This note is a preliminary account of research undertaken jointly with G. Marmo ofNapoli and P. Urbanski of Warsaw.
We propose a new description of dynamics of autonomous mechanical systemswhich includes the momentum-velocity relation. This description is formulated asa variational principle of virtual action more complete than the Hamilton Principle.The inclusion of constraints in this description is the main topic of the present note.We give examples and models ofconstraints in variational formulation s of statics anddynamics of autonomous systems.
A complete description of the dynamics of a mechanical system involves both external forces and momenta. In a fixed time interval the dynamics is a relation betweenthe motion of the system in configuration space, external forces applied to the systemduring the time interval, and the initial and final momenta. This relation is derivedfrom a variational principle which involves variations of the end points . Constrainedsystems are idealized representations ofunconstrained systems. Such idealizations areappropriate when forces at our disposal are unable to move the configuration of thesystem away from a subset of the configuration space by a perceptible distance. Thisdescription fits at least the case of holonomic constraints . We believe that constraintsshould be imposed on virtual displacements . Holonomic constraints are usually interpreted as restrictions on configurations of a mechanical system. Nonholonomic constraints are additional restrictions imposed on velocities. This traditional terminologyis not adapted to our concept of constraints as imposed on virtual displacements andonly indirectly affecting configurations and velocities. Our concept of nonholonomicconstraints makes perfect sense for static systems even if velocities do not appear inthe description of such systems. We will use the terms configuration constraints andvelocity constraints instead of holonomic and nonholonomic constraints.
2. GEOMETRIC STRUCTURES
Let Q be the Euclidean affine space ofNewtonian mechanics. The model space for Qis a vector space V ofdimension 3. The Euclidean structure is represented by a metrictensor g: V --> V *. The space V * is the dual of the model space. The canonical
We denote by qi - qo the vector associated with the points qo and qi . We writeqi = qo + v if v = q1 - qo. The norm Ilvll of a vector v E V is defined by
II vll = ) (g(v),v ).
The derivative of a function F : Q ----+ lR is the mapping
dDF : Q x V ----+ lR : (q, v) f---> dsF(q + sv) ls=o.
(2)
(3)
The first and second derivatives of a differentiable curve "(: lR ----+ Q are mappings'"Y: lR ----+ V and i: lR ----+ V .
The tangent bundle TQ is identified with Q x V , the cotangent bundle T*Q isidentified with Q x V*, the second tangent bundle T 2Q is identified with Q x V x V ,the iterated tangent bundle TTQ is identified with Q x V x V x V, and the tangentof the cotangent bundle TT*Q is identified with Q x V * x V x V *. We have theprojections
TC = {(q , 8q) E TQ ; there is a curve T lR ----+ Q
such that "((0) = q, D"((O) = Sq, (8)
and "((s) E C if s 2: O}
The spaceo 0
TQ = Q x V = {(q ,v) E TQ ; v # O}
is the tangent bundle with the zero section removed.
3. STATICS OF A MATERIAL POINT
(9)
We consider the statics of a material point in the Euclidean affine space Q of Newtonian physics.
An element (q, 8q) ofTQ is a virtual displac ement and an element (q, I) ofT*Qrepresents an external force . The evaluation
((q, I) , (q, 8q)) = (1, 8q) (10)
A NOTE ON HOLONOMIC CONSTRAINTS 405
ofan external force (q,1) E T *Q on a virtual displacement (q, oq) E T Q is the virtualwork performed by an external device controlling the configuration of the system.
Admissible displacements form a subset C 1 C TQ . If (q, oq) is an admissibledisplacement, then (q, koq) is again an admissible displacement for each number k 2:0. The set C 1 represents constraints imposed on virtual displacements. Implicitly itrestricts admissible configurations to the set
CO= {q E Q; (q,oq) E C 1 forsome oq E V} . (11)
The inclusion C 1 C T CO is usually satisfied. We say that constraints are configurationconstraints if C 1 = TCo. The set C = CO itself is called a configuration constraint.A simple two-sided configuration constraint is an embedded submanifold C eQ
There is a function(12)
assigning to each admissible virtual displacement the virtual work that an externaldevice has to perform in order to effect this displacement. This virtual work function
o
is differentiable on C 1 n TQ and positive homogeneous in the sense that
O'(q, kOq) = kO'(q , oq)
if k 2: 0. A typical example of a virtual work function is the mapping
0' : C 1 ~ 1R: (q,oq) f-* DU(q,oq)
(13)
(14)
derived from an internal energy function U defined in a domain large enough to makethe derivative DU(q, oq) meaningful. In the case of a configuration constraint C eQit is enough to have the internal energy defined on C . The function
0': C 1 ~ 1R : (q,oq) f-* p(q)lloq ll (15)
represents virtual work due to friction.The response of the system to external control is represented by a set S C T *Q of
external forces satisfying the principle ofvirtual work
(1,oq) :::; O'(q,oq) for each virtual displacement (q, oq) E C 1. (16)
The set S is the constitutive set of the system. It can be viewed as the list of possible configurations of the system together with external forces compatible with theseconfigurations. If dq, -oq) = -O'(q,oq) and the constraints are two-sided, then theprinciple of virtual work assumes the simpler form
(1,oq ) = O'(q , oq) for each virtual displacement (q,oq) E C 1. (17)
Example I : Let a material point be constrained to a circular hoop with the centerat qo E Q and radius a in the plane orthogonal to a unit vector n E V . We havetwo-sided configuration constraints
{q E Q; (g(q - qo) ,n ) = 0, Ilq- qo ll = a},
TCo = {(q,oq) E TQ ; (g(q - qo), n ) = 0, Il q- qo ll = a,
(g(oq), n) = 0, (g(oq), u(q) ) = O},
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(19)
406 WLODZIMIERZ M. T ULCZYJEW
where u(q) is the unit vector (q - qo)IIq - qoll-l. The constit utive set
S {(q, f ) E T*Q; q E c o, (j, 8q) = °for each (q, 8q) E C 1} (20)
{(q, f) E T*Q; (g(q - qo), n ) = 0, Ilq- qoll = a,
(j, u(q)) = 0, (j, n) = o}
represents the statics of the system without friction and the constitu tive set
S { (q,f) E T*Q; q E Co, (j, 8q):S pI18qll foreach (q,8q) E C 1} (21)
{(q , f) E T*Q; (g(q - qo), n ) = 0, Ilq- qoll = a,
(j,U (q))2+ (j,n)2 :s p2}
takes cons tant friction into account.
Example 2: Let a material poin t be constrained to the exterior of a solid ball wit h thecentre at qo E Q and radius a. In its displacements on the surface of the ball the pointencounters fric tion proport ional to the component of the external force pressing thepoint against the surface. Correct represe ntation of the statics of the point is obtainedwith one-sided constraints
if (g(q - qo) , i) = 0 and (g(q - qo) , j ) = O} .
408 WLODZIMIERZ M. TuLCZYJEW
4. MODELING CONFIGURATION CONSTRAINTS IN STATICS
We believe that constraint static systems are idealized representations ofunconstrainedsystems. The magnitude of the force that an external device can apply to a staticsystem is limited and instruments used to observe displacements have a limited resolution. Idealizations take these limitations into account. We restrict the analysis toconfiguration constraints. A definition of configuration constraints will be based onthe assumption that the norms of external forces at our disposal have an upper boundF and that displacements of distances less than d can not be detected.
Let C be a subset of Q. We denote by d(q, C ) the distance of a point q E Q fromC . If C e Q is an embedded submanifold or a submanifold with smooth boundary,then for each configuration q sufficiently close to C there is a unique point qc E Cnearest to q. The distance d(q, C) = Ilq - qc ll ofq from C is a well defined functionin a neighbourhood of C . If q is not in C , then qc -I- q and the unit vector e(q) =(q - qc )llq - qc ll- 1 is orthogonal to C or the boundary of C at qc .
Let C e Q be an embedded submanifold or a submanifold with smooth boundaryand let S c T*Q be the constituti ve set of a static system derived from the principleof virtual work
(j, bq) ::; O" (q, bq)
for each virtual displacement (q, bq) E TC .
(31)
A model of this static system is constructed by choosing a function (f on TQ such that0" is the restriction of (f to TC and replacing the original principle of virtual work bythe principle
(j, bq) ::; (f(q, bq) + kd(q, C )(g(e(q)), bq)
for each virtual displacement (q, bq).
(32)
The term kd(q, C) (g(e(q)), bq) is the directional derivative DK (q, bq) of the elasticinternal energy function
K (q) = ~ (d(q , C ))2 (33)
defined in the neighbourhood of C in which the distanc e function d(q, C) is welldefined. The inequalities
and
kd(q, C) ::; (j, e(q)) + O"(q , - e(q))
kd(q, C) ::; l(j, e(q))1 + 100 (q, - e(q))1
(34)
(35)
are derived from the principle of virtual work by setting bq = - e(q). We will assumethat F « kd and expect that the inequality 100 (q, - e(q))1« kd is satisfied. Theseinequalit iestogetherwith l(j, e(q)) I ::; Il fll::; Fresultin d(q,C ) « d. It follows thatusing external forces at our disposal we can not induce the material point to assumeconfigurations at noticeable distances away from C . It also follows that within the
A NOTE ON HOLONOMIC CONSTRAINTS 409
limits imposed by 11 111 :S F the component (f, e(q)) is arbitrary, Examples will beused to clarify details and present variations of this construction.
Example 5: Let Cbe the set COof example I. We obtain the equation
for the distance d(q,C) of q from Cif this distance is less than a. If d(q, C) =1= 0, then
is the unit vector orthogonal to C at the point q' E C closes t to q pointing from q'to q. Let a function (j : TQ --4 lR be defined by (j(q, n ) = 0, (j(q, q - qo) = 0, and(j(q,8q) = pl18qll if (g(8q),n) = 0 and (g(8q), q - qo) = O. The unconstrainedsystem represented by the principle of virtual work
(f,8q) :S (j(q,8q)+ kd(q, C) (g(e(q)) , 8q)
for each virtual displacement (q,8q)
(38)
is a model of the constrai ned system of example 1. It follows from the principle ofvirtual work that 1 = kd(q,C) g(e(q))+1',where the component l' satisfies relations(f' ,n) = 0, (f' , q - qo) = 0, and 11 1'11 :S p. If k --4 00, then d(q,C) --4 O. Any valuecan be obtai ned forthe component kd(q,C)g (e(q)) as k --4 00 and d(q, C) --4 O. Thisis in agreement with the principle of virtual work of example I.
Example 6: Let Cbe the set Co of example 2 and let u(q) = (q - qo)IIq - qo11 -1.The distance d(q,C) is equal to II q - qoll - a. A function (J: TQ --4 lR is defined by(J (q, 8q) = 0 if Il q- qoll 2: a and
(f, 8q) :S atq , 8q) for each virtual displacement (q, 8q) E TQ (40)
implies the following relations for the external force f. If Il q- qoII 2: a, then 1 = O.If Ilq- qoll < a, then 1 = -kd(q, C )g(u(q))+l' with 'J', u(q)) = 0 and (f' , 8q) :Skd(q, C)v 118qll if (g(u(q)), 8q) = O. If k --4 00, then d(q, C) --4 O. The component(f, u(q) ) = - kd(q, C) can have any negative limit and (f', 8q) :S - (f ,u(q) )v I18qll·This is in agreement with the principle of virtual work of example 2.
Example 7: A model for the system in example 3 can be easily constructed even ifthe boundary of the set C= COis not smooth. The distance d(q, C) is defined by
d(q,C) = - (g(i) ,q - qo)
if (g(j ), q - qo) 2: 0 and (g(i ), q - qo) < 0,
d(q,C) = - (g(j ), q - qo)
(4 1)
(42)
410 WLODZIM IERZ M . TULCZYJEW
if (g(i ), q - qo) ~ 0 and (g(j ), q - qo) < 0, and
if (g(i ),q - qo) < 0 and (g(j ), q - qo) < O. A vector field e(q) is defined is definedoutside ofC by e(q) = -i if (g(j ), q - qo) ~ 0 and (g(i ), q - qo) < 0,
e(q) = - j (44)
if (g(i ), q - qo) ~ 0 and (g(j ), q - qo) < 0 and
if (g(i ), q - qo ) < 0 and (g(j ), q - qo) < O. A function a on TQ is defined bya (q, 8q) = 0 if q E C and a (q, 8q) = k(g(e(q)), 8q) if q rf: C. The constitutive set ofexample 3 is obtained from the principle of virtual work
(j ,8q) = a(q ,8q) for each virtual displacement (q,8q) E TQ (46)
with k ----+ 00 and d(q, C ) ----+ O.
Example 8: The construction of the model in the preceding example followed exactlythe prescription given at the beginning of the present section. This construction cannot be directly applied to the set C = CO of example 4. It can be applied to themodified set
((g(i ), q - qo) - r)2 + ((g(j) , q - qo) - r )2 > r }. (47)
The original set Co is obtained as the limit as r ----+ O.
5. KINEMATICS OF AUTONOMOUS SYSTEMSAND SCLERONOMIC CONSTRAINTS
Motio ns of a material point in the Euclidean affine space Q are curves ~: I ----+ Qparameterized by time t in an open interval I c R We have the tangentprolongation«.».I ----+ T Q : t f---+ (~ ( t), ~ (t ) ) and the second tangent prolongation (~, ~ , ~) : 1 ----+T 2 Q : t f---+ (~(t) ,~(t) ,~(t ) ) of a motion f .
Variational formulations of analytical mechanics require the concept of a virtualdisplacement ofa motion. A virtual displacement of a motion ~ is a mapping (~ , 8~) :1 ----+ T Q: t f---+ (~(t) , 8~(t ) ) . This mapping is obtained from a homotopy
x : lR x I----+ Q .
The base curve X(O, .) is the motion F, The virtua l displacement is the mapping
(~ , 8~): I ----+ TQ : t f---+ tX(-, t)( O).
(48)
(49)
A mapping (~ , ~ , 8~ , 8~) : I ----+ TTQ : t f---+ (~(t) , ~ (t ) , 8~(t) , 8~(t)) .is obtained froma virtua l displacement (~ , 8~) as the composition "-q 0 (~, 8~ , ~ , 8~) of the tangent
A NOTE ON HOLONOMIC CONSTRAI NTS 411
prolongation (~, 8~ , ~, 8~) with the involution KQ. Virtual displ acements are subjectto constraints. All considered versions of constraints can eventuall y be reduced todifferential equations formulated in terms of a subset C ( l ,l ) C TTQ such that if(q, q, Sq, 8q) E C (1 ,I ), then (q, q, k8q, k8q) E C ( l ,l ) for each number k 2: O. Anadmissible virtual displacement (~ , 80 is required to satisfy the condition
(~(t), ~ (t ) , 8~ (t ) , 8~ (t ) ) E C (l ,l )
for each t c I. This cond ition implies conditions
(~( t) , ~ (t)) E C (O,I ) ,
( ~( t ), 8~ (t)) E C(1 ,O) ,
and~ (t ) E C (O,O)
for each t e I , Sets C (O,! ), C( I ,O), and C (O,O) are defined by
(50)
(51)
(52)
(53)
C (O,I ) = { (q,q) E TQ; (q,q, 8q,8q) E C (l ,l ) for some (8q,8q) E V x V} ,
(54)
C (I ,O) = { (q,8q) E TQ; (q,q,8q, 8q) E C (l ,l ) for some (q,8q) E V x V } ,
(55)and
C (O,O) = {q E Q; (q,q) E C (O,I ) for some q E V}. (56)
Condition (51) is a differential equation for the mot ion ~: I -t Q. Constraints areusually discussed it terms of this equation. The inclusion
C (O, I ) c T C (O,O) (57)
must be satisfied since it is a necessary integrability condition for the equation (51) .Constraints will be called configuration constraints if
(58)
Velocity constraints are said to be linear if the set C (O,O) is a submanifold of Q andC(O ,I ) is a distribution on this submanifold. Linear constraints are said to be holonomic if C(O,I) is integrable in the sense of Frobenius. Configuration constraints area special case of holonomic constraints. Sets C (I ,O) and C(1 ,I ) are not usually discussed directly even if information contained in the velocity constraints C (O, I ) is notsufficient for the application of variational methods . The condition (50) is equivalentto
(~ ( t ) , 8~(t) , ~ (t ) , 8~ (t)) = t(~ , 8~)(t ) E KdC( I ,I ) ) c TTQ. (59)
It is a differential equation for the virtual displacement (~ , 8~): I -t TQ. The inclusion
(60)
412 W LODZIMIERZ M . TuLCZYJEW
is a necessary integrability condition for this equation.Two different methods of constru ctin g the set C CI ,I) from the velocity constraints
C CO,I ) are found in an art icle of Arn old , Kozlov, and Neis htadt (1993).
1. In vaconomic mechanics the natural constru ction
C CI ,I ) = TC CO,I)
is used. This con stru ction is the resul t of the differenti al equation
tx (s , ·)(t ) E C CO,I)
(6 1)
(62)
imposed on curves X(s,·) also with s -I- O. See [Am] for modifi cations of thisconstruction necessary when virtual displacements vanishing at the ends of atime interval are requ ired. With these modifications the formula (61) is stillvalid. The set CCI,O) is the tangent set T C CO ,O) of CCO,O).
2. The d 'Alembert-Lagrange principl e is based on the inclusion
CC I,I) C C Cl,l ) = { (q, q,8q, 8q) E TT Q; (q, q, 0, 8q) E TC CO,I )} . (63)
Thi s construction derives from the condition
tx (' t)( s) E C CI ,O)
for each i e t and each s and the condition
tx (O, ·)(t) E C CO,I)
(64)
(65)
imposed on the base curve ~ = X(O , .) but not the curves X(s, .) for s -I- O. The
set C CI ,I ) may not repre sent an integrable differential equation (59) for (~ , 8~ ) .
The set C (l ,l ) is the integrable part OfC CI ,I ). If C CO,I) is a vector subbundle ofTQ, then C CI ,O) = C CO,I). If C CO,I ) is an affine subbundle, then C CI,O) is the
model bundle. In both cases
C CI,I) = { (q, q,8q, 8q) E TTQ; (q, q) E C CO ,I), (q,8q) E C CI ,O) } (66)
and
For configuration constraints both construction give the same result
C CI ,l ) = TTC CO,O).
(67)
(68)
A NOTE ON HOLONOMIC CONSTRAINTS
6. DYNAMICS OF UNCONSTRAINED AUTONOMOUS SYSTEMS
We see four possible fonnu1ations of dynamics:
413
A. Dynamics of a material point can be specified as a collection of boundary valuerelations with externalforces associated with time intervals. A boundary value relationfor a time interval [a,b] C JR is a relation between an arc ~: [a ,b] ---+ Q, a mapping(~ , cp) : [a,b] ---+ T*Q , and two covectors (~(a),7f(a)) and (~(b) , 7f(b)) . The mapping(~, cp) represents the external force applied to the material point along the arc ~ thecovectors (~(a),7f(a)), (~(b) , 7f(b)) are the initial momentum andfinal momentum. Itis convenient to consider the arc ~ : [a, b] ---+ Q and the mapping (~ , cp ): [a,b] ---+ T* Qthe restrictions to the interval [a ,b]of mappings ~: I ---+ Q and (~ , cp ) : I ---+ T *Qdefined on an open interval I C JR containing [a,b]. The covectors (~(a), 7f(a)) and(~(b),7f(b)) will be considered values ofa mapping (~ , 7f) : I ---+ T*Q at the endsof the interval. The two mappings (~, cp ): I ---+ T* Q and (~ , 7f) : I ---+ T*Q can becombined in a single mapping
for each virtual displacement (~, 8~) : [a,b] ---+ TQ obtained as a restriction to [a,b] ofa virtual displacement (~, 8~): 1---+ TQ. The term
m(g(~(t)) ,8~(t) )
is the derivative DT(~(t) , ~(t) , 8~(t) , 8~(t)) of the kinetic energy function
T : TQ ---+ JR: (q,q) f-' ; llql12
(72)
(73)
of a material point with mass m . The function A: TTQ ---+ JR represents the virtualaction of internal forces . For the sake of simplicity we assume that A is a linear form (alinear function of (8q, 8q)). The proposed principle of virtual action is more generalthan the Hamilton Principle. Note that the virtual displacements (~(a) , 8~(a)) and(~(b) , 8~(b)) of the end points of the arc do not vanish. Variational principles withvariations of end points but without external forces were considered by Schwinger.The momentum-velocity relation is a law of physics and is a part of dynamics of a
414 WLODZIMIERZ M . T ULCZYJEW
material point. This relation is included in the Schwinger version of the principle ofvirtual action but not in the Hamilton Principle.
Examples of the virtual action function A include the function
A: TTQ --> JR. : (q,q,6q,6q) f-4 e(A(q),6q) +e(DA(q,6q),q) (74)
for a charged particle in a magnetic field derived from the vector potential A : Q -->
V* and the function
A: TTQ --> JR. : (q,q,6q,6q) f-4 '1 (g(q),6q) (75)
for a material point immersed in a viscous medium. In the first of these examplesthe function A(q,q, Sq, 6q) is the derivative Do:(q,q, Sq, 6q) of the function 0: : TQ -->
JR. : (q,q) f-4 e(A(q),q). We will continue the analysis assuming that the function A isof the simpler type
B. Dynamics can be specified as the collection 'D of curves
(~, <p,7f): 1 -* Q x V* x V *
415
(82)
defined on open intervals I c JR with the property that for each time interval [a,b] C Ithe arc (~ , <p )I[a,b] and the covectors (~(a),7f(a)) and (~(b) , 7f (b)) are in the boundaryrelation D [a,bJ'
C. Dynamics can be specified as differential equations
and
<p(t ) - ir(t) = fL(~(t)) (83)
7f(t) = g(~(t)) (84)
for mappings (~ , <p ,7f) : I -* Q x V * x V *. These equations will be denoted by D.D. Dynamics can be specified as differential equations
and
<p(t ) = mg(~(t)) + fL(~(t)) (85)
7f(t) = g(~(t)) (86)
for mappings (~ , sp , 7f) : I -* Q x V * x V *. These equations will be denoted by E.Of the four formulations of dynamics version A is fundamental. The family of
curves 'D and the differential equations D and E introduced in B, C , and D are auxiliary objects.
Differential equations D and E are obviously equivalent.We show that the family 'D is the set of solutions of the equations D. The proof
is based on version (79) of the principle of virtual action . If a curve (~ , ip , 7f) : I -*
Q x V * x V * is in 'D and (~, 8~): 1-* TQ is an arbitrary virtual displacement, thenthe equality (79) holds for all intervals [a,b] C I. It follows that the equality
holds at each t e I, This implies that equations D are satisfied due to arbitrariness ofthe vectors 8~(t) and 8~(t) . Conversely if (~ , <p, zr} : 1 -* Q x V * x V * is a solutionof D, then the equality (87) holds in I with an arbitrary displacement (~, 8~): I -*
TQ. The validity of the principle of virtual action for each time interval [a,b] C I isestablished by integration. Hence, (~ , <p, 7f) is in 'D.
It follows from the definition of'D that this family is constructed from the boundaryvalue relations. We show that elements ofboundary value relations can be constructedfrom elements of 'D. Let [a,b]be a time interval included in an open interval I C JRand let (~ , ip , 7f): 1 -* Q x V * x V * be a mapping such that
(~ , <p )I[a,b], (~(a) , 7f(a)) , (~(b) , 7f(b)))
is in D [a,bj. It follows from version (78) of the principle of virtual action that themapping (~ , <p ) satisfies the equation
<p(t ) = mg(~(t)) + fL(~(t)) (88)
416 WLODZIMIERZ M . T ULCZYJEW
in [a, b] and thatn(a) = g(~(a)) and n(b) = g(~(b)) .
Let mappings <p' : I ----7 V* and n ' : I ----7 V* be defined by
<p' (t ) = mg(~(t)) + f-L(~(t))
andn'(t) = mg(~(t)) .
(89)
(90)
(91)
The mapping (~ , <p' ,n/) : I ----7 Q x V* x V* is in 2) since it is a solution of E . Theboundary value data extracted from this mapping are in the boundary value relationsince (~ , <p' )I[a,b] = (~ , <p )I[a, b], n' (a) = n(a), and n' (b) = n(b).
7. DYNAMICS WITH CONFIGURATION CONSTRAINTS
We list four possible formulations of dynamics with constraints analogous to the fourformulations in the preceding section. As defined in Section 5 an admissible virtualdisplacement is a mapping (~ , o~) : 1 ----7 TQ satisfying the condition
(~(t), ~(t) , o~(t) , o~(t)) E C (1,l ) = TTC(O ,O) (92)
for each t e I,
A. Dynamics of a material point can be considered a collection of boundary valuerelations associated with time intervals. An element
( (f-L(~(t)),o~(t) ) - m(g (~ ( t ) ) , o~ ( t )) ) dt . (94)
for each admissible virtual displacement (~ , o~) : [a,b] ----7 TQ obtained as a restrictionto [a, b] of an admissible virtual displacement (~, o~): I ----7 TQ. The mapping f-L isdefined on C(1,Q) = TC(O ,O) . The condition ~(t) E C (O,O) for each t e I is implied.
There are again the equivalent versions of this variational principle
B. Dynamics can be specified as the collection 'D of curves
(~ , cp ,IT): 1 ---+ Q x V* x V * (97)
defined on open intervals I c IR such that for each time interval [a, b] C I the arc(~, cp )I[a, b]and the covectors (~(a), IT(a)) and (~(b), IT (b)) are in the boundary relation D [a,bj.
C. Dynamics can be specified as the differential equation
to be satisfied by a curve (~ , cp, IT): I ---+ Q x V* x V * at each t E l and for each(q,~(t) , 8~(t) , 8~(t) ) E C ( l ,l ). This is equivalent to equations
satisfied by a mapping (~ , ip , IT) : I ---+ Q x V * x V * at each t E l and each(~(t), 8~( t)) E C (1 ,O) .
The four formulations are valid for configuration constraints and are equivalent asin the case ofunconstrained systems. The situation is much more complex in the caseof more general constraints. The method of models could be a tool for testing thevalidity of different formulati ons. We will apply this tool to the momentum- velocityrelation. Note that the usual momentum- velocity relation
IT( t) = g(~(t)) (103 )
is replaced by the equation (102). We will attempt a justification of this modificationof the momentum- velocity relation based on models of configuration constraints.
418 W LODZIMIERZ M. TuLCZYJEW
8. MODELS OF AUTONOMOUS SYSTEMSWITH CONFIGURATION CONSTRAINTS
Let a material point of mass m be constrained to a plane C (O,O) C Q passing througha point qo and orthogonal to a unit vector n. We will assume that there are no internalforces and no external forces are applied. The constraint will be modeled by a stronginternal elastic force k(g(q - qo),n)g(n). Let an initial momentum (~(a) , 7l" (a)) suchthat ~(a) E C (O,O) and (7l" (a), 11, ) = °be applied to the point. The solution of thedynamical equations will be the mapping (~ , ip, zr}: R ---4 Q x V * x V* with ~ (t) =~(a) + m- 1g- 1(7l"(a)) (t - a), cp(t ) = 0, and 7l"(t) = 7l" (a). If the constraint isreplaced by the elastic force the solution mapping will be the same. Let now theinitial momentum have a non zero component (7l" (a), 11, ). For the unconstra ined modelthe solution is the mapping (~ , cp, 7l") with
and w = Jk/m. The oscillati on may be invisible since the amplitude w- 1(7l" (a), 11, )may be small due to the high value of w. The component (7l" (a),n)g(n) cos w(t - a) ofthe momentum transverse to the plane C (O,O) depends on the initial value (7l" (a),n)g(n )changes rapidly and is arbitrary within certain limits. This component can be detectedby making the material point collide with an unconstrained mass. Time dependentexternal forces and curvature of constraint set C (O,O) may even cause the transversecomponent of momentum influence the visible part of the motion along the constraint.The element of the boundary value relation for the idealized constrained system iscomposed of the mapping (~, cp ) : [a, b] ---4 Q x V* with
cp(t ) = 0, and covectors (~(a) , 7l" (a)) and (~( b) , 7l" (b)) satisfying the equality 7l"(b) (7l" (b), n)g(n) = 7l"(a) - (7l" (a), n)g(n) . The transverse component (7l" (a), n)g(n) ofthe initial momentum is arbitrary. Due to the rapidity of oscillations the final valueof the transverse component (7l" (b), n)g(n) of final momentum should be consideredarbitrary and independent of the initial value.
This analysis based on a purely elastic model suggests that the virtual action principle
- m(g(~(b)), 8~ (b) ) +m(g (~ (a ) ) , 8~ (a ) ) (106)
A NOTE ON HOLONOMIC CONSTRAINTS 419
for each admissible virtual displacement (~ , 8~) : [a,b] -* TQ is appropriate for material points subject to configuration constraints . If this formulation of dynamics withconfiguration constraints is adopted, then the momentum-velocity relation
momentum = mass x velocity
is no longer valid. Velocity is the rate of change of configuration. The component of velocity transverse to the constraint set is zero. This is not true of the transverse component of momentum. Along directions tangent to the constraint the usualmomentum-velocity relation holds. Other models may be found appropriate in certainsituations . When a rigid bar is struck with a hammer it starts an invisible vibrationdetectable through the sound it emits. The sound is due to momentum (and energy)transfer to air molecules colliding with the surface of the bar. The vibration will eventually die away although it may continue for a long time if the bar is placed in vacuum.The inevitable damping can be taken into account by supplementing the elastic forcek(g(q - qo),n)g(n) with a viscous damping force I(g (~ ( t ) ) , n)g(n). It may be appropriate to consider the transverse component of the initial momentum arbitrary andthe transverse component of the final momentum effectively reduced to zero. In casesof relatively short time intervals the effect of the damping can be ignored. In cases ofstrong damping and relatively long time intervals. It may be correct to assume that thetransverse component of the initial momentum is completely absorbed in an essentially inelastic collision with the suspension and not transferred to the material point.In such cases the boundary value relation will be a solution of the Hamilton principlewith external forces:
for each admissible virtual displacement (~, 8~): [a,b] -* TQ with 8~(a) = 0 and8~(b) = O. The equalities 7r(a) = g(~(a)) and 7r(b) = g(~(b)) supplement thevariational principle.
Istituto Nazionale di Fisica Nucleare
NOTE
* In honour of my friend John Stachel.
REFERENCE
Arnold, V. I., V. V. Kozlov, and A. I. Neishtadt. 1993. "Aspects of Classical and Celestia l Mechanics." InDynamical Systems lIl, ed. V. I. Arnold. New York: Springer-Verlag .
