of motion of the rheonomic system considered, whose proof is based on Hagedorn’s variational methodology applied to the case of scleronomic systems. Section 3 is considered with the case of relative motion of a mechanical system, with given transport motion (which makes the system rheonomic), relative motion being restricted by the presence of scleronomic constraints. The condition for the existence of the Painleve integral in this case reduces to the condition that the given Pfaff form is a perfect differential of the function of coordinates and time. If this condition is satisfied, the problem of the application of Hagedorn’s variational methodology to the stability of motion reduces to the one considered in Section 2. In Section 4, the conditions for the existence of the Painleve integral of energy in the case of a mechanical system moving in the rheonomic field of potential forces and Lorentz forces are founded. Also, in this part of the paper the case of a mechanical system moving in the field of forces for which Mayer’s potential can be determined is considered. In both cases the condition for the existence of the Painleve integral reduces to the condition that the given Pfaff form is a perfect differential of the function depending on coordinates and time. Section 5 is considered with the mechanical system with a rheonomic Lagrangian function containing a term linear with respect to generalized velocities. It is assumed that the differential equations of motion of that system allow the existence of the Painleve integral and of a class of integrals linear with respect to the generalized velocities. It is shown that the linear integrals are cyclic ones in the case when, in accordance with the conditions of existence of the Painleve integral, the differential equations are obtained from the Lagrangian function, gauge invariant with respect to the original one. It is proved that, at the difference from the classical case, under the given conditions the existence of cyclic integrals is possible also when the original Lagrangian function depends on cyclic coordinates. By applying Routh’s method of ignoring the cyclic coordinates, the differential equations of motion of the system are formed with respect to positional coordinates, demonstrating that the existence of the Painleve integral allows the substitution of the original Routh function by the guage invariant one. In this last function, at the difference from the classical case, the Routh–Painleve potential is present. Further, it is shown that it is also possible, in this case, to formulate the Hamilton principle in Jacobi’s form, which yields Routh’s equations. This introduces the Hagedorn variational methodology in the study of instability of the steady motion of the system (Theorem 2 is formulated), allowed by its differential equations of motion. Finally, by introducing a new explicit form of the relation between Routh’s function and Lagrange’s function, a form of Lagrange’s function independent from time is found, allowing the formulation of a Hagedorn’s tipe theorem (Theorem 3) on the instability of steady motion with respect to positional coordinates. In Section 6 the results are illustrated by an example of steady motion of a mechanical system with rheonomic constraints, in a rheonomic field of potential forces.
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2. The case of mechanical systems with rheonomic constraints Hagedorn’s theorem establishes that, in the case of the motion of a discrete mechanical system in a potential field of forces, subject to holonomic scleronomic ideal constraints, the trivial solution q ¼ 0 of the differential equations of motion is instable if these equations allow the existence of generalized Jacobi’s energy integral, and if, in the position q ¼ 0, the part (H0 ) of the Hamiltonian function independent of the generalized momentum components, has a strict maximum. It has further been shown (Liubushin, 1980) that this theorem, in the case when that part of the Hamiltonian function of the system considered has a non strict maximum at q ¼ 0, still holds. Generalization of Hagedorn’s variational method to the class of mechanical systems subject to the rheonomic holonomic ideal constraints is presented in this paper. Let us consider a mechanical system moving in the field of potential forces, subject to ideal holonomic rheonomic constraints. Let the configuration of the system be defined by the set of Lagrangian coordinates q ¼ ðq1 ; q2 ; . . . ; qn Þ. The Lagrangian function of the system considered has, in the general case, the form (Pars, 1968) _ tÞ þ T1 ðq; q; _ tÞ þ T0 ðq; tÞ Pðq; tÞ; _ tÞ ¼ T2 ðq; q; Lðq; q; where
ð1Þ
1
1 _ tÞ ¼ aij q_ i q_ j ; T1 ¼ ai q_ i T2 ðq; q; ð2Þ 2 and where T ¼ T2 þ T1 þ T0 is the kinetic energy of the system, P––its potential energy, and q_ ¼ ðq_ 1 ; q_ 2 ; . . . ; q_ n Þ––the generalized velocities of the system. Let the conditions aij ðq; tÞ; ai ðq; tÞ 2 C2 ;
T0 ðq; tÞ; Pðq; tÞ 2 C2 ;
hold. The differential equations of motion of the system d oL oL i ¼ 0; dt oq_ i oq
ð3Þ
have as a consequence the well-known relation dðT2 þ P T0 Þ ¼
oL dt: ot
ð4Þ
If the condition oL dt ¼ dV ðq; tÞ; ot
V ðq; tÞ 2 C2 ;
ð5Þ
holds, Eq. (3) lead, by (4), to the first quadratic integral of the differential equations of motion, known as Painleve’s integral (see Painleve, 1897, p. 89; Appell, 1911, p. 288): T2 þ P T0 þ V ¼ h;
h ¼ const:
ð6Þ
Let us further show that time t does not explicitly appear in Painleve’s integral. If (5) holds, we have 1 oaij i j oai i o oV oV dt; q_ dq þ dq ðP T0 Þ dt i dqi þ 2 ot ot oq ot ot
ð7Þ
1 Einstein summation convention is used throughout the paper. Unless the different is stated explicitly, indices take the following values: i; j; k ¼ 1; . . . ; n; a; b ¼ 1; . . . ; m; m; q; p; h ¼ m þ 1; . . . ; n.
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wherefrom oaij ¼ 0 ! aij ¼ aij ðqÞ; ot
ð8Þ
oai oV ¼ i; oq ot
ð9Þ
oðP T0 Þ oV ¼ : ot ot
ð10Þ
The last equation leads to oðP þ V T0 Þ ¼ 0 ! ðP þ V T0 Þ ¼ UðqÞ: ot
ð11Þ
Relations (8) and (11) confirm that time t does not appear explicitly in the integral (6). Let us notice that Painleve’s integral differs from the Jacobi generalized integral of energy by the presence of the function V ¼ V ðq; tÞ in the dynamical potential of the system. Let us further show that, if Painleve’s integral exists, it is always possible to substitute the original Lagrangian function (1) by the corresponding gauge invariant one, whose dynamical potential (the part of the Lagrangian function which does not depend on the generalized velocities) includes the function V ¼ V ðq; tÞ, generated by a part of the original Lagrangian function which is linear with respect to the generalized velocities and which explicitly depends on time. This function we call the potential of rheonomic constraints. By the relation (9) one obtains ai ¼ ~ ai ðqÞ þ ai ðq; tÞ;
ð12Þ
where ai ¼
Z
or ai ¼
o oqi
oV ðq; tÞ dt; oqi Z
V ðq; tÞ dt:
The last relation leads to Z Z o d V ðq; tÞ dt V ; ai q_ i ¼ i V ðq; tÞ dt q_ i ! ai q_ i ¼ oq dt
ð13Þ
ð14Þ
ð15Þ
wherefrom, taking into account (12), the expression (1) for the Lagrangian function can be written in the form Z 1 d L ¼ aij q_ i q_ j þ ~ V ðq; tÞ dt ðP þ V T0 Þ: ai ðqÞq_ i þ ð16Þ 2 dt The Lagrangian function gauge invariant to (1) has, in virtue of the above expression, the form 1 e L ¼ aij q_ i q_ j þ ~ ai q_ i ðP þ V T0 Þ; 2 which allows the existence of an energy integral identical to Painleve’s integral (6).
ð17Þ
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If the condition oðP þ V T0 Þ ¼0 oqi in the position q ¼ 0 holds, the differential equations of motion (3) have the trivial solution q ¼ 0. Theorem 1. If the motion of a mechanical system in the field of potential forces is subject to the rheonomic ideal constraints, while its differential equations (3) allow the existence of Painleve’s integral (6) and have trivial e 0 of the Hamiltonian function independent solution q ¼ 0, this solution will be unstable if, at q ¼ 0, the part H from generalized momentum components, e 0 ¼ 1 aij ~ H aj þ ðP þ V T0 Þ; ai ; ~ 2
ð18Þ
which corresponds to the Lagrangian function (17) gauge invariant to the original Lagrangian function (1), has a strict maximum. Proof. After substituting (17) to (1), it is obvious that, in the case of the system considered, Hamilton’s principle, in virtue of (6), can be written in the Jacobi form: Z t1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i j 2½h ðP þ V T0 Þaij q_ q_ þ ~ ð19Þ ai q_ dt ! inf : t0
By solving (19) we obtain the differential equations of motion of the system considered in which, obviously, time t does not appear explicitly. The remaining part of the proof coincides with the proof given in (Hagedorn, 1976). h Note. Let us notice that the coordinates aij (see (18)) satisfy aij ajk ¼ dik ;
i ¼ k ! dik ¼ 1;
i 6¼ k ! dik ¼ 0;
where dik denotes the Kronecker symbol. Let us also remark that the condition (see Hagedorn, 1976, case V 0) ðP þ V T0 Þðq¼0Þ ¼ 0;
ð20Þ
implies (see (18) and (19)) ~ ai ðq ¼ 0Þ ¼ 0:
ð21Þ
The condition (20) can be obtained by an appropriate choice of the level of the zero potential, while the condition (21), if it is not fulfilled, can be reached by substitute the Lagrangian function (17) for its gauge invariant Lagrangian function: e L)e L~ ai ðq ¼ 0Þq_ i :
ð22Þ
3. The case of the relative motion with a given transport motion Let us now consider the case of the motion of a mechanical system subject to the scleronomic constraints, with Lagrangian function in which time t appears explicitly. Such a situation arises, e.g., in the case of mechanical systems having relative motion with scleronomic constraints imposed to it, and with the equations of transport motion which are known functions of time. For such a motion of a mechanical system the condition (8) for existence of Painleve’s integral is satisfied always.
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Let us consider a system of points M1 ; . . . ; MN , with masses m1 ; . . . ; mN respectively, which moves relatively with respect to the coordinate system Angf, this motion being subject to the ideal holonomic constraints of the form f r ðn1 ; . . . ; nN ; g1 ; . . . ; gN ; f1 ; . . . ; fN Þ ¼ 0;
r ¼ 1; . . . ; k < 3N ;
ð23Þ
where ni , gi , fi are Cartesian rectangular coordinates of Mi , with respect to Angf. If the transformations of the coordinates ðn ¼ 3N kÞ ni ¼ ni ðq1 ; . . . ; qn Þ;
gi ¼ gi ðq1 ; . . . ; qn Þ;
fi ¼ fi ðq1 ; . . . ; qn Þ;
make that (23) become identities, then the coordinates ðq1 ; . . . ; qn Þ are independent (Lagrangian) and completely determine the position of the system considered with respect to Angf. Taking into account that the equations of the transport motion are given, it is obvious that the velocity of the point A as well as the ~ of the transport motion, are known functions of time, too: angular velocity x ~ vA ¼ ~ vA ðtÞ;
~¼x ~ðtÞ: x
ð24Þ
The kinetic energy of this system has the form N N X 1X onk onk ogk ogk ofk ofk onk og i j _ _ mk þ m q þ þ q þ vg ki T ¼ k vn i 2 k¼1 oqi oqj oqi oqj oqi oqj oq oq k¼1 of of og on of og on þ vf ki þ xn gk ki fk ki þ xg fk ki nk ki þ xf nk ki gk ki q_ i oq oq oq oq oq oq oq N X 1 1 þ mk ðen nk þ eg gk þ ef fk Þ þ xn ðJn xn Jng xg Jnf xf Þ þ xg ð Jgn xn þ Jg xg Jgf xf Þ 2 2 k¼1 1 ð25Þ þ xf ð Jfn xn Jfg xg þ Jf xf Þ ; 2 where vn ¼ vn ðtÞ, vg ¼ vg ðtÞ, vf ¼ vf ðtÞ; xn ¼ xn ðtÞ, xg ¼ xg ðtÞ, xf ¼ xf ðtÞ; en ¼ en ðtÞ, en ¼ en ðtÞ, ~, ~ ~ on axes An , Ag , Af respectively, and Jn , Jg , Jf , en ¼ en ðtÞ––are the projections of vectors ~ vA , x e ¼~ vA x Jng , Jgf , Jfn ––the coordinates of the tensor of inertia of the mechanical system, calculated with respect to Angf, which are functions of Lagrangian coordinates. It is obvious that the condition (8) is always satisfied in the case considered (see the expression for the part of the kinetic energy, appearing in (25), which represents quadratic form with respect to the generalized velocities). Taking into consideration the last remark, the necessary and sufficient conditions for the existence of Painleve’s integral are satisfied in this case if the following Pfaff’s form ! N X oP ~r þ~ ~ Pf ¼ ~ v_A K e_ mk~ e JA x dt ð26Þ qk þ~ e ~ LAr þ ~ ot k¼1 is a total differential of the function depending on time and on the Lagrangian coordinates. In the last ~r ––linear momentum of the system considered, corresponding expression the following quantities appear: K to its relative motion; LAr ––the angular momentum corresponding to the relative motion of the system, taken with respect to the origin A; JA ––the tensor of inertia of the mechanical system, taken with respect to Angf; ~ e––the angular acceleration vector of the transport motion; ~ v_A , ~ e_ ––the relative derivatives with respect to time of vectors ~ vA and ~ e; P ¼ Pðq; tÞ––the potential energy of the system considered; ~ qi ––radius vector of the material point Mi with respect to A. Let us remark that the coordinates of all vectors in that expression are taken with respect to Angf.
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So, if the condition (26) is satisfied, it is possible to apply Hagedorn’s methodology using the considerations identical with the ones presented in Section 1 of the paper.
4. Mayer’s rheonomic potential Lagrangian function can include a part linear with respect to generalized velocities, which depend on time explicitly, in other cases of motion of the system subject to the scleronomic constraints, too. Such a case appears when a mechanical system moves in a field forces having the structure of the Lorentz force (see Pars, 1968, p. 184; Gantmaher, 1960, p. 57). In the case of a material point that force has the form o~ Aðx; y; z; tÞ ~ F ¼ þ~ v rot~ Aðx; y; z; tÞ; ~ Aðx; y; z; tÞ 2 C2 ; ð27Þ ot where ~ v denotes the velocity of the point, and where x,y,z are the Cartesian rectangular coordinates of the point. The force (27) can be presented by the generalized (Mayer’s) potential VM ¼ ~ A ~ v;
ð28Þ
as follows (Gantmaher, 1960) d oVM oVM ; X ¼ þ dt o_x ox
Y ¼
d dt
oVM o_y
þ
oVM ; oy
Z¼
d dt
oVM o_z
þ
oVM ; oz
ð29Þ
where ~ F ¼ ðX ; Y ; ZÞ:
ð30Þ
For the motion of a scleronomic mechanical system in the field forces (27) and in the rheonomic field of potential forces, the Lagrangian function has the form ðVM ¼ ai ðq; tÞq_ i Þ: 1 L ¼ aij ðqÞq_ i q_ j þ ai ðq; tÞq_ i Pðq; tÞ; 2
ð31Þ
which corresponds to the class of rheonomic mechanical systems satisfying the condition (8) for existence of Painleve’s integral. So, having in mind the foregoing considerations, the necessary and sufficient conditions for the existence of Painleve’s integral are satisfied in this case if following Pfaff’s form (see (9) and (10)) Pf0 ¼
oai i oP dt dq ot ot
ð32Þ
c is a total differential of the function depending on time and on the Lagrangian coordinates. Also, in (Covi and Lukacevic, 1997) it is shown that in the case of nonconservative generalized forces of the form ð0Þ
Qi ¼ bij ðq; tÞq_ j þ binþ1 ðq; tÞ;
ð33Þ
for which the conditions ð0Þ
ð0Þ
bij ðq; tÞ ¼ bji ðq; tÞ; ð0Þ ð0Þ ð0Þ obsp obrp obrs þ ¼ ; p r oq oq oqs
ð34Þ
r ¼ 1; . . . ; n 2;
s ¼ i þ 1; . . . ; n 1;
p ¼ j þ 1; . . . ; n;
ð35Þ
c, M.M. Veskovic / Mechanics Research Communications 32 (2005) 265–280 V.M. Covi
272 ð0Þ
ð0Þ
ð0Þ ob ob obrs þ snþ1 ¼ rnþ1 ; r ot oq oqs
r ¼ 1; . . . ; n 1;
s ¼ i þ 1; . . . ; n 1
ð36Þ
hold, Mayer’s rheonomic potential can always be found in the form ! Z q1 Z qi1 Z q2 n n Z X X ð0Þ ð1Þ 1 2 i2 i1 i VM ¼ b1i dq þ b2i dq þ þ bi1i dq q_ þ i¼2
k1
k2
ki1
i¼1
qi
ði1Þ
binþ1 dqi ;
ð37Þ
ki
where ðiÞ
ð0Þ
biþ1j ¼ biþ1j ðq1 ¼ k1 ; q2 ¼ k2 ; . . . ; qi ¼ ki ; qiþ1 ; . . . ; qn ; tÞ;
k1 ; . . . ; ki ¼ const:;
i ¼ 1; . . . ; n 1;
j ¼ i þ 2; . . . ; n: This leads to the conclusion that, in the case of the motion of a scleronomic system in the field forces (33) satisfying the conditions (34)–(36), the Lagrangian function has the form corresponding to the rheonomic systems: 1 L ¼ aij ðqÞq_ i q_ j þ ax ðq; tÞq_ x Pðq; tÞ; 2
x ¼ 2; . . . ; n:
ð38Þ
It is obvious that this case, in further considerations of application of Hagedorn’s methodology, reduces to the previous one (see (31)).
5. The case of the steady motion of a system If the expression for the Lagrangian function has the form corresponding to the systems subject to the rheonomic constraints, it will also hold its rheonomic form if, in the case of existence of cyclic integrals, one applies Routh’s method of ignoration of cyclic coordinates. Let us further consider the motion of mechanical system with rheonomic Lagrangian function _ tÞ ¼ T2 ðq0 ; q; _ tÞ þ T1 ðq0 ; q; _ tÞ þ T0 ðq0 ; tÞ Pðq0 ; tÞ; Lðq; q;
ð39Þ
in which q0 ¼ ðq1 ; q2 ; . . . ; qm Þ denotes the set of the positional coordinates. To the cyclic coordinates q00 ¼ ðqmþ1 ; qmþ2 ; . . . ; qn Þ correspond the cyclic integrals oL ¼ cq ; oq_ q
cq ¼ const:;
ð40Þ
or aqi q_ i þ aq ¼ cq :
ð41Þ
However, in the case of existence of Painleve’s integral, the existence of the first integrals of the form (41) is possible, under the fixed conditions, in the case when coordinates q00 appear in the Lagrangian function. This statement can be proved as follows. Let the Lagrangian function of a mechanical system which differential equations allow Painleve’s integral is given in the form 1 _ tÞ ¼ aij ðq0 Þq_ i q_ j þ ai ðq0 ; q00 ; tÞq_ i þ T0 ðq0 ; q00 ; tÞ Pðq0 ; q00 ; tÞ; Lðq; q; 2
ð42Þ
while the condition ai ðq0 Þ þ ai ðq0 ; q00 ; tÞ: ai ðq0 ; q00 ; tÞ ¼ ~
ð43Þ
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holds. The differential equations of motion corresponding to the coordinates q00 have the form d oa o ðaqi q_ i þ ~ aq þ aq Þ qi q_ i þ q ðP T0 Þ ¼ 0; dt oq oq wherefrom, having in mind the relations Z Z oaq o2 oai o2 V ðq; tÞ dt; V ðq; tÞ dt; ¼ ¼ oqi oqi oqq oqq oqq oqi
ð44Þ
oaq oV ¼ q; oq ot
ð45Þ
we obtain d o ðaqi q_ i þ ~ aq Þ þ q ðP þ V T0 Þ ¼ 0: dt oq
ð46Þ
Then, under the condition that o ðP þ V T0 Þ ¼ 0 oqq
ð47Þ
holds, the following integrals result: aqi q_ i þ ~ aq ¼ c q :
ð48Þ
Thus, in the case when the differential equations of motion allow the existence of the Painleve energy integral, the existence of cyclic integrals is possible even in the case when coordinates q00 , to which correspond the cyclic integrals (48), satisfy the condition oL 6¼ 0: ð49Þ oqq In the case considered the existence of Painleve’s integral, as mentioned, allows to substitute the original Lagrangian function (42) by its gauge invariant Lagrangian function which, with (47) satisfied, has the form 1 e _ tÞ ¼ aij ðq0 Þq_ i q_ j þ ~ Lðq; q; ai ðq0 Þq_ i ðP þ V T0 Þ; ð50Þ 2 which leads to the cyclic integrals (48). Let us remark that, with respect to the Lagrangian function (50), the concept of the cyclic coordinate has its usual meaning. The complete system of the differential equations of motion of the mechanical system considered is given by the cyclic integrals (48) and, in accordance with Routh’s method of ignoration of cyclic coordinates, by the equations ! e e d oR oR ð51Þ i ¼ 0; i dt oq_ oq e is given by the expression (the case of existence of Painleve’s integral) where the Routh function R e¼e ð52Þ R L cq q_ q ; in which the cyclic velocities have to be eliminated using the following relations, obtained from (41): ð53Þ q_ q ¼ bqm ðdm þ ama q_ a Þ; where bqm amh ¼ dqh ;
dm ¼ ~ am c m :
In that way one obtains the result 1 1 mq a b a e R ¼ bab q_ q_ þ ba q_ P þ V þ b dm dq T0 ; 2 2
ð54Þ
ð55Þ
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274
where bab ¼ aab bmq ama aqb ;
ba ¼ ~ aa bmq dm aqa :
ð56Þ
Let us notice that the expression (55) for the Routh function differs from the one related to scleronomic system (see e.g. Hagedorn, 1971) by the presence of the ‘‘rheonomic potential’’ V in the expression of Routh’s potential of the system. It is the reason for which the potential e ¼ P þ V þ 1 bmq dm dq T0 ; P 2
ð57Þ
we shall name the Routh–Painleve potential. In the case when the condition ! e oP ¼0 oqa 0
ð58Þ
ðq ¼0Þ
holds, the differential equations of motion (48) and (51) allow steady motion of the form qa ¼ 0;
qq ¼ qq0 þ q_ q0 ðt t0 Þ;
qq0 ; q_ q0 ; t0 ¼ const:
ð59Þ
Notice. From the fact that the Lagrangian function (50) is gauge invariant to the original Lagrangian function L, given by (42), it results that the Routh function (52) is gauge invariant to the original Routh function R ¼ L cq q_ q : The first relation in (59) represents the trivial solution of Eq. (51). Obviously (see (55), (56), (54), and (50)) e oR ¼ 0; ot
ð60Þ
holds, which leads to the conclusion that the differential equations of motion have a quadratic first integral (the integral of energy in which the Routh–Painleve potential appears) of the form 1 e ¼ h; bab q_ a q_ b þ P 2
h ¼ const:;
ð61Þ
which, in fact, represents Painleve’s integral (6) in which the cyclic velocities q_ 00 are excluded in virtue of (53). From foregoing consideration it follows that the differential equations (51) can be deduced from the Jacobi form of Hamilton’s principle, too, i.e. from Z
t1 t0
ffi ) (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 mq a a b 2 h P þ V þ b dm dq T0 bab q_ q_ þ ba q_ dt ! inf : 2
ð62Þ
On the basis of the previous considerations we formulate the following Theorem 2. If, for the mechanical system considered, the following conditions are fulfilled: • Lagrangian function of the system has the form (42), whereas the condition (43) hold; • the differential equations of motion (51) of the system allow the existence of Painleve’s integral (6) and trivial solution q0 ¼ 0;
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• dynamic potential of the system, including also potential V of the rheonomic constraints, does not depend on the coordinates q00 (condition (47) satisfied); the trivial solution will be unstable if the part of Hamilton’s function, constructed over the Routh function (55) which is gauge invariant to the original Routh function, which does not depend on the generalized momentum components, has a strict maximum in the position q0 ¼ 0, i.e. if the condition 1 ab 1 b ba bb þ P þ V þ bmq dm dq T0 < 0; 2 2
8q0 6¼ 0;
ð63Þ
holds, while it is (see (56)) bab bbc ¼ dac :
ð64Þ
Proof. It was demonstrated that, in the case of the existence of Painleve’s integral, the Lagrangian function (42), together with (43) and (47), generates the cyclic integrals (48). Also, in this case it is possible to replace the original Lagrangian function by its gauge invariant one, given by (50), to which corresponds the Routh function (55), which is gauge invariant to the original one. The function (50) permits, thanks to (60) and (61), that the differential equations of motion of the system considered, corresponding to the positional coordinates, can be deduced from the Jacobi form of the Hamilton’s principle (62). The remaining part of the proof coincides with the proof given in (Hagedorn, 1976). h We notice that in the case of scleronomic systems considered in (Hagedorn, 1976), a criterion, obtained if one puts in (63) V 0; holds. In order that the criterion (63) can be applied, the conditions 1 mq ba ðq ¼ 0Þ 0; P þ V þ b dm dq T0 0; 2 ðq¼0Þ
ð65Þ
ð66Þ
must be fulfilled. The second condition in the last expression is trivially satisfied by subtracting from the Routh–Painleve potential its value at the point q0 ¼ 0. In the general case, however, first condition in (66) is not satisfied. In order to overcome this problem the Lagrangian function (50) can be substituted by its gauge invariant Lagrangian function 1 _ tÞ ¼ aij ðq0 Þq_ i q_ j þ ~ ai ðq0 Þq_ i ba ðq0 ¼ 0Þq_ a ðP þ V T0 Þ; L ðq; q; 2 which makes that the Routh function takes the form 1 1 mq a b a R ¼ bab q_ q_ þ ba q_ P þ V þ b dm dq T0 ; 2 2
ð67Þ
ð68Þ
where ba ¼ ba ðq0 Þ ba ðq0 ¼ 0Þ ! ba ðq0 ¼ 0Þ 0:
ð69Þ
The mentioned problem can be overcome in the following way, too. The Lagrangian function in the Routh form (55) can be transformed to the form
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276
e ¼ 1 bab q_ a q_ b þ d ðba qa Þ R 2 dt
obb b a 1 mq _ q P þ V þ d d T b q m q 0 : 2 oqa
ð70Þ
Removing the second term at the right-hand side in the last relation, we obtain again the relation (68) in which we have in this case ba ¼
obb b q ; oqa
ð71Þ
where ba ðq0 ¼ 0Þ ¼ 0:
ð72Þ
With such modifications of the Lagrangian function in the Routh form, Hagedorn’s criterion of instability of the trivial solution q0 ¼ 0 of differential equations of motion (51) (which preserve their form after the substitution of (55) by (68) or (71)) gets the form 1 ab 1 b ba bb þ P þ V þ bmq dm dq T0 < 0 2 2
8q0 6¼ 0:
ð73Þ
Let us show now that the tensor bab of the reduced system (see (68)) represents the inverse of the matrix obtained by omitting in the tensor aij the rows and columns corresponding to the cyclic coordinates, i.e. let us show that (see (64)) bab ¼ aab :
ð74Þ
holds. Indeed, as, in virtue of aij ajk ¼ dki ; it follows aab abc þ aaq aqa ¼ dca ;
aab abq þ aam amq ¼ 0;
ð75Þ
aqb aba þ aqm ama ¼ 0;
aqb abm þ aqp apm ¼ dmq ;
ð76Þ
one obtains (see (56) and (54)) ðaab bmq ama aqb Þabc ¼ dca ; ! bab abc ¼ dca ;
ð77Þ
which proves the validity of (74). Moreover, (75) and (76) lead to the relation amq ¼ bmq bmp apa aab abh bmh ;
ð78Þ
which, together with (75)–(77), to Hagedorn’s criterion (73) gives the form 1 ij a ð~ ai ci Þð~aj cj Þ þ P þ V T0 < 0; 2
8q0 6¼ 0;
ð79Þ
where (see (52)) ca ¼ ba ðq0 ¼ 0Þ and cm ––the value of the cyclic generalized momentum components (see (40)).
ð80Þ
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As the potential energy P is undeterminated to an additive constant, this constant has to be chosen so as to nullify the expression at the left-hand side of the inequality (79) in q0 ¼ 0. The criterion (79) gives the possibility to formulate the following Theorem 3. Under the conditions providing the validity of the Theorem 2, the trivial solution q0 ¼ 0 will be unstable if the part of Hamilton’s function corresponding to the Lagrangian function gauge invariant to (50), ¼e L L ci ; q_ i ; which does not depend on generalized momentum components, has in q0 ¼ 0 a strict maximum. Proof. It is obvious that the expression (79) represents Hamilton’s function which corresponds to the As (79) represents a new form of the criterion (63), the remaining part of the proof Lagrangian function L. is the same as the proof in Theorem 2. h Obviously, in this theorem one does not use the Routh method of ignoration of cyclic coordinates, although it refers to the instability with respect to positional coordinates. Notice. It is obvious that in Theorems 1–3 the condition for strict maximum of the part of Hamilton’s function, independent on the generalized momentum components, can be, having in mind the results from (Liubushin, 1980), replaced by the condition for non strict maximum. 6. Example The mechanical system consisting of two points M1 and M2 , each of mass m ¼ 1, moves in the field of potential forces with potential energy (x1 , y1 , z1 ; x2 , y2 , z2 ––Cartesian rectangular coordinates of the respective material points): 1 ðe1 Þ P ¼ ðz41 þ x42 Þ ðz61 þ x62 Þ þ ðz21 þ x22 Þ cos 2t þ z1 x2 sin 2t: 2 The motion of the system is subject to the rheonomic constraints given by y2 ¼ ðx22 þ z21 Þ cos t þ x2 z1 sin t þ cos t; z2 ¼ ðx22 þ z21 Þ sin t x2 z1 cos t sin t; x1 ¼ const:
ðe2 Þ
(a) Prove that the differential equations of motion of the system considered allow the existence of Painleve’s integral; (b) Determine the corresponding cyclic integral and examine the stability of the trivial solution of the equations of motion in the Routh form. Solution. The kinetic energy of the system considered has the form 2 1 X T ¼ m ð_x2 þ y_ i2 þ z_ 2i Þ; 2 i¼1 i
ðe3 Þ
which, if one introduces independent generalized coordinates q1 ¼ z 1 ;
q2 ¼ x 2 ;
q3 ¼ y1 ;
ðe4 Þ
can be written, in virtue of (e2 ), in the form 1 T ¼ aij ðqÞq_ i q_ j þ ~ ai ðqÞq_ i þ ai ðq; tÞq_ i þ T0 ðq; tÞ; 2
ðe5 Þ
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278
where a11 ¼ 1 þ 4ðq1 Þ2 þ ðq2 Þ2 ; 1 2
2 2
a22 ¼ 1 þ ðq Þ þ 4ðq Þ ; 2
1 2
2 2
~ a1 ¼ q ½ðq Þ ðq Þ ; a1
a12 ¼ 5q1 q2 ; a23 ¼ 0;
a13 ¼ 0;
a33 ¼ 1;
2 2
2 ~ a2 ¼ q ½ðq Þ ðq1 Þ ; 1
a2
~a3 ¼ 0;
ðe6 Þ
¼ q cos 2t 2q sin 2t; ¼ q cos 2t 2q sin 2t; a3 ¼ 0; 1 1 2 2 2 2 2 2 T0 ¼ ½ðq1 Þ þ ðq2 Þ þ ðq1 q2 Þ ½ðq1 Þ þ ðq2 Þ cos 2t q1 q2 sin 2t: 2 2 2
1
1
2
Hence oL dt ¼ f1 dq1 þ f2 dq2 þ f0 dt; ot where f1 ¼ ð2q2 sin 2t þ 4q1 cos 2tÞ; 2
f2 ¼ ð2q1 sin 2t þ 4q2 cos 2tÞ;
2
f0 ¼ 4½ðq1 Þ þ ðq2 Þ sin 2t 4q1 q2 cos 2t;
ðe7 Þ
so that, having in mind that the conditions of1 of2 ¼ ; oq2 oq1
of1 of0 ¼ 1; ot oq
of2 of0 ¼ 2; ot oq
ðe8 Þ
hold, one obtains oL dt ¼ dV ðq1 ; q2 ; tÞ; ot
ðe9 Þ
and 2
2
V ¼ 2bðq1 Þ þ ðq2 Þ c cos 2t 2q1 q2 sin 2t:
ðe10 Þ
In virtue of (4) and (5) one obtains Painleve’s integral (see (6)): 1 1 2 2 2 2 2 2 ½1 þ 4ðq1 Þ þ ðq2 Þ ðq_ 1 Þ þ ½1 þ ðq1 Þ þ 4ðq2 Þ ðq_ 2 Þ þ 5q1 q2 q_ 1 q_ 2 2 2 1 3 2 6 6 2 2 þ ðq_ 3 Þ ðq1 Þ ðq2 Þ ðq1 Þ ðq2 Þ ¼ h: 2 2
ðe11 Þ
To the cyclic coordinate q3 corresponds the cyclic integral q_ 3 ¼ q_ 30 ;
q_ 30 ¼ const:;
ðe12 Þ
with the help of which, using (50) and (52), we form the expression for the Lagrangian function in the Routh form (we omit the constant terms): e ¼ 1 ½1 þ 4ðq1 Þ2 þ ðq2 Þ2 ðq_ 1 Þ2 þ ð5q1 q2 Þq_ 1 q_ 2 þ 1 ½1 þ 4ðq1 Þ2 þ ðq2 Þ2 ðq_ 2 Þ2 þ q2 ½ðq1 Þ2 ðq2 Þ2 q_ 1 R 2 2 3 1 2 2 2 1 2 2 1 2 2 1 6 2 6 þ q ½ðq Þ ðq Þ q_ þ ðq Þ þ ðq Þ þ ðq Þ ðq Þ : 2
ðe13 Þ
It follows, by the last expression, that the Routh-Painleve potential is (see (57)) e ¼ ðq1 Þ6 ðq2 Þ6 3 ðq1 Þ2 ðq2 Þ2 ; P 2
ðe14 Þ
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279
and that it has a strict maximum with respect to positional coordinates in q1 ¼ q2 ¼ 0. Thus, the conditions for existence of the following steady motion q1 ¼ 0;
q2 ¼ 0;
q3 ¼ q_ 30 t þ q30 ;
q30 ¼ const:;
ðe15 Þ
are fulfilled, and these conditions obviously do not depend on the value of the cyclic momentum. In order to apply the criterion (79), we find the coordinates of the tensor aij : a22 a12 a11 a11 ¼ ; a12 ¼ ; a13 ¼ 0 a22 ¼ ; a23 ¼ 0; a33 ¼ 1 D D D
ðe16 Þ
where 4
2
4
2
2
2
D ¼ 1 þ 4ðq1 Þ þ 5ðq1 Þ þ 4ðq2 Þ þ 5ðq2 Þ 8ðq1 Þ ðq1 Þ :
ðe17 Þ
Finally, it follows that the left-hand side of (79), multiplied by a positive quantity D, has the value 1 1 2 2 a22 ð~ a1 Þ þ a11 ð~ a2 Þ a12 a1 a2 þ DðP þ V T0 Þ 2 2 3 1 1 ¼ ðq1 Þ2 ðq2 Þ2 ðq1 Þ6 ðq2 Þ6 8ðq1 Þ4 ðq2 Þ2 8ðq1 Þ2 ðq2 Þ2 9ðq1 Þ6 ðq2 Þ2 9ðq1 Þ2 ðq2 Þ6 2 2 2 1 4 2 4 1 6 2 4 þ 4ðq Þ ðq Þ 4ðq Þ ðq Þ 4ðq1 Þ4 ðq2 Þ6 3ðq1 Þ8 3ðq2 Þ8 þ 8ðq1 Þ8 ðq2 Þ2 þ 8ðq1 Þ2 ðq2 Þ8 10
10
4ðq1 Þ 4ðq2 Þ
ðe18 Þ
wherefrom we conclude that the criterion (79) of instability of the trivial solution q1 ¼ q2 ¼ 0 of the differential equations of motion of the system considered d dt
e oR oq_ 1
!
e oR 1 ¼ 0; oq
d dt
e oR oq_ 2
!
e oR ¼ 0; oq2
ðe19 Þ
is satisfied.
7. Conclusion The main idea of this paper refers to the enlargement of Hagedorn’s variational approach in stability of motion of mechanical systems to the case of rheonomic systems. This idea is carried out in several cases of rheonomic systems, while the results obtained are formulated in the form of three theorems. By an example referring to the steady motion of the mechanical system the results are illustrated.
References Appell, P., 1911. Traite de Mecanique Rationnelle, T II, Dynamique des Systemes––Mecanique Analitique. Gauthier-Villar, Paris. c, V., Lukacevic, M., 1997. On the existence of Mayer’s potential. J. Appl. Mech. 64 (3). Covi Gantmaher, F.R., 1960. Lektsii po analititcheskoj mekhanike, Gos. izdat. fiz.-mat. lit., Moskva. Hagedorn, P., 1971. Die Umkehrung der Stabilit€atss€atze von Langrange–Dirichlet und Routh. Arch. Rational Mech. Anal. 42 (4), 281–316. € Hagedorn, P., 1976. Uber die Instabilit€at conservativer Systeme mit gyroskopischen Kr€aften. Arch. Rational Mech. Anal. 58 (1), 1–9.
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Liubushin, A.E., 1980. Ob neustojtchivosti ravnovesiya, kogda silovaya funktsiya ne est maksimum. Prikl. Mat. i Mekh. 44 (2), 211–220. Painleve, P., 1897. Lecßons sur l’integration des equations de la Mecanique, Paris. Pars, L.A., 1968. A Treatise on Analytical Dynamics. Heinemann, London.
Mechanics Research Communications 32 (2005) 265–280 www.elsevier.com/locate/mechrescom
Hagedorn’s theorem in some special cases of rheonomic systems c V.M. Covi
a,*
, M.M. Veskovic
b
a b
Faculty of Mechanical Engineering, University of Belgrade, 27 Marta 80, 11000 Belgrade, Serbia and Montenegro Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, 36000 Kraljevo, Serbia and Montenegro Available online 25 February 2004
Abstract Hagedorn’s theorem on instability [Arch. Rational Mech. Anal. 58 (1976) 1], deduced from Jacobi’s form of Hamilton’s principle, refers to scleronomic mechanical systems. In this paper we shall prove that Hagedorn’s methodology can be generalized to a class of rheonomic mechanical systems with differential equations of motion which allow the existence of Painleve’s integral of energy. The application of this methodology to the case of rheonomic systems which allow, together with Painleve’s integral, cyclic integrals, as well as to the mechanical systems having resultant motion, with prescribed transport motion, and, finally, to the systems having Mayer’s rheonomic potential, are also considered. Obtained results are illustrated by an example. 2004 Elsevier Ltd. All rights reserved. Keywords: Rheonomic systems; Stability of motion; Variational method; Hagedorn’s theorem
1. Introduction In order to apply Hagedorn’s variational methodology, used analysing the instability of motion of a scleronomic system (see (Hagedorn, 1976)), in this paper are identified the classes of rheonomic systems for which the differential equations of motion can be deduced from a variational principle of Hamilton’s type, with the condition in the form of the Painleve integral. This problem is solved by finding special classes of rheonomic mechanical systems (in Lagrangian function of the system time appears explicitly either through the rheonomic holonomic constraints, or by means of the given equations of motion with respect to the part of generalized coordinates of the system, or, finally, thanks to the rheonomic field of potential forces and forces having rheonomic Mayer’s potential)
*
Corresponding author. Fax: +381-11-3246-382. c). E-mail address: [email protected] (V.M. Covi
0093-6413/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2004.02.009
266
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for which we establish the conditions providing the existence of the Painleve integral of energy. By analogy with the case of the Jacobi integral, this integral (which, in contrast to the scleronomic case, contains also the potential of rheonomic constraints) transforms Hamilton’s principle to the form which is equivalent to Jacobi’s principle. By this the connection with Hagedorn’s variational methodology is established. Section 2 of the paper are exposed to the conditions of existence of the Painleve integral of energy in the case of the motion of a mechanical system in a potential rheonomic field forces, this motion being subject to ideal holonomic rheonomic constraints. It is shown that in such a case the Painleve integral does not depend explicitly on time and that, as the consequence of the existence of that integral, the differential equations of motion can be deduced from the Hamilton principle in Jacobi’s form, which corresponds to a Lagrangian function which is guage invariant with respect to the original one. In this new Lagrangian function the potential of rheonomic constraints is added to the potential forces. The fact that, in the case considered, Hamilton’s principle can be represented in Jacobi’s form, made it possible to formulate the theorem on instability (Theorem 1) of the trivial solution of differential equations of motion of the rheonomic system considered, whose proof is based on Hagedorn’s variational methodology applied to the case of scleronomic systems. Section 3 is considered with the case of relative motion of a mechanical system, with given transport motion (which makes the system rheonomic), relative motion being restricted by the presence of scleronomic constraints. The condition for the existence of the Painleve integral in this case reduces to the condition that the given Pfaff form is a perfect differential of the function of coordinates and time. If this condition is satisfied, the problem of the application of Hagedorn’s variational methodology to the stability of motion reduces to the one considered in Section 2. In Section 4, the conditions for the existence of the Painleve integral of energy in the case of a mechanical system moving in the rheonomic field of potential forces and Lorentz forces are founded. Also, in this part of the paper the case of a mechanical system moving in the field of forces for which Mayer’s potential can be determined is considered. In both cases the condition for the existence of the Painleve integral reduces to the condition that the given Pfaff form is a perfect differential of the function depending on coordinates and time. Section 5 is considered with the mechanical system with a rheonomic Lagrangian function containing a term linear with respect to generalized velocities. It is assumed that the differential equations of motion of that system allow the existence of the Painleve integral and of a class of integrals linear with respect to the generalized velocities. It is shown that the linear integrals are cyclic ones in the case when, in accordance with the conditions of existence of the Painleve integral, the differential equations are obtained from the Lagrangian function, gauge invariant with respect to the original one. It is proved that, at the difference from the classical case, under the given conditions the existence of cyclic integrals is possible also when the original Lagrangian function depends on cyclic coordinates. By applying Routh’s method of ignoring the cyclic coordinates, the differential equations of motion of the system are formed with respect to positional coordinates, demonstrating that the existence of the Painleve integral allows the substitution of the original Routh function by the guage invariant one. In this last function, at the difference from the classical case, the Routh–Painleve potential is present. Further, it is shown that it is also possible, in this case, to formulate the Hamilton principle in Jacobi’s form, which yields Routh’s equations. This introduces the Hagedorn variational methodology in the study of instability of the steady motion of the system (Theorem 2 is formulated), allowed by its differential equations of motion. Finally, by introducing a new explicit form of the relation between Routh’s function and Lagrange’s function, a form of Lagrange’s function independent from time is found, allowing the formulation of a Hagedorn’s tipe theorem (Theorem 3) on the instability of steady motion with respect to positional coordinates. In Section 6 the results are illustrated by an example of steady motion of a mechanical system with rheonomic constraints, in a rheonomic field of potential forces.
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2. The case of mechanical systems with rheonomic constraints Hagedorn’s theorem establishes that, in the case of the motion of a discrete mechanical system in a potential field of forces, subject to holonomic scleronomic ideal constraints, the trivial solution q ¼ 0 of the differential equations of motion is instable if these equations allow the existence of generalized Jacobi’s energy integral, and if, in the position q ¼ 0, the part (H0 ) of the Hamiltonian function independent of the generalized momentum components, has a strict maximum. It has further been shown (Liubushin, 1980) that this theorem, in the case when that part of the Hamiltonian function of the system considered has a non strict maximum at q ¼ 0, still holds. Generalization of Hagedorn’s variational method to the class of mechanical systems subject to the rheonomic holonomic ideal constraints is presented in this paper. Let us consider a mechanical system moving in the field of potential forces, subject to ideal holonomic rheonomic constraints. Let the configuration of the system be defined by the set of Lagrangian coordinates q ¼ ðq1 ; q2 ; . . . ; qn Þ. The Lagrangian function of the system considered has, in the general case, the form (Pars, 1968) _ tÞ þ T1 ðq; q; _ tÞ þ T0 ðq; tÞ Pðq; tÞ; _ tÞ ¼ T2 ðq; q; Lðq; q; where
ð1Þ
1
1 _ tÞ ¼ aij q_ i q_ j ; T1 ¼ ai q_ i T2 ðq; q; ð2Þ 2 and where T ¼ T2 þ T1 þ T0 is the kinetic energy of the system, P––its potential energy, and q_ ¼ ðq_ 1 ; q_ 2 ; . . . ; q_ n Þ––the generalized velocities of the system. Let the conditions aij ðq; tÞ; ai ðq; tÞ 2 C2 ;
T0 ðq; tÞ; Pðq; tÞ 2 C2 ;
hold. The differential equations of motion of the system d oL oL i ¼ 0; dt oq_ i oq
ð3Þ
have as a consequence the well-known relation dðT2 þ P T0 Þ ¼
oL dt: ot
ð4Þ
If the condition oL dt ¼ dV ðq; tÞ; ot
V ðq; tÞ 2 C2 ;
ð5Þ
holds, Eq. (3) lead, by (4), to the first quadratic integral of the differential equations of motion, known as Painleve’s integral (see Painleve, 1897, p. 89; Appell, 1911, p. 288): T2 þ P T0 þ V ¼ h;
h ¼ const:
ð6Þ
Let us further show that time t does not explicitly appear in Painleve’s integral. If (5) holds, we have 1 oaij i j oai i o oV oV dt; q_ dq þ dq ðP T0 Þ dt i dqi þ 2 ot ot oq ot ot
ð7Þ
1 Einstein summation convention is used throughout the paper. Unless the different is stated explicitly, indices take the following values: i; j; k ¼ 1; . . . ; n; a; b ¼ 1; . . . ; m; m; q; p; h ¼ m þ 1; . . . ; n.
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wherefrom oaij ¼ 0 ! aij ¼ aij ðqÞ; ot
ð8Þ
oai oV ¼ i; oq ot
ð9Þ
oðP T0 Þ oV ¼ : ot ot
ð10Þ
The last equation leads to oðP þ V T0 Þ ¼ 0 ! ðP þ V T0 Þ ¼ UðqÞ: ot
ð11Þ
Relations (8) and (11) confirm that time t does not appear explicitly in the integral (6). Let us notice that Painleve’s integral differs from the Jacobi generalized integral of energy by the presence of the function V ¼ V ðq; tÞ in the dynamical potential of the system. Let us further show that, if Painleve’s integral exists, it is always possible to substitute the original Lagrangian function (1) by the corresponding gauge invariant one, whose dynamical potential (the part of the Lagrangian function which does not depend on the generalized velocities) includes the function V ¼ V ðq; tÞ, generated by a part of the original Lagrangian function which is linear with respect to the generalized velocities and which explicitly depends on time. This function we call the potential of rheonomic constraints. By the relation (9) one obtains ai ¼ ~ ai ðqÞ þ ai ðq; tÞ;
ð12Þ
where ai ¼
Z
or ai ¼
o oqi
oV ðq; tÞ dt; oqi Z
V ðq; tÞ dt:
The last relation leads to Z Z o d V ðq; tÞ dt V ; ai q_ i ¼ i V ðq; tÞ dt q_ i ! ai q_ i ¼ oq dt
ð13Þ
ð14Þ
ð15Þ
wherefrom, taking into account (12), the expression (1) for the Lagrangian function can be written in the form Z 1 d L ¼ aij q_ i q_ j þ ~ V ðq; tÞ dt ðP þ V T0 Þ: ai ðqÞq_ i þ ð16Þ 2 dt The Lagrangian function gauge invariant to (1) has, in virtue of the above expression, the form 1 e L ¼ aij q_ i q_ j þ ~ ai q_ i ðP þ V T0 Þ; 2 which allows the existence of an energy integral identical to Painleve’s integral (6).
ð17Þ
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If the condition oðP þ V T0 Þ ¼0 oqi in the position q ¼ 0 holds, the differential equations of motion (3) have the trivial solution q ¼ 0. Theorem 1. If the motion of a mechanical system in the field of potential forces is subject to the rheonomic ideal constraints, while its differential equations (3) allow the existence of Painleve’s integral (6) and have trivial e 0 of the Hamiltonian function independent solution q ¼ 0, this solution will be unstable if, at q ¼ 0, the part H from generalized momentum components, e 0 ¼ 1 aij ~ H aj þ ðP þ V T0 Þ; ai ; ~ 2
ð18Þ
which corresponds to the Lagrangian function (17) gauge invariant to the original Lagrangian function (1), has a strict maximum. Proof. After substituting (17) to (1), it is obvious that, in the case of the system considered, Hamilton’s principle, in virtue of (6), can be written in the Jacobi form: Z t1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i j 2½h ðP þ V T0 Þaij q_ q_ þ ~ ð19Þ ai q_ dt ! inf : t0
By solving (19) we obtain the differential equations of motion of the system considered in which, obviously, time t does not appear explicitly. The remaining part of the proof coincides with the proof given in (Hagedorn, 1976). h Note. Let us notice that the coordinates aij (see (18)) satisfy aij ajk ¼ dik ;
i ¼ k ! dik ¼ 1;
i 6¼ k ! dik ¼ 0;
where dik denotes the Kronecker symbol. Let us also remark that the condition (see Hagedorn, 1976, case V 0) ðP þ V T0 Þðq¼0Þ ¼ 0;
ð20Þ
implies (see (18) and (19)) ~ ai ðq ¼ 0Þ ¼ 0:
ð21Þ
The condition (20) can be obtained by an appropriate choice of the level of the zero potential, while the condition (21), if it is not fulfilled, can be reached by substitute the Lagrangian function (17) for its gauge invariant Lagrangian function: e L)e L~ ai ðq ¼ 0Þq_ i :
ð22Þ
3. The case of the relative motion with a given transport motion Let us now consider the case of the motion of a mechanical system subject to the scleronomic constraints, with Lagrangian function in which time t appears explicitly. Such a situation arises, e.g., in the case of mechanical systems having relative motion with scleronomic constraints imposed to it, and with the equations of transport motion which are known functions of time. For such a motion of a mechanical system the condition (8) for existence of Painleve’s integral is satisfied always.
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Let us consider a system of points M1 ; . . . ; MN , with masses m1 ; . . . ; mN respectively, which moves relatively with respect to the coordinate system Angf, this motion being subject to the ideal holonomic constraints of the form f r ðn1 ; . . . ; nN ; g1 ; . . . ; gN ; f1 ; . . . ; fN Þ ¼ 0;
r ¼ 1; . . . ; k < 3N ;
ð23Þ
where ni , gi , fi are Cartesian rectangular coordinates of Mi , with respect to Angf. If the transformations of the coordinates ðn ¼ 3N kÞ ni ¼ ni ðq1 ; . . . ; qn Þ;
gi ¼ gi ðq1 ; . . . ; qn Þ;
fi ¼ fi ðq1 ; . . . ; qn Þ;
make that (23) become identities, then the coordinates ðq1 ; . . . ; qn Þ are independent (Lagrangian) and completely determine the position of the system considered with respect to Angf. Taking into account that the equations of the transport motion are given, it is obvious that the velocity of the point A as well as the ~ of the transport motion, are known functions of time, too: angular velocity x ~ vA ¼ ~ vA ðtÞ;
~¼x ~ðtÞ: x
ð24Þ
The kinetic energy of this system has the form N N X 1X onk onk ogk ogk ofk ofk onk og i j _ _ mk þ m q þ þ q þ vg ki T ¼ k vn i 2 k¼1 oqi oqj oqi oqj oqi oqj oq oq k¼1 of of og on of og on þ vf ki þ xn gk ki fk ki þ xg fk ki nk ki þ xf nk ki gk ki q_ i oq oq oq oq oq oq oq N X 1 1 þ mk ðen nk þ eg gk þ ef fk Þ þ xn ðJn xn Jng xg Jnf xf Þ þ xg ð Jgn xn þ Jg xg Jgf xf Þ 2 2 k¼1 1 ð25Þ þ xf ð Jfn xn Jfg xg þ Jf xf Þ ; 2 where vn ¼ vn ðtÞ, vg ¼ vg ðtÞ, vf ¼ vf ðtÞ; xn ¼ xn ðtÞ, xg ¼ xg ðtÞ, xf ¼ xf ðtÞ; en ¼ en ðtÞ, en ¼ en ðtÞ, ~, ~ ~ on axes An , Ag , Af respectively, and Jn , Jg , Jf , en ¼ en ðtÞ––are the projections of vectors ~ vA , x e ¼~ vA x Jng , Jgf , Jfn ––the coordinates of the tensor of inertia of the mechanical system, calculated with respect to Angf, which are functions of Lagrangian coordinates. It is obvious that the condition (8) is always satisfied in the case considered (see the expression for the part of the kinetic energy, appearing in (25), which represents quadratic form with respect to the generalized velocities). Taking into consideration the last remark, the necessary and sufficient conditions for the existence of Painleve’s integral are satisfied in this case if the following Pfaff’s form ! N X oP ~r þ~ ~ Pf ¼ ~ v_A K e_ mk~ e JA x dt ð26Þ qk þ~ e ~ LAr þ ~ ot k¼1 is a total differential of the function depending on time and on the Lagrangian coordinates. In the last ~r ––linear momentum of the system considered, corresponding expression the following quantities appear: K to its relative motion; LAr ––the angular momentum corresponding to the relative motion of the system, taken with respect to the origin A; JA ––the tensor of inertia of the mechanical system, taken with respect to Angf; ~ e––the angular acceleration vector of the transport motion; ~ v_A , ~ e_ ––the relative derivatives with respect to time of vectors ~ vA and ~ e; P ¼ Pðq; tÞ––the potential energy of the system considered; ~ qi ––radius vector of the material point Mi with respect to A. Let us remark that the coordinates of all vectors in that expression are taken with respect to Angf.
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So, if the condition (26) is satisfied, it is possible to apply Hagedorn’s methodology using the considerations identical with the ones presented in Section 1 of the paper.
4. Mayer’s rheonomic potential Lagrangian function can include a part linear with respect to generalized velocities, which depend on time explicitly, in other cases of motion of the system subject to the scleronomic constraints, too. Such a case appears when a mechanical system moves in a field forces having the structure of the Lorentz force (see Pars, 1968, p. 184; Gantmaher, 1960, p. 57). In the case of a material point that force has the form o~ Aðx; y; z; tÞ ~ F ¼ þ~ v rot~ Aðx; y; z; tÞ; ~ Aðx; y; z; tÞ 2 C2 ; ð27Þ ot where ~ v denotes the velocity of the point, and where x,y,z are the Cartesian rectangular coordinates of the point. The force (27) can be presented by the generalized (Mayer’s) potential VM ¼ ~ A ~ v;
ð28Þ
as follows (Gantmaher, 1960) d oVM oVM ; X ¼ þ dt o_x ox
Y ¼
d dt
oVM o_y
þ
oVM ; oy
Z¼
d dt
oVM o_z
þ
oVM ; oz
ð29Þ
where ~ F ¼ ðX ; Y ; ZÞ:
ð30Þ
For the motion of a scleronomic mechanical system in the field forces (27) and in the rheonomic field of potential forces, the Lagrangian function has the form ðVM ¼ ai ðq; tÞq_ i Þ: 1 L ¼ aij ðqÞq_ i q_ j þ ai ðq; tÞq_ i Pðq; tÞ; 2
ð31Þ
which corresponds to the class of rheonomic mechanical systems satisfying the condition (8) for existence of Painleve’s integral. So, having in mind the foregoing considerations, the necessary and sufficient conditions for the existence of Painleve’s integral are satisfied in this case if following Pfaff’s form (see (9) and (10)) Pf0 ¼
oai i oP dt dq ot ot
ð32Þ
c is a total differential of the function depending on time and on the Lagrangian coordinates. Also, in (Covi and Lukacevic, 1997) it is shown that in the case of nonconservative generalized forces of the form ð0Þ
Qi ¼ bij ðq; tÞq_ j þ binþ1 ðq; tÞ;
ð33Þ
for which the conditions ð0Þ
ð0Þ
bij ðq; tÞ ¼ bji ðq; tÞ; ð0Þ ð0Þ ð0Þ obsp obrp obrs þ ¼ ; p r oq oq oqs
ð34Þ
r ¼ 1; . . . ; n 2;
s ¼ i þ 1; . . . ; n 1;
p ¼ j þ 1; . . . ; n;
ð35Þ
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272 ð0Þ
ð0Þ
ð0Þ ob ob obrs þ snþ1 ¼ rnþ1 ; r ot oq oqs
r ¼ 1; . . . ; n 1;
s ¼ i þ 1; . . . ; n 1
ð36Þ
hold, Mayer’s rheonomic potential can always be found in the form ! Z q1 Z qi1 Z q2 n n Z X X ð0Þ ð1Þ 1 2 i2 i1 i VM ¼ b1i dq þ b2i dq þ þ bi1i dq q_ þ i¼2
k1
k2
ki1
i¼1
qi
ði1Þ
binþ1 dqi ;
ð37Þ
ki
where ðiÞ
ð0Þ
biþ1j ¼ biþ1j ðq1 ¼ k1 ; q2 ¼ k2 ; . . . ; qi ¼ ki ; qiþ1 ; . . . ; qn ; tÞ;
k1 ; . . . ; ki ¼ const:;
i ¼ 1; . . . ; n 1;
j ¼ i þ 2; . . . ; n: This leads to the conclusion that, in the case of the motion of a scleronomic system in the field forces (33) satisfying the conditions (34)–(36), the Lagrangian function has the form corresponding to the rheonomic systems: 1 L ¼ aij ðqÞq_ i q_ j þ ax ðq; tÞq_ x Pðq; tÞ; 2
x ¼ 2; . . . ; n:
ð38Þ
It is obvious that this case, in further considerations of application of Hagedorn’s methodology, reduces to the previous one (see (31)).
5. The case of the steady motion of a system If the expression for the Lagrangian function has the form corresponding to the systems subject to the rheonomic constraints, it will also hold its rheonomic form if, in the case of existence of cyclic integrals, one applies Routh’s method of ignoration of cyclic coordinates. Let us further consider the motion of mechanical system with rheonomic Lagrangian function _ tÞ ¼ T2 ðq0 ; q; _ tÞ þ T1 ðq0 ; q; _ tÞ þ T0 ðq0 ; tÞ Pðq0 ; tÞ; Lðq; q;
ð39Þ
in which q0 ¼ ðq1 ; q2 ; . . . ; qm Þ denotes the set of the positional coordinates. To the cyclic coordinates q00 ¼ ðqmþ1 ; qmþ2 ; . . . ; qn Þ correspond the cyclic integrals oL ¼ cq ; oq_ q
cq ¼ const:;
ð40Þ
or aqi q_ i þ aq ¼ cq :
ð41Þ
However, in the case of existence of Painleve’s integral, the existence of the first integrals of the form (41) is possible, under the fixed conditions, in the case when coordinates q00 appear in the Lagrangian function. This statement can be proved as follows. Let the Lagrangian function of a mechanical system which differential equations allow Painleve’s integral is given in the form 1 _ tÞ ¼ aij ðq0 Þq_ i q_ j þ ai ðq0 ; q00 ; tÞq_ i þ T0 ðq0 ; q00 ; tÞ Pðq0 ; q00 ; tÞ; Lðq; q; 2
ð42Þ
while the condition ai ðq0 Þ þ ai ðq0 ; q00 ; tÞ: ai ðq0 ; q00 ; tÞ ¼ ~
ð43Þ
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holds. The differential equations of motion corresponding to the coordinates q00 have the form d oa o ðaqi q_ i þ ~ aq þ aq Þ qi q_ i þ q ðP T0 Þ ¼ 0; dt oq oq wherefrom, having in mind the relations Z Z oaq o2 oai o2 V ðq; tÞ dt; V ðq; tÞ dt; ¼ ¼ oqi oqi oqq oqq oqq oqi
ð44Þ
oaq oV ¼ q; oq ot
ð45Þ
we obtain d o ðaqi q_ i þ ~ aq Þ þ q ðP þ V T0 Þ ¼ 0: dt oq
ð46Þ
Then, under the condition that o ðP þ V T0 Þ ¼ 0 oqq
ð47Þ
holds, the following integrals result: aqi q_ i þ ~ aq ¼ c q :
ð48Þ
Thus, in the case when the differential equations of motion allow the existence of the Painleve energy integral, the existence of cyclic integrals is possible even in the case when coordinates q00 , to which correspond the cyclic integrals (48), satisfy the condition oL 6¼ 0: ð49Þ oqq In the case considered the existence of Painleve’s integral, as mentioned, allows to substitute the original Lagrangian function (42) by its gauge invariant Lagrangian function which, with (47) satisfied, has the form 1 e _ tÞ ¼ aij ðq0 Þq_ i q_ j þ ~ Lðq; q; ai ðq0 Þq_ i ðP þ V T0 Þ; ð50Þ 2 which leads to the cyclic integrals (48). Let us remark that, with respect to the Lagrangian function (50), the concept of the cyclic coordinate has its usual meaning. The complete system of the differential equations of motion of the mechanical system considered is given by the cyclic integrals (48) and, in accordance with Routh’s method of ignoration of cyclic coordinates, by the equations ! e e d oR oR ð51Þ i ¼ 0; i dt oq_ oq e is given by the expression (the case of existence of Painleve’s integral) where the Routh function R e¼e ð52Þ R L cq q_ q ; in which the cyclic velocities have to be eliminated using the following relations, obtained from (41): ð53Þ q_ q ¼ bqm ðdm þ ama q_ a Þ; where bqm amh ¼ dqh ;
dm ¼ ~ am c m :
In that way one obtains the result 1 1 mq a b a e R ¼ bab q_ q_ þ ba q_ P þ V þ b dm dq T0 ; 2 2
ð54Þ
ð55Þ
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where bab ¼ aab bmq ama aqb ;
ba ¼ ~ aa bmq dm aqa :
ð56Þ
Let us notice that the expression (55) for the Routh function differs from the one related to scleronomic system (see e.g. Hagedorn, 1971) by the presence of the ‘‘rheonomic potential’’ V in the expression of Routh’s potential of the system. It is the reason for which the potential e ¼ P þ V þ 1 bmq dm dq T0 ; P 2
ð57Þ
we shall name the Routh–Painleve potential. In the case when the condition ! e oP ¼0 oqa 0
ð58Þ
ðq ¼0Þ
holds, the differential equations of motion (48) and (51) allow steady motion of the form qa ¼ 0;
qq ¼ qq0 þ q_ q0 ðt t0 Þ;
qq0 ; q_ q0 ; t0 ¼ const:
ð59Þ
Notice. From the fact that the Lagrangian function (50) is gauge invariant to the original Lagrangian function L, given by (42), it results that the Routh function (52) is gauge invariant to the original Routh function R ¼ L cq q_ q : The first relation in (59) represents the trivial solution of Eq. (51). Obviously (see (55), (56), (54), and (50)) e oR ¼ 0; ot
ð60Þ
holds, which leads to the conclusion that the differential equations of motion have a quadratic first integral (the integral of energy in which the Routh–Painleve potential appears) of the form 1 e ¼ h; bab q_ a q_ b þ P 2
h ¼ const:;
ð61Þ
which, in fact, represents Painleve’s integral (6) in which the cyclic velocities q_ 00 are excluded in virtue of (53). From foregoing consideration it follows that the differential equations (51) can be deduced from the Jacobi form of Hamilton’s principle, too, i.e. from Z
t1 t0
ffi ) (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 mq a a b 2 h P þ V þ b dm dq T0 bab q_ q_ þ ba q_ dt ! inf : 2
ð62Þ
On the basis of the previous considerations we formulate the following Theorem 2. If, for the mechanical system considered, the following conditions are fulfilled: • Lagrangian function of the system has the form (42), whereas the condition (43) hold; • the differential equations of motion (51) of the system allow the existence of Painleve’s integral (6) and trivial solution q0 ¼ 0;
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• dynamic potential of the system, including also potential V of the rheonomic constraints, does not depend on the coordinates q00 (condition (47) satisfied); the trivial solution will be unstable if the part of Hamilton’s function, constructed over the Routh function (55) which is gauge invariant to the original Routh function, which does not depend on the generalized momentum components, has a strict maximum in the position q0 ¼ 0, i.e. if the condition 1 ab 1 b ba bb þ P þ V þ bmq dm dq T0 < 0; 2 2
8q0 6¼ 0;
ð63Þ
holds, while it is (see (56)) bab bbc ¼ dac :
ð64Þ
Proof. It was demonstrated that, in the case of the existence of Painleve’s integral, the Lagrangian function (42), together with (43) and (47), generates the cyclic integrals (48). Also, in this case it is possible to replace the original Lagrangian function by its gauge invariant one, given by (50), to which corresponds the Routh function (55), which is gauge invariant to the original one. The function (50) permits, thanks to (60) and (61), that the differential equations of motion of the system considered, corresponding to the positional coordinates, can be deduced from the Jacobi form of the Hamilton’s principle (62). The remaining part of the proof coincides with the proof given in (Hagedorn, 1976). h We notice that in the case of scleronomic systems considered in (Hagedorn, 1976), a criterion, obtained if one puts in (63) V 0; holds. In order that the criterion (63) can be applied, the conditions 1 mq ba ðq ¼ 0Þ 0; P þ V þ b dm dq T0 0; 2 ðq¼0Þ
ð65Þ
ð66Þ
must be fulfilled. The second condition in the last expression is trivially satisfied by subtracting from the Routh–Painleve potential its value at the point q0 ¼ 0. In the general case, however, first condition in (66) is not satisfied. In order to overcome this problem the Lagrangian function (50) can be substituted by its gauge invariant Lagrangian function 1 _ tÞ ¼ aij ðq0 Þq_ i q_ j þ ~ ai ðq0 Þq_ i ba ðq0 ¼ 0Þq_ a ðP þ V T0 Þ; L ðq; q; 2 which makes that the Routh function takes the form 1 1 mq a b a R ¼ bab q_ q_ þ ba q_ P þ V þ b dm dq T0 ; 2 2
ð67Þ
ð68Þ
where ba ¼ ba ðq0 Þ ba ðq0 ¼ 0Þ ! ba ðq0 ¼ 0Þ 0:
ð69Þ
The mentioned problem can be overcome in the following way, too. The Lagrangian function in the Routh form (55) can be transformed to the form
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e ¼ 1 bab q_ a q_ b þ d ðba qa Þ R 2 dt
obb b a 1 mq _ q P þ V þ d d T b q m q 0 : 2 oqa
ð70Þ
Removing the second term at the right-hand side in the last relation, we obtain again the relation (68) in which we have in this case ba ¼
obb b q ; oqa
ð71Þ
where ba ðq0 ¼ 0Þ ¼ 0:
ð72Þ
With such modifications of the Lagrangian function in the Routh form, Hagedorn’s criterion of instability of the trivial solution q0 ¼ 0 of differential equations of motion (51) (which preserve their form after the substitution of (55) by (68) or (71)) gets the form 1 ab 1 b ba bb þ P þ V þ bmq dm dq T0 < 0 2 2
8q0 6¼ 0:
ð73Þ
Let us show now that the tensor bab of the reduced system (see (68)) represents the inverse of the matrix obtained by omitting in the tensor aij the rows and columns corresponding to the cyclic coordinates, i.e. let us show that (see (64)) bab ¼ aab :
ð74Þ
holds. Indeed, as, in virtue of aij ajk ¼ dki ; it follows aab abc þ aaq aqa ¼ dca ;
aab abq þ aam amq ¼ 0;
ð75Þ
aqb aba þ aqm ama ¼ 0;
aqb abm þ aqp apm ¼ dmq ;
ð76Þ
one obtains (see (56) and (54)) ðaab bmq ama aqb Þabc ¼ dca ; ! bab abc ¼ dca ;
ð77Þ
which proves the validity of (74). Moreover, (75) and (76) lead to the relation amq ¼ bmq bmp apa aab abh bmh ;
ð78Þ
which, together with (75)–(77), to Hagedorn’s criterion (73) gives the form 1 ij a ð~ ai ci Þð~aj cj Þ þ P þ V T0 < 0; 2
8q0 6¼ 0;
ð79Þ
where (see (52)) ca ¼ ba ðq0 ¼ 0Þ and cm ––the value of the cyclic generalized momentum components (see (40)).
ð80Þ
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As the potential energy P is undeterminated to an additive constant, this constant has to be chosen so as to nullify the expression at the left-hand side of the inequality (79) in q0 ¼ 0. The criterion (79) gives the possibility to formulate the following Theorem 3. Under the conditions providing the validity of the Theorem 2, the trivial solution q0 ¼ 0 will be unstable if the part of Hamilton’s function corresponding to the Lagrangian function gauge invariant to (50), ¼e L L ci ; q_ i ; which does not depend on generalized momentum components, has in q0 ¼ 0 a strict maximum. Proof. It is obvious that the expression (79) represents Hamilton’s function which corresponds to the As (79) represents a new form of the criterion (63), the remaining part of the proof Lagrangian function L. is the same as the proof in Theorem 2. h Obviously, in this theorem one does not use the Routh method of ignoration of cyclic coordinates, although it refers to the instability with respect to positional coordinates. Notice. It is obvious that in Theorems 1–3 the condition for strict maximum of the part of Hamilton’s function, independent on the generalized momentum components, can be, having in mind the results from (Liubushin, 1980), replaced by the condition for non strict maximum. 6. Example The mechanical system consisting of two points M1 and M2 , each of mass m ¼ 1, moves in the field of potential forces with potential energy (x1 , y1 , z1 ; x2 , y2 , z2 ––Cartesian rectangular coordinates of the respective material points): 1 ðe1 Þ P ¼ ðz41 þ x42 Þ ðz61 þ x62 Þ þ ðz21 þ x22 Þ cos 2t þ z1 x2 sin 2t: 2 The motion of the system is subject to the rheonomic constraints given by y2 ¼ ðx22 þ z21 Þ cos t þ x2 z1 sin t þ cos t; z2 ¼ ðx22 þ z21 Þ sin t x2 z1 cos t sin t; x1 ¼ const:
ðe2 Þ
(a) Prove that the differential equations of motion of the system considered allow the existence of Painleve’s integral; (b) Determine the corresponding cyclic integral and examine the stability of the trivial solution of the equations of motion in the Routh form. Solution. The kinetic energy of the system considered has the form 2 1 X T ¼ m ð_x2 þ y_ i2 þ z_ 2i Þ; 2 i¼1 i
ðe3 Þ
which, if one introduces independent generalized coordinates q1 ¼ z 1 ;
q2 ¼ x 2 ;
q3 ¼ y1 ;
ðe4 Þ
can be written, in virtue of (e2 ), in the form 1 T ¼ aij ðqÞq_ i q_ j þ ~ ai ðqÞq_ i þ ai ðq; tÞq_ i þ T0 ðq; tÞ; 2
ðe5 Þ
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where a11 ¼ 1 þ 4ðq1 Þ2 þ ðq2 Þ2 ; 1 2
2 2
a22 ¼ 1 þ ðq Þ þ 4ðq Þ ; 2
1 2
2 2
~ a1 ¼ q ½ðq Þ ðq Þ ; a1
a12 ¼ 5q1 q2 ; a23 ¼ 0;
a13 ¼ 0;
a33 ¼ 1;
2 2
2 ~ a2 ¼ q ½ðq Þ ðq1 Þ ; 1
a2
~a3 ¼ 0;
ðe6 Þ
¼ q cos 2t 2q sin 2t; ¼ q cos 2t 2q sin 2t; a3 ¼ 0; 1 1 2 2 2 2 2 2 T0 ¼ ½ðq1 Þ þ ðq2 Þ þ ðq1 q2 Þ ½ðq1 Þ þ ðq2 Þ cos 2t q1 q2 sin 2t: 2 2 2
1
1
2
Hence oL dt ¼ f1 dq1 þ f2 dq2 þ f0 dt; ot where f1 ¼ ð2q2 sin 2t þ 4q1 cos 2tÞ; 2
f2 ¼ ð2q1 sin 2t þ 4q2 cos 2tÞ;
2
f0 ¼ 4½ðq1 Þ þ ðq2 Þ sin 2t 4q1 q2 cos 2t;
ðe7 Þ
so that, having in mind that the conditions of1 of2 ¼ ; oq2 oq1
of1 of0 ¼ 1; ot oq
of2 of0 ¼ 2; ot oq
ðe8 Þ
hold, one obtains oL dt ¼ dV ðq1 ; q2 ; tÞ; ot
ðe9 Þ
and 2
2
V ¼ 2bðq1 Þ þ ðq2 Þ c cos 2t 2q1 q2 sin 2t:
ðe10 Þ
In virtue of (4) and (5) one obtains Painleve’s integral (see (6)): 1 1 2 2 2 2 2 2 ½1 þ 4ðq1 Þ þ ðq2 Þ ðq_ 1 Þ þ ½1 þ ðq1 Þ þ 4ðq2 Þ ðq_ 2 Þ þ 5q1 q2 q_ 1 q_ 2 2 2 1 3 2 6 6 2 2 þ ðq_ 3 Þ ðq1 Þ ðq2 Þ ðq1 Þ ðq2 Þ ¼ h: 2 2
ðe11 Þ
To the cyclic coordinate q3 corresponds the cyclic integral q_ 3 ¼ q_ 30 ;
q_ 30 ¼ const:;
ðe12 Þ
with the help of which, using (50) and (52), we form the expression for the Lagrangian function in the Routh form (we omit the constant terms): e ¼ 1 ½1 þ 4ðq1 Þ2 þ ðq2 Þ2 ðq_ 1 Þ2 þ ð5q1 q2 Þq_ 1 q_ 2 þ 1 ½1 þ 4ðq1 Þ2 þ ðq2 Þ2 ðq_ 2 Þ2 þ q2 ½ðq1 Þ2 ðq2 Þ2 q_ 1 R 2 2 3 1 2 2 2 1 2 2 1 2 2 1 6 2 6 þ q ½ðq Þ ðq Þ q_ þ ðq Þ þ ðq Þ þ ðq Þ ðq Þ : 2
ðe13 Þ
It follows, by the last expression, that the Routh-Painleve potential is (see (57)) e ¼ ðq1 Þ6 ðq2 Þ6 3 ðq1 Þ2 ðq2 Þ2 ; P 2
ðe14 Þ
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and that it has a strict maximum with respect to positional coordinates in q1 ¼ q2 ¼ 0. Thus, the conditions for existence of the following steady motion q1 ¼ 0;
q2 ¼ 0;
q3 ¼ q_ 30 t þ q30 ;
q30 ¼ const:;
ðe15 Þ
are fulfilled, and these conditions obviously do not depend on the value of the cyclic momentum. In order to apply the criterion (79), we find the coordinates of the tensor aij : a22 a12 a11 a11 ¼ ; a12 ¼ ; a13 ¼ 0 a22 ¼ ; a23 ¼ 0; a33 ¼ 1 D D D
ðe16 Þ
where 4
2
4
2
2
2
D ¼ 1 þ 4ðq1 Þ þ 5ðq1 Þ þ 4ðq2 Þ þ 5ðq2 Þ 8ðq1 Þ ðq1 Þ :
ðe17 Þ
Finally, it follows that the left-hand side of (79), multiplied by a positive quantity D, has the value 1 1 2 2 a22 ð~ a1 Þ þ a11 ð~ a2 Þ a12 a1 a2 þ DðP þ V T0 Þ 2 2 3 1 1 ¼ ðq1 Þ2 ðq2 Þ2 ðq1 Þ6 ðq2 Þ6 8ðq1 Þ4 ðq2 Þ2 8ðq1 Þ2 ðq2 Þ2 9ðq1 Þ6 ðq2 Þ2 9ðq1 Þ2 ðq2 Þ6 2 2 2 1 4 2 4 1 6 2 4 þ 4ðq Þ ðq Þ 4ðq Þ ðq Þ 4ðq1 Þ4 ðq2 Þ6 3ðq1 Þ8 3ðq2 Þ8 þ 8ðq1 Þ8 ðq2 Þ2 þ 8ðq1 Þ2 ðq2 Þ8 10
10
4ðq1 Þ 4ðq2 Þ
ðe18 Þ
wherefrom we conclude that the criterion (79) of instability of the trivial solution q1 ¼ q2 ¼ 0 of the differential equations of motion of the system considered d dt
e oR oq_ 1
!
e oR 1 ¼ 0; oq
d dt
e oR oq_ 2
!
e oR ¼ 0; oq2
ðe19 Þ
is satisfied.
7. Conclusion The main idea of this paper refers to the enlargement of Hagedorn’s variational approach in stability of motion of mechanical systems to the case of rheonomic systems. This idea is carried out in several cases of rheonomic systems, while the results obtained are formulated in the form of three theorems. By an example referring to the steady motion of the mechanical system the results are illustrated.
References Appell, P., 1911. Traite de Mecanique Rationnelle, T II, Dynamique des Systemes––Mecanique Analitique. Gauthier-Villar, Paris. c, V., Lukacevic, M., 1997. On the existence of Mayer’s potential. J. Appl. Mech. 64 (3). Covi Gantmaher, F.R., 1960. Lektsii po analititcheskoj mekhanike, Gos. izdat. fiz.-mat. lit., Moskva. Hagedorn, P., 1971. Die Umkehrung der Stabilit€atss€atze von Langrange–Dirichlet und Routh. Arch. Rational Mech. Anal. 42 (4), 281–316. € Hagedorn, P., 1976. Uber die Instabilit€at conservativer Systeme mit gyroskopischen Kr€aften. Arch. Rational Mech. Anal. 58 (1), 1–9.
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Liubushin, A.E., 1980. Ob neustojtchivosti ravnovesiya, kogda silovaya funktsiya ne est maksimum. Prikl. Mat. i Mekh. 44 (2), 211–220. Painleve, P., 1897. Lecßons sur l’integration des equations de la Mecanique, Paris. Pars, L.A., 1968. A Treatise on Analytical Dynamics. Heinemann, London.