Motion of Nonholonomous Rheonomous Systems in the Lagrangian Formalism

Journal of Applied Mathematics and Physics, 2015, 3, 295-309 Published Online March 2015 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2015.33043 Motion of Nonholonomous Rheonomous Systems in the Lagrangian Formalism F. Talamucci DIMAI, Dipartimento di Matematica e Informatica “Ulisse Dini”, Università Degli Studi di Firenze, Florence, Italy Email: federico.talamucci@math.unifi.it Received 4 March 2015; accepted 19 March 2015; published 23 March 2015 Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract The main purpose of the paper consists in illustrating a procedure for expressing the equations of motion for a general time-dependent constrained system. Constraints are both of geometrical and differential type. The use of quasi-velocities as variables of the mathematical problem opens the possibility of incorporating some remarkable and classic cases of equations of motion. Afterwards, the scheme of equations is implemented for a pair of substantial examples, which are presented in a double version, acting either as a scleronomic system and as a rheonomic system. Keywords Nonholonomous Systems, Rheonomic Constraints, Quasi-Velocites, Appell and Boltzmann-Hamel Equations 1. Introduction Nonholonomous systems are beyond a doubt more and more considered, mainly in view of the important implementations they exhibit for mechanical models. From the mathematical point of view, the draft of the equations for such systems commonly matches the introduction of the quasi-velocities and, starting from the Euler-Poincaré equations [1], several sets of equations have been formulated. The time-dependent case is probably more disregarded in literature: we direct here our attention especially to rheonomic systems, admitting the holonomic and nonholonomic constraints and the applied forces to depend explicitly on time. The nonholonomous restrictions are assumed to be linear, so that the equations of motion can be written in the linear space of the admissible displacements of the system, eliminating the Lagrangian multipliers connected to the constraints. How to cite this paper: Talamucci, F. (2015) Motion of Nonholonomous Rheonomous Systems in the Lagrangian Formalism. Journal of Applied Mathematics and Physics, 3, 295-309. http://dx.doi.org/10.4236/jamp.2015.33043 F. Talamucci If on the one hand the use of quasi-velocities formally complicates calculations, on the other hand the final form of the system allows computing the equations merely by means of a list of particular matrices, once the Lagrangian function has been written and the quasi-velocities have been chosen. We pay attention to keep separated the various contributions to the mobility of the system; the customary stationary case can be easily recovered from the general equations we will write. An energy balance-type equation, which will be proposed in terms of the quasi-velocities, affirms the conservation of the energy in the full stationary case and shows the contributions of the different terms in the rheonomic context. We will conclude by presenting some applications of the developed system of equations. Most of the formal notation used onward is explained just below. For a given a list of variables y = ( y1 , , yn ) ,  ∂f ∂f  , , the operator ∇ y will compute the gradient ∇ y f =   of a scalar funcion f , and J y calculates ∂yn   ∂y1 ∂v the m × n Jacobian matrix of a vector v ( y ) = ( v1 ( y ) , , vm ( y ) ) : ( J y v ) = i , i = 1, , m , j = 1, , n . i, j ∂y j Anywhere, vectors are in bold type and are meant as columns: row vectors will be written by means of the 0 T . Moreover, 0n is the null column vector   ∈  n , n , m is the n × m null matrix, transposition symbol 0   n the n × m null matrix and  n the unit matrix of size n . 2. Modelling the System The theoretical frame we point and expand is contained in [2]. Let us consider a system of n point particles ( P1 , m1 ) ,  , ( Pn , mn ) restricted both by µ geometrical constraints and by ν kinematic constraints, µ ≥ 0 , ν ≥ 0 , µ + ν < 3n : Y ( X,t ) = 0 µ (1)  + G ( X, t ) = 0ν  ( X, t ) X 1 (2) fixed t, Y (Y1 , , Yµ ) :  3n →  µ ,  where X ∈  3n is the representative vector of the system and, for each = ν is a matrix of size 3n ×ν , G1 a vector in  . The constraint equations are assumed to be independent: = rank J X Y µ= , rank  ν (3) We first make use of the ν integer relations (1) in order to write the system configuration by means of the = q ( q1 , , q ) ∈ ⊆   , = parametrisation X = X ( q,t ) , where  3n − µ are the local Lagrangian coordiX ∂  ( J X ) q + agrees with (1), but it must be consistent also with the nates. The velocity of the system X = q ∂t differential constraints (2) which are rewritten, in terms of the Lagrangian coordinates q and of the generalized velocities q , as α ( q, t ) q + β ( q, t )= 0ν , α ( q, t )=  ( X ( q, t ) , t ) ( J q X ) ∈ ν × , β ( q, t )=  ∂X + G1 ∈ ν ∂t (4)  − F − Φ =0 , and β = 0ν in case of fixed constraints. The dynamics of the system is summarized in  3n by Q 3n where Q = ( mS P1 , , mn Pn ) represents the momentum of the system, F , Φ ∈  3n respectively all active forces and all constraint reactions (the i-th triplet concerning Pi ). The virtual displacements of the system at ˆ ∈  3n such that [2] each time t and at each position X are the vectors in X ˆ = 0  J Y ( X, t ) X  X µ  ˆ  =0  ( X, t ) X ν 296 (5) F. Talamucci giving in each X , t the 3n − ( µ + ν ) dimensional linear space (  = span ∇ XY1 , , ∇ XYµ , 1 , , ν ) ⊥ ⊂  3n , ˆ = 0 where 1 , , ν are the rows of  . At the same time, the assumption of smooth constraints Φ ⋅ X ˆ  ∀ X ∈  make us write µ ν ∑λ j ∇ X f j + ∑λ j  j = Φ (6) =j 1 =j 1 where λ j , λ j are unknown multipliers. ∂X , i = 1, ,  (the ∂qi columns of J q X ), although such as space strictly includes  , if ν > 0 , is anyhow noteworthy: The projection of the dynamics equation on the subspace generated by the  vectors ( J X) ( T q )  − F − Φ= Q  λ1    d T λ 0 , = λ  ∇q  ) − ∇q  − α = ( dt λ   ν (7) where we assumed F = ∇ X  ( X,t ) and we defined the Lagrangian function  = T + U , T ( q, q , t ) = 1 q ⋅ A ( q, t ) q + b ( q, t ) ⋅ q + c ( q, t ) 2 (8) with A symmetric and positive definite matrix of size  and U ( q, t ) =  ( X ( q, t ) , t ) . The  Equation (7) written for the  + ν unknown quantities q1 , , q , λ1 , , λν have to be considered together with the ν Equations (4). In order to improve (7), we see from (4) and (5) that  (virtual displacements) is the set of vectors  γ1   ˆ ˆ = ∑γ ∂X such that αγ = 0 , γ =    . X i ν   ∂qi i =1 γ    Owing to (3) and recalling (4), it is rankα = ν , hence the solution of the come last linear system, which ex plicitly writes ∑α i , j ( q, t ) γ j = 0 , j =1 i = 1, ,ν is σ γi = 1, ,  ∑Γi , j ( q, t )η j , i = (9) j =1 with Γi , j appropriate coefficients and η = (η1 , ,ησ ) arbitrary factors in σ , σ =  −ν . We conclude that ˆ  σ  ∂X ∂X ˆ X = Γ η ∑∑ i , j j ∂q , or, equivalently, the σ vectors ∑Γi ,k ∂q , k = 1, , σ form a basis for  . =i 1 =j 1 i =1 i i At this stage, calling Γ the matrix of size  × σ and elements Γi , j and noticing that the columns of ( J q X ) Γ give the basis for  , the projection of the dynamics equation on  gives, by virtue also of (6): (( J X ) Γ ) (Q − F − Φ ) = Γ ( J X ) ( Q − F ) = Γ T q T T q T d   ( ∇q  ) − ∇q   = 0σ d t   (10) where the effect of the nonholonomic constraints (through Γ ) on the ordinary Lagrangian equations for hold onomic systems is evident (in the absence of (2), say ν = 0 , both (10) and (7) are ( ∇q  ) − ∇q  = 0 ). dt The σ differential Equation (10) are for the  unknown quantities q and they have to be combined to- 297 F. Talamucci gether with the ν Equation (4). With respect to (7), they have the advantage of not exhibiting the multipliers λ. Remark 2.1 Either Equation (7) or (10) can be employed not necessarily for discrete systems of point particles: once the Lagrangian coordinates have been selected and the Lagrangian function has been written, they can be the same calculated. The expedience of introducing quasi-velocities (or pseudovelocities) which have to be chosen in a suitable way in order to disentangle the mathematical problem, is by custom performed in nonholonomic systems. Following the adopted standpoint, the definition of the quasi-velocities steps in establishing a specific (and convenient) connection between η and q   η1 = Z ( q, t ) q + ψ ( q, t ) ∑z1, j ( q, t ) q j +ψ 1 ( q, t ) ,  , ησ = ∑zσ , j ( q, t ) q j +ψ σ ( q, t ) or η = (11) =j 1 =j 1  z1,1  z1,        zσ ,1  zσ ,   Z  where zi , j are required to guarantee that the square matrix of size    =   is invertible. In  α1,1  α1,   α       α   ν ,1  αν ,  this way, each set of kinetic variables q is linked to a singular set of quasi-velocities η , and vice versa. More precisely, (11) and (4) give −1 Z  η − ψ   Z  η − ψ  η − ψ    q =  and q =    =( Γ Θ )   α   −β   α   −β   −β  (12) where Γ is the same as (9) and Θ ( q,t ) is a  ×ν matrix. The first system in (12) shows both the selection on the coordinates q of the tangent space J q X necessary to fulfill the restrictions on the system’s velocity (leading to the subspace  ) and the kinematic conditions themselves. In order to express (10) as a function of the variables q , η and to eliminate q , it suffices to extract from (12) q ( q,η , t ) = Γ ( q, t ) (η −ψ ( q, t ) ) − Θ ( q, t ) β ( q, t ) (13) and to define  ( q,η , t ) =  ( q, q ( q,η , t ) , t ) = 1 (η −ψ ) ⋅ AΓ (η −ψ ) + β ⋅ AΘ β − 2 (η −ψ ) ⋅ AΓ ,Θ β  + b ⋅ ( Γ (η −ψ ) − Θβ ) + c + U , 2 (14) where AΓ ( q, t ) = Γ T AΓ, AΓ ,Θ ( q, t ) = Γ T AΘ, AΘ ( q, t ) = ΘT AΘ (15) By using the formulae (see (11)) ∇q  = Z T ∇η  , ( ∇q L = ∇q  + J qTη ) q =Γ (η −ψ ) −Θβ ∇η  where J qTη ( q,η , t ) is the  × σ matrix whose elements are, for each i = 1, ,  , j = 1, , σ  J Tη  q  ( ) σ ν  ∂z  ∂ψ j j ,k   = Γ k , s (η s −ψ s ) − ∑Θ k , p β p  +  ∑ ∑  q =Γ (η −ψ ) −Θβ= ∂qi  s 1 i , j k 1 = = p 1  ∂qi we can write (10) in terms of the demanded variables (we use Z Γ =Iσ , see (12)): 298 (16) F. Talamucci ( ) ( ) d ∇η  − Γ T ∇q  + Γ T Z T − J qTη ∇η  = 0σ dt (17) Remark 2.2 Multiplying both sides of (17) by η and performing the customary steps leading to the energy balance one finds T   d ∂ ∂Z    η ⋅∇η  −  + − ( Θβ + Γψ ) ⋅ ∇q  +  J qη +  ∇η   dt ∂t ∂t     ( ) (18) d ∂ψ  ∂Z  +  Γη + − Z ( Θβ + Γψ )  ⋅∇η  = 0 dt ∂t  ∂t  In the stationary circumstance β = 0 , ψ = 0 ,  =  ( q,η ) and Z = Z ( q ) the Legendre transform η ⋅∇η  −  of  is conserved. Our next step is writing (17) explicitly, sorting the terms in a suitable way: we start from the calculation ∇η  ( q,η= , t ) AΓ (η −ψ ) − AΓ ,Θ β + Γ T b, 1 T  1 T  T ∇q  ( q,η , t ) = − J q ψ AΓ + J q ( AΓ (η −ψ ) ) − J q ( AΓ ,Θ β )  (η −ψ ) 2  2  1 1  +  J qT β AΘ + J qT ( AΘ β ) + J qTψ AΓ ,Θ  β 2 2   ( ) ( ( ) ( (19) ) ) + J qT b ( Γ (η −ψ ) − Θβ ) +  J qT ( Γ (η −ψ ) − Θβ )  b + ∇q c + ∇qU . so that (17) takes the structure AΓ ( q, t )η + Q ( q,η , t ) + Λ ( q,η , t ; β ,ψ , b ) + N ( q, t ; β ,ψ , b ) − AΓ ∂ψ ∂β ∂b   − AΓ ,Θ − Γ T  ∇q c + ∇qU −  = 0σ (20) ∂t ∂t ∂t   Provided that M ( k ) means the k -th column of any matrix M and defining for any k = 1, ,  the operation   ∂Z ∂M  k  = − Jq Z ( )  Γ + k ( M )  M   ∂qk     ∂qk T (21) for a matrix M ( q ) of size N × σ , the terms in (20) are defined by the following expressions, where means the k -th component of any vector v and AΘ,Γ = AΓ ,Θ : 1 σ  Q ( q, η , t ) = ∑ ( Γη )k k ( AΓ ) − ∑ηr 2r 1 = k 1= (( J A ) Γ ) η q (( ) ) ) (( (r ) Γ T T 1 σ   r Λ ( q,η , t ; β ,ψ , b ) = − ∑ ( Γψ )k k ( AΓ ) − ∑ψ r J q AΓ( ) Γ + AΓ ( J qψ ) Γ η 2r 1 = k 1=  ν  T T   p −  AΓ ,Θ ( J q β ) Γ − AΓ ,Θ ( J q β ) Γ − ∑β p J q AΓ( ,Θ) Γ + ∑ ( Θβ )k k ( AΓ ) η = p 1= k 1   σ   T 1   r − ∑ ( Γη )k k ( AΓ ) − ∑ηr J q AΓ( ) Γ ψ − ∑ ( Γη )k k ( AΘ,Γ ) β 2 k =1 =  k 1 =r 1  ( (( ) ) ) ) ( ) σ T T   r + Γ T ( J q b ) − ( J q b )  Γη + ∑ ( Γη )k k ( Γ ) − ∑ηr  J q Γ( ) Γ   b      =  k 1 =r 1  T  ∂A  ∂Z   +  Γ +  AΓ Γ  η , ∂t    ∂t  299 ( v )k F. Talamucci ( ) (( ) ) ( ) (( ) )  T T   1 ν N ( q, t ;= β ,ψ , b )  AΓ ,Θ ( J q β ) Θ − AΘ ( J q β ) Γ − ∑β p J q AΘ( p ) Γ + ∑ ( Θβ )k k ( AΘ,Γ )  β 2 p 1 =k 1 =   σ  T T 1   r +  AΓ ( J qψ ) Γ − AΓ ( J qψ ) Γ − ∑ψ r J q AΓ( ) Γ + ∑ ( Γψ )k k ( AΓ ) ψ 2 r 1= k 1 =   (( ) ) ν  T T   p +  AΓ ,Θ ( J q β ) Γ − AΓ ,Θ ( J q β ) Γ − ∑β p J q AΓ( ,Θ) Γ + ∑ ( Θβ )k k ( AΓ ) ψ p =1 k =1    T   +  AΓ ( J qψ ) Θ − AΘ,Γ ( J qψ ) Γ + ∑ ( Γψ )k k ( AΘ,Γ )  β k =1   ( ) ( ) {( T − Γ T  J q b − ( J q b )  ( Γψ + Θβ ) + Γ ( J qψ ) Γ   ) + ( Θ ( J β ) Γ ) }b T T q ν  T T σ  r p + ∑ψ r  J q Γ( ) Γ  + ∑βν  J q Θ( ) Γ  − ∑ ( Γψ + Θβ )k k ( Γ )  b      r =1=p 1 =k 1  T T  ∂A  ∂Z    ∂Γ T  ∂Z T  ∂Z    ∂AΓ ,Θ  − +  AΘ,Γ Γ   β −  Γ +  AΓ Γ  ψ +  + Γ Γ   b. ∂t   ∂t     ∂t  ∂t    ∂t  ∂t  ( ) ( ) Equation (20) is sorted on the strength of the quasi-velocities η : Q is quadratic with respect to η1 , ,ησ , Λ is linear with respect to the same variables and Z does not contain η . Since A is a positive-definite square matrix and rank Γ =σ , even AΓ = Γ T AΓ is a positive-definite σ × σ η  symmetric matrix. Hence, system (20) + (13) can be written in the normal form   = Y (η , q,t ) , where Y is q a list of  + σ functions, whose regularity allows us to apply the standard theorems on existence and uniqueness of solutions to first-order equations with given initial conditions. Before commenting Equation (20), we remark that the σ × N entries of the matrix k ( M ) defined in (21) are, for each k = 1, ,  :   ∂z σ ∂z  = ) )i , j ∑∑M j ,r  r ,h − r ,k  Γ h,i + ( k ( M ∂q ∂q h= 1 = r 1  k h  ∂M j ,i = , i 1,= , σ , j 1, , N ∂qk (22) We see now that a certain number of significant cases are encompassed by (20): • merely geometric constraints, corresponding to σ =  , ν = 0 , so that (4) are not present and all the terms containing β , Θ and the related quantities AΘ AΓ ,Θ must be dropped in (20). Furthermore: ○ selecting η = q (quasi-velocities are the generalized velocities) in (11) and (13) means Z = Γ =  , AΓ = A, ψ = 0 so that in (20) are written with as = Q ∑ qk k ( A) − 2 qk ( J q A( k ) )  k =1   1 T T ∂A    Λ  J q b − ( J q b ) +  q , = N 0  q, = ∂t    thus the Lagrangian equations for geometric constraints (bearing in mind (22))    ∂a   ∂bi ∂br ∂ai , r  1 ∂ar , s  ∂c ∂U ∂bi 0 − + − + =  qr qs + ∑   qr − ∂qi = ∂qi ∂t  ∂qi ∂qi ∂t r 1  ∂qr  ∑ai ,r qr + ∑  ∂qi ,r − 2 r 1= r,s 1 =  s i = 1, ,  , are achieved. ○ establishing (11) as η = ∇q  = Aq + b (quasi-velocities are the generalized momenta) means Z = A, Γ = A−1 , AΓ = Γ, ψ = b In this case (13) together with (20) are the Hamiltonian equations for 300 F. Talamucci 1 H ( q,η , t ) = η ⋅ q ( q,η , t ) −  ( q,η , t ) = Γ (η − b ) ⋅ (η − b ) − c − U : 2 indeed the first one is q = Γ (η − b ) = ∇η H , whereas (20) reduces to Γ ( q, t )η + Q ( q,η , t ) + Λ ( q,η , t ; b ) + N ( q, t ; b ) − Γ ( ∇q c + ∇qU ) = 0 (23) with  ( = k ) ( 1  k ∑ηk Γ J q Γ( ) 2 1= k 1 Q ( q,η , t ) =−∑ ( Γη )k Γ J q A( k) T Γη − ( ) )η T (  T 1  ηk Γ J q Γ( k ) b + ∑ ( Γb )k Γ J q A( k ) ∑ 2 k 1 =k 1 = Λ ( q,η , t ; β ,ψ , b ) = −Γ ( J q b ) Γη − T ( 1  k ∑bk Γ J q Γ( ) 2 k =1 N ( q, t ; b ) =Γ ( J q b ) Γb + T ( ) k (actually from ΓA =   one deduces k ( Γ ) = −Γ J q A( ) b = ψ , many terms are cancelled). Since ( J A( ) ) k q T ( Γ( ) = − J q Γ( h h) ) T A( k) T Γ and Γ ) T ) T Γη Γ ∂A ∂Γ A so that, also considering = − ∂t ∂t for any h, k = 1, ,  , it is ( 1  k Q + Λ + N = Γ  ∑ (ηk − bk ) J q Γ( )  2 k =1 ) − ( J b) T T q  1  Γ  (η − b ) = Γ∇ q  Γ (η − b ) ⋅ (η − b )  2   therefore (23) is Γη = −Γ∇q H , as stated. • Stationary case, where the different contributions producing the dependence on t must be dropped. If one is dealing with a scleronomic system (covering many of common instances), the constraints (1), (2) reduce to Y ( X ) = 0µ (24)  ( X ) X = 0ν (25) 1 Conditions (24) entail X = X ( q ) and  ( q, q ) =q ⋅ A ( q ) q + U ( q ) (if even the forces are independent of 2 time), on the other hand (25) implies β = 0 . Equation (11), if one reasonably chooses ψ = 0 and Z independent of t (otherwise, changes will be obvious), is η = Z ( q ) q . Since Λ= N= 0σ , system (20) + (13) drastically simplifies to T  AΓ ( q )η + Q ( q,η ) − Γ ∇qU = 0σ ,  q = Γ ( q )η , (26) or, index by index, calling bi , j the entries of the matrix AΓ , i, j = 1, , σ and having in mind (22) σ σ  ∂U ∑bi ,rηr + ∑r(, s)ηrηs − ∑Γ h,i ∂q= = r 1= r,s 1 i = h 1 i 1, , σ ; 0, = h σ q p = ∑Γ p, j ( q1 , , q )η j , j =1 (27) p= 1, , . where  ( i ) is, for each index i , the square matrix of order σ q ) r(, s) ( q1 , , = i   σ  ∂z j , k ∑  Γ k ,i Γ h, s ∑br , j   = h , k 1 =j 1  ∂qh 301 − ∂z j , h  ∂br ,i 1 ∂br , s  − Γ h ,i   + Γ h, s ∂qk  ∂qh 2 ∂qh  F. Talamucci Equations (27) are identified with the Boltzmann-Hamel Equations (17) for the Lagrangian function 1 1  ( q,η ) = η ⋅ AΓη + U (see [3] [4]). In this case the Legendre transform η ⋅ AΓη − U is a first integral of 2 2 motion, see Remark 1.2. • Reduced Lagrangian function for geometric constraints: in case of ν cyclic variables qσ +1 , , q , σ =  −ν , ∂ , = (4) can play the role of the ν relations derived from the first integral of motion pi = i σ + 1, ,  , ∂qi  that is 0, ∑ai , j q j + bi − pi = j =1 = i σ + 1, ,  . Assuming that det ( ar , s ) =/ 0 , r , s= σ + 1, ,  , it is possible to acquire, according to (13), qi= σ ∑Γi , j q j − Θi , j ( b j − p j ) , j =1 = i σ + 1, ,  , where Γi , j , Θi , j and bj depend only on ( q1 , , qσ , t ) . At this point, setting η1 = q1 ,  , ησ = qσ we have, with respect to (11) and (12), δ r , s (Kronecker’s delta), r , s = 1, , σ . Equation (20), which writes simply Z = ( σ ×ν σ ) and Γ r , s = d ∇η  =Γ T ∇q  , are the equations of motion for the reduced Lagrangian dt ( ) ( ) ( ( ) ( ) ) ( ) σ σ σ σ  q( ) ,η , t =  q( ) ,η , qσ +1 q( ) ,η , t , , q q( ) ,η , t , t , with η = (η1 , ,ησ ) , q(σ ) = ( q1 , , qσ ) ; on the other hand, qi q(σ ) ,η , t = σ ∑Γi , jη j − Θi , j ( b j − p j ) for j =1 = i σ + 1, ,  , are the so called reconstruction equations. 3. Some Applications We adopt now Equation (20) in order to formulate a couple of remarkable mechanical systems, each of them in a double form, as scleronomous and rheonomous model. 3.1. Pendulum on a Skate Consider a system of four points { PF , PB , PS , PD } , PF and PB equidistant and lying on a horizontal plane, PS equidistant from PF and PB , PD oscillating around O1 , equidistant from PF and PB and coplanar to the latter points and PB (see Figure 1). The system represents a simple model for the motion of a bicycle, as exhibited in [5]: the mass in PS is added on order to sketch the rigid structure of the bicycle (just as PF and PB represent the front and the back wheels), as well as the pendulum PD simulates the movement of a driver. Let O be a fixed point on the horizontal plane containing PF and PB , k the ascending vertical versor, C the midpoint of the segment PB PF and O1C perpendicular to the same segment: the geometrical constraints (1) are written by means of the constant assigned values ρ , κ , κ1 as ( PF − O= )⋅k PB ) 0 ( PD − C ) ⋅ ( PF − = 2ρ , = P= PS C κ1 , O= κ, F PB 1 PD PB ) ( PS − C ) ⋅ ( PF − = 0, 0, ( PB − O= )⋅k 0, (28) Since the constraints are independent and n = 4 , we have µ = 7 ,  = 5 . Setting a fixed reference system O { , i, j, k } and the angle φ between PF − PB and i , the angle θ between PS − C and k , the angle θ1 between PD − O1 and −k , one defines the orthonormal versors −sinθ sinφ i + sinθ cosφ j + cosθ k , eθ1 = = eφ cosφ i + sinφ j , eθ = −sinθ1sinφ i + sinθ1cosφ j − cosθ1k so that PF − PB =e κ1 θ , O1 − C = (κ1 + κ 2 ) eθ , P2 − O1 = 2 ρ φ , PS − C =e κ eθ1 and choose the five parameters q = ( xC , yC , φ , θ , θ1 ) as Lagrangian coordinates, where xC i + yC j =C − O . Opting for considering the segment PB PF as a rigid bar of mass M (instead of a discrete point system, although not significant), the Lagrangian function (8) is written with b = 0 , c = 0 , U = −υ1 gcosθ + κ mD gcosθ1 and 302 F. Talamucci Figure 1. A simple model for the motion of a bicycle.  MT  0   A = − F1 (θ , θ1 ) cosφ   −υ1cosθ sinφ  − m κ cosθ sinφ 1  D 0 MT − F1 (θ , θ1 ) sinφ υ1cosθ cosφ mDκ cosθ1cosφ − F1 (θ , θ1 ) cosφ −υ1cosθ sinφ −mDκ cosθ1sinφ   mDκ cosθ1cosφ  − F1 (θ , θ1 ) sinφ υ1cosθ cosφ  0 0 F2 (θ , θ1 )  υ2 υ3 cos (θ + θ1 )  0  mDκ 2 υ3 cos (θ + θ1 ) 0  where M T = M + mS + mD is the total mass and υ1 =mS κ1 + mD (κ1 + κ 2 ) , υ2 =mS κ12 + mD (κ1 + κ 2 )2 , υ3 =mDκ (κ1 + κ 2 ) ,  = , θ1 ) υ1sinθ + mDκ sinθ1 ,  F1 (θ  F (θ , θ ) = I C + mDκ 2 sin 2θ1 + υ2 sin 2θ + 2mDκ (κ1 + κ 2 ) sinθ sinθ1 . 1  2 (29) The only one kinetic constraint concerns with the velocity of the back “wheel” PB , to be aligned with the segment: PB ∧ eφ = 0 R  R   φ= 0 , that is (4) for ν = 1= , α  sinφ , −cosφ , , 0, 0  , β = 0 . 2 2   Hence σ = 4 and the four quasi-velocities (11) are selected by setting or xC sinφ − yC cosφ +  1  ρ cosφ   1 Z =  − sinφ  ρ  0  0  1 ρ 1 ρ sinφ cosφ 0 0 303  0 0 0   0 0 0  and ψ = 0 .  0 1 0  0 0 1 (30) F. Talamucci Furthermore, (12) gives  ρ cosφ   ρ sinφ Γ = 0   0  0  − ρ sinφ ρ cosφ 1 0 0 0 0  0 0 0 0  1 0 0 1  so that  MT ρ 2  − ρ F1 AΓ =   0   0  − ρ F1 F2 + ρ 2 M T 0 ρυ1cosθ ρυ1cosθ υ2 ρ mDκ cosθ1 υ3 cos (θ + θ1 )  0  ρ mDκ cosθ1  υ3 cos (θ + θ1 )    mDκ 2  By computing the first line in (26) one finds the four equations of motion   ρ 2 M Tη1 − ρ F1η2 − ( F2 + ρ 2 M T )η22 + ρ F1η1η2 − ρ υ1cosθ + (  ∂F1  ∂F1  0, η η = η 2η3 − ρ  mDκ cosθ1 + ∂θ  ∂ θ1  2 4  ) − ρ F1η1 + F2 + ρ 2 M T η2 + ρυ1η3cosθ + ρ mDκη4 cosθ1 + ρ F1η12 − 2 ρ F1η 22 − ρυ1η32sinθ − ρ mDκη 42sinθ1 ( ) + ρ 2 M T − F2 η1η 2 − ρ ∂F1 ∂F 0, (η1η3 + η1η4 ) + 2 (η2η3 + η2η4 ) = ∂θ ∂θ   ρυ1η2 cosθ + υ2η3 + mDκ (κ1 + κ 2 )η4 cos (θ + θ1 ) +  ρυ1cosθ − ∂F1 1 ∂F2  2 2 η1η2 η 2 − υ3η 4 sin (θ + θ1 ) + ρ ∂θ 2 ∂θ  ρ   + υ2 − υ1sinθ η 2η3 + υ3η 2η 4 cos (θ + θ1 ) − υ1 gsinθ = 0, 2    ∂F1 1 ∂F2  2 2 mDκρ cosθ1η2 + υ3η3sin (θ + θ1 ) + mDκ 2η4 +  ρ mDκ cosθ − ηη η 2 − υ3η3 sin (θ + θ1 ) + ρ ∂ ∂ 2 θ θ1 1 2  1  + υ3η 2η3cos (θ + θ1 ) + mDκ 2η 2η 4 + κ mD gsinθ1 = 0. joined with the conservation of the quantity η ⋅ ∇η  −  . 3.2. Assignment of the Front Motion We modify the previous model by forcing the velocity of the front “wheel” to be a known function of time (a simpler version was considered in [6] for the motion of a bike): PF ( t ) −= O xF ( t ) i + yF ( t ) j . With respect to (28), time t enters explicitly the geometrical constraints and the fourth one has to be removed. Hence, in this example we have n = 3 , µ = 6 ,  = 3 and we choose q = (φ , θ , θ1 ) . The midpoint C is located by C −= O ( xF ( t ) − ρ cosφ ) i + ( yF ( t ) − ρ sinφ ) j and the Lagrangian function (8) is written with  F2 + ρ 2 M T − ρυ1cosθ − ρ mDκ cosθ1    υ2 υ3 cos (θ + θ1 )  ,= A= b  − ρυ1cosθ1 2  − ρ m κ cosθ υ cos (θ + θ )  mDκ 1 3 1 D   ∂β  1  2 ρ M T β − F1 ∂φ    1 M T x F2 ( t ) + y F2 ( t )  , c θ −υ1 β cos= 2    −mDκβ cosθ1      ( ) whereas U is the same function. 0 , that is (4) for ν = 1 , α = ( 2 ρ , 0, 0 ) The constraint (30) is now x F ( t ) sinφ − y F ( t ) cosφ + 2 ρφ = 304 F. Talamucci = β x F ( t ) sinφ − y F ( t ) cosφ . Choosing η1θ , η2 = θ1 we have simply 0 0 0 1 0   = = 0, Θ Z  , Γ 1 = 0 0 1   0 1 1 ( 2 ρ )   υ2 υ3 cos (θ + θ1 )      0=  AΓ υ cos (θ + θ ) mDκ 2 1  3   0    Equation (20) are written with    ∂F2    η22  1  1  ∂θ  ∂β  υ1cosθ   β β Q = −υ3sin (θ + θ1 )  2  , Λ = 0, N = − +   4 ρ  2 ρ  ∂F2  ∂φ  mDκ cosθ1   η1     ∂θ   1   and correspond to υ2η1 + υ3η2 cos (θ + θ1 ) − υ3sin (θ + θ1 )η22 − 1 ( xF ( t ) sinφ − y F ( t ) cosφ ) 2 ∂F2 ∂θ 8ρ 1 0, − ( xF ( t ) sinφ − y F ( t ) cosφ ) ( xF ( t ) cosφ + y F ( t ) sinφ )υ1cosθ − υ1 gsinθ = 4ρ 2 ∂F2 υ3η1cos (θ + θ1 ) + mDκ 2η2 − υ3sin (θ + θ1 )η12 − ( xF ( t ) sinφ − y F ( t ) cosφ ) ∂θ1 − 2 1 0. ( xF ( t ) sinφ − y F ( t ) cosφ ) ( xF ( t ) cosφ + y F ( t ) sinφ ) mDκ cosθ1 + mDκ gsinθ1 = 4ρ The energy balance (18) writes d ∂ and the function in the right side of the η ⋅∇η  −  = Θβ ⋅∇q  − ∂t dt ( ) latter equality is F 1 1  Ψ  β  M T − 22 2 2  2ρ  ∂Ψ 1  ∂β 2   1  F1 ∂β   + M T ( x F  − − − − + β cos cos m F xF + y F  yF ) η υ θ η κ θ  1 1 2 D 1 1  ∂φ 2 ρ  ∂φ    ρ ∂φ  2ρ    ∂β β ∂β  with = Ψ  − .  ∂t 2 ρ ∂φ  3.3. Rolling Disk with Pendulum A different version of the model 3.1 lies in replacing the bar with a disk and obtaining the unicycle with rider model presented in [7] (see Figure 1 again, replacing the bar with the disk). The system we consider here is a disk of diameter 2R and mass M , in addition to the same points PS (with mass mS ) and PD (with mass PD ). We directly choose the coordinates (see Remark 2.1) q = ( xC , yC , φ , φ1 , θ , θ1 ) where the new parameter φ1 is the angle of rotation of the disk around the axis perpendicular to the disk and passing through the centre. The Lagrangian function is written with U = −υˆ1 gcosθ + mD gκ cosθ1 and  MT  0   ˆ − F1cosφ  A= 0    −υˆ1cosθ sinφ   −mDκ cosθ1sinφ − Fˆ1cosφ − Fˆ sinφ 0 −υˆ1cosθ sinφ 0 υˆ1cosθ cosφ − I D sinθ ID 0 0 Fˆ2 − I D sinθ υˆ1cosθ cosφ 0 0 −mDκ cosθ1cosφ 0 0 0 MT − Fˆ sinφ 1 1 305 0 1 υˆ2 + I D 2 υ3 cos (θ + θ1 ) −mDκ cosθ1sinφ   − mDκ cosθ1cosφ   0   0   υ3 cos (θ + θ1 )   mDκ 2  F. Talamucci where I D = 1 MR 2 and (see (29)) 2 υˆ1 = υ1 + MR, υˆ2 = υ2 + MR 2 , 1  1  , θ1 ) F1 (θ , θ1 ) + MRsinθ , Fˆ2 (θ= , θ1 ) F2 (θ , θ1 ) − I C +  MR 2 + I D  sin 2θ + I D . Fˆ1 (θ= 2 2   The kinematic constraint of rolling without sliding entails the zero velocity of the contact point C : = xC φ= yC φ1 Rsinφ 1 Rcosφ , 1 0 0 − Rcosφ which is (4) with ν = 2 , α =   0 1 0 − Rsinφ This time σ = 4 and the choice (31) 0 0  and β = 0 . 0 0 ∂L ∂L  C cosφ ) , η2 ==− Fˆ2φ I Dφ1sinθ − Fˆ1 ( xC cosφ + xy I Dφ1 I Dφsinθ , η3 = η1 ==− θ, η4 = θ1 ∂φ ∂φ1 leads to  − Fˆ1cosφ  0 Z =  0  0  − Fˆ1sinφ 0 Fˆ2 − I D sinθ 0 0 − I D sinθ ID 0 0 0 0  I D Rsinθ cosφ   I D Rsinθ sinφ  1 ID Γ=   δ  I D sinθ  0  0  ( RFˆ2 cosφ RFˆ sinφ 2 0 0 I D sinθ + RFˆ1 Fˆ 0 2 0 0 δ 0 0 0 0  0 0 , 1 0  0 1  0  0  0  0 0  δ ) where δ (θ , θ1 ) = I D Fˆ2 − I D sinθ I D sinθ + RFˆ1 > 0 . Moreover 0 0  b1,1 b1,2    0 0  b2,1 b2,2    1 AΓ =  0 0 υˆ2 + I D υ3 cos (θ + θ1 )  2   1   0 υ3 cos (θ + θ1 ) υˆ2 + I D   0 2   with  I D  I D sinθ  1+ M T R 2 sinθ − RFˆ1  , b1,1 (θ , θ1 ) =  δ  δ    I D sinθ  R  ˆ ˆ ˆ  b1,2 (θ , θ1 )= b2,1 (θ , θ1 )= 1 +  M T RF2 − I D sinθ + RF1 F1   , δ δ     Fˆ R b2,2 (θ , θ1 ) =2 1 +  M T RFˆ2 − I D sinθ + RFˆ1 Fˆ1   .  δ  δ  ( ) ( ( and the corresponding equations of motion (20) are 306 ) ) F. Talamucci ∂b1,1  ∂b1,1  ∂b2,1     b1,1η1 + b1,2η2 + η1η3  b1,1G1 − b1,2 G5 +  + η2η3  b2,1G1 − b2,2 G5 +  + η1η4  b1,1G2 +  θ θ ∂ ∂ ∂θ     1   ∂b2,1   0, + η2η4  b2,1G2 + = ∂θ1   ∂b2,2  ∂b1,2  ∂b1,2     b2,1η1 + b2,2η2 − η1η3  b1,1G3 + b1,2 G6 −  − η2η3  b2,1G4 + b2,2 G6 −   + η1η4  b1,1G4 − ∂θ1  ∂θ  ∂θ     ∂b2,2   0, + η2η4  −b2,1G4 + = ∂θ1   1  1 ∂b1,1  2  1 ∂b2,2   2  + η2  b2,1G3 + b2,2 G6 −  υˆ2 + I D η3 − υ3η4 cos (θ + θ1 ) − η1  b1,1G1 + 2  2 ∂θ  2 ∂θ     ∂b2,1   2 0, − η1η2  b2,1G1 − b1,1G3 − b2,2 G5 − b1,2 G6 +  − υ3η4 sin (θ + θ1 ) − υˆ1 gsinθ = ∂θ    1 ∂b1,1  2  1 ∂b2,2  −υ3η3 cos (θ + θ1 ) + mDκ 2η4 − η12  b1,1G2 +  + η2  b2,1G4 −  2 ∂θ1  2 ∂θ1    ∂b2,1   2 + η1η2  b2,1G2 − b1,1G4 + 0.  − υ3η3 sin (θ + θ1 ) + mDκ sinθ1 = ∂ θ  1  where = G1 (θ , θ 1)  I D  ∂Fˆ2 − ( I D + Rυˆ1 ) cosθ sinθ  ,  δ  ∂θ  G2= (θ ,θ1 )  I D  ∂Fˆ2 − mDκ Rcosθ1sinθ  ,  δ  ∂θ1  ∂Fˆ2  1 G3 (θ , θ1 ) =  ( I D + Rυˆ1 ) Fˆ2 cosθ − I D sinθ + RFˆ1 , ∂θ  δ ∂Fˆ2  1 G=  mDκ RFˆ2 cosθ1 − I D sinθ + RFˆ1 , 4 (θ , θ1 ) ∂θ1  δ I2 G5 (θ , θ1 ) = D cosθ , ( ( ) ) δ G6 (θ , θ1 ) = ID δ (I D ) sinθ + RFˆ1 cosθ . 3.4. Assigned Rotational Velocity of the Disk We finally consider the same system with the differential constraint (31), but φ1 = φ1 ( t ) assigned (we may think about an engine-driven motor bike or electric bike): in that case q = ( xC , yC , φ , θ , θ1 ) and (4) is setted  −φ1 ( t ) Rcosφ  1 0 0 0 0  with ν = 2 and α =  . , β =   0 1 0 0 0  −φ1 Rsinφ  The Lagrangian fucntion (8) is written with A the same as in the previous Example 3.1, except for removing 0     0   1 the fourth row and the fourth column, and b =  − I D sinθ  , c = I Dφ12 ( (t ) ) . In the matter of (11), which has to 2   0     0   307 F. Talamucci ∂L ˆ  ˆ  C cosφ ) − I Dφ1 ( t ) sinθ , be written for σ = 3 , if one defines the quasi-velocities η1 = = F2φ − F1 ( xC cosφ + xy ∂φ η2 = θ , η3 = θ1 one gets ψ = 0 and  − Fˆ1cosφ  0  = Z  0  1   0  − Fˆ1sinφ 0 0 0 1 Fˆ2 0 0   0 1 0 0= 0 1, Γ  0 0 0 0 0 0   0 0 0    0 0 0  1  0 0, Θ  ˆ=  F2   0 1 0  0 0 1        Fˆ  1  Fˆ2     1 0 cosφ 0 0     Fˆ1 sinφ  Fˆ2   0  0  0 1 Calculating the products in (15) gives  1  0 0  ˆ   F2    1 υˆ2 + I D υ3 cos (θ + θ1 )  , AΓ =  0 2   0 cos υ θ mDκ 2   ( + θ1 ) 3     0   AΓ ,= Θ  −υˆ1cosθ sinφ  −m κ cosθ sinφ 1  D  Fˆ12 2  M T − ˆ cos φ F2  AΘ =  2 ˆ  − F1 sinφ cosφ  Fˆ  2  υˆ1cosθ cosφ  , mDκ cosθ1cosφ  0  Fˆ12 sinφ cosφ  ˆ F2   2 ˆ F M T − 1 sin 2φ  Fˆ2  − and the computation of (20) gives the three equations of motion I Fˆ 1 1 1 0, η1 + I Dφ1 ( t )η2 cosθ + φ1 ( t ) R (υˆ1η2 cosθ + mDκη3 cosθ1 ) − D φ1 ( t )η2 cosθ + 12 I Dφ12 ( t ) Rcosθ = ˆ ˆ ˆ ˆ F2 F2 F2 F2 Fˆ2 Fˆ ∂Fˆ 1  1  1 ∂Fˆ 1  υˆ2 + I D η2 + υ3 cos (θ + θ1 )η3  − I D 2 2 φ1 ( t )η1sinθ + Rφ1 ( t ) η1υˆ1cosθ − Rφ1 ( t ) 12 η1 2   2 2  ∂θ Fˆ2   Fˆ2 ∂θ Fˆ2 Fˆ2  1  ID  Fˆ ∂Fˆ ∂  Fˆ12  I D  φ1 ( t )η1cosθ + φ12 ( t )  R 2 +  ( 2 I D + Rυˆ1 ) sinθ cosθ − 2 1 2 Rsinθ − Fˆ1 Rcosθ      2 ∂θ  Fˆ2  Fˆ2  Fˆ2 Fˆ2 ∂θ   0, − υˆ1 gsinθ = +  1 1 1 ∂Fˆ2  υ3 cos (θ + θ1 )η2 + mDκ 2η3 + φ1 ( t )  mDκ Rcosθ1 − I D sinθ + RFˆ1 η1 2 Fˆ2 Fˆ2 Fˆ2 ∂θ1   ( ( ) ) 1  Fˆ1 ∂Fˆ2   ∂  Fˆ12  I D 0. + φ12 ( t )  R 2  ˆ  + ˆ sinθ  mDκ cosθ1 − 2 R ˆ   + mDκ gsinθ1 = 2 ∂θ1  F2  F2 F2 ∂θ1     4. Conclusions The paper aims at formulating a general scheme of equations for rheonomic mechanical systems exposed to ei- 308 F. Talamucci ther geometrical (1) and differential (2) constraints. We pay special attention to tell apart the different contributions due to the explicit dependence on time, deriving from the holonomous constrictions (via b and c of (8)), the nonholonomous constrictions (via β of (4)) and the definition of quasi-velocities (via ψ ) of (11)). Since the equations of motion are projected in the subspace of the velocities allowed by the constraints (both holonomous and nonholonomous), the Lagrange multipliers are absent from the equations. The procedure proposed by (20) requires only calculation of the Jacobian matrix of vectors and the algebraic multiplication of matrices and vectors. Making use of quasi-velocities renders the equations versatile to more than one formalism and, as it is known, the appropriate choice of them meets the target of facilitating the mathematical resolution of the problem. The last point is part of the matters listed below and which will be dealt with in the future: -Find an appropriate choice of the quasi-velocities in order to disentangle (20) from (13) as much as possible, -Make use of the structure of the equations and of the properties of the various matrices involved in order to study the stability of the system, -Take advantage of some peculiarity of the system in order to refine the set of equations and achieve information. The latter subject is faced in [8] [9] for the stationary case by means of a robust and complex theory in connection with symmetries in nonholonomic systems. References [1] Poincaré, H. (1901) Sur une forme nouvelle des èquations de la méchanique. Comptes Rendus de l’Académie des Sciences, 132, 369-371. 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Systems and Control Letters, 46, 293-302. http://dx.doi.org/10.1016/S0167-6911(01)00187-6 [8] Bloch, A.M., Krishnaprasad, P.S., Mardsen, J.E. and Murray, R. (1996) Nonholonomic Mechanical Systems with Symmetry. Archive for Rational Mechanics and Analysis, 136, 21-99. http://dx.doi.org/10.1007/BF02199365 [9] Bloch, A.M., Mardsen, J.E. and Zenkov, D.V. (2009) Quasivelocities and Symmetries in Non-Holonomic Systems. Dynamical Systems, 24, 187-222. http://dx.doi.org/10.1080/14689360802609344 309