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• W1993472591 abstract "Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, March 25, 2003, final revision, February 6, 2004. Associate Editor: I. Mezic. In Lagrangian mechanics, under certain conditions, the Jacobi energy integral exists and plays a fundamental role (see 123456). More generally, when Jacobi’s integral does not exist, it is still possible to gain useful engineering information from a consideration of power versus rate-of-energy relations. In the present note, we are concerned with a system of N ⩾1 particles subject to general holonomic and non-holonomic constraints. The unconstrained physical system may be represented by an abstract particle P in a 3N-dimensional Euclidean configuration space. In the presence of holonomic constraints, the motion of P is confined to a submanifold M whose dimension is equal to the number of generalized coordinates needed to describe the system. In general, M moves through configuration space and may also change its shape with time.1 Now, the velocity v of P can always be expressed as the vector sum of two components v′ and v″ such that v″ is the velocity of the point A (say) of M that P occupies at time t, and v′ is the velocity of P relative to A. It will be shown that when this decomposition is employed, the corresponding portions P′ and P″ of the total power P of the forces acting on the particles, can be expressed as time derivatives (partial and total) of portions of the kinetic energy.2 These expressions furnish a convenient means for calculating the power expended in moving the manifold M, and in moving P relative to M. This is particularly useful in the former case, because the constraint forces that move M would have been eliminated from the Lagrangian analysis. The discussion is presented both in terms of physical quantities and abstract variables in configuration space. A few remarks regarding the desirability of the latter geometrical representation are in order: Several different approaches to Lagrange’s equations can be found in the literature. These vary both in generality and in the degree of physical insight that they provide. Some are based on d’Alembert’s principle and the principle of virtual work, while some others use variational principles. With the advent of Riemannian geometry and tensor calculus in the 19th century, a new, abstract approach to dynamical theory arose, represented most cogently, perhaps, by Hertz 7. In 1927, Synge 8 argued passionately for an approach to dynamics that is phrased in geometrical terms using the analytical apparatus of the tensor calculus (see also Section 186 of 1, as well as Synge’s address, 9, to the American Mathematical Society in 1935). For constrained particle systems, a derivation of Lagrange’s equations based on such an approach was first published by Synge and Schild 10, who proved that Lagrange’s equations are just the covariant components, in the configuration manifold M of the constrained system, of Newton’s second law. In this kind of derivation, no appeal to the concepts of virtual displacements, virtual velocities, nor virtual work is necessary (although the majority of authors of monographs on tensor calculus still prefer to employ them (see, e.g., 1112)). Moreover, the exact physical content of Lagrange’s equations is revealed clearly. Recently, Casey 13 showed that, in Synge and Schild’s type of derivation, it is actually possible to bypass the cumbersome manipulations of Christoffel symbols, while maintaining the logical thrust of the original argument.3 An additional major advantage of the geometrical approach to dynamics is that it places the subject in the rich mathematical environment of the global theory of differential equations on manifolds, a theory which is undoubtedly one of the most beautiful and powerful in all of mathematics.4Consider a system consisting of particles Pi i=1,…,N moving relative to a Newtonian frame of reference under the influence of forces and possibly subject to time-dependent holonomic and non-holonomic constraints. Let ri, with rectangular Cartesian components xi1,xi2,xi3, be the position vector of the particle Pi relative to the origin o of the Newtonian frame, let Mi >0 be the mass of Pi, and let the total mass of the system be m. Let the resultant force vector acting on Pi be Fi, and denote its rectangular components by Fi1,Fi2,Fi3. We represent the physical system by an abstract particle P of mass m moving in a fixed 3N-dimensional Euclidean vector space, configuration space, E3N, as follows: The coordinates ui i=1,2,…,3N of P, taken along mutually orthogonal axes through an arbitrarily chosen origin O, are identified as u3i−2,u3i−1,u3i=xi1,xi2,xi3,i=1,2,…,N. Correspondingly, the position of P may be represented by its position vector r. A metric on E3N may be defined by (1)md2P,O=∑i=1NMiris˙ri=∑i=13Nmiui2,where m3i−2=m3i−1=m3i=Mi i=1,2,…,N. Thus, the distance d of P from O is defined to be the radius of gyration of the particle system about the origin o in physical space.5 The corresponding inner product is (2)[r,r*]=1m ∑i=1NMiris˙ri*,where the asterisk denotes a second set of position vectors for the system. The position vector of P can be expressed as r=∑i=13Nuiei, where ei 1=1,2,…,3N are pairwise orthogonal basis vectors, whose magnitudes are determined by (2).6 A reciprocal basis ei i=1,2,…,3N may be defined by the conditions [ei,ej]=δji,i=1,2,…,3N;j=1,2,…,3N, where δji is the Kronecker delta, having the value unity if i=j, and zero otherwise. We introduce an abstract force vector Φ in E3N by Φ=∑i=13NΦiei, with Φ3i−2,Φ3i−1,Φ3i=Fi1,Fi2,Fi3,i=1,…,N. Newton’s second law, written for each particle Pi, is equivalent to the vector equation (3)Φ=mr¨in E3N (see 13). Thus, the dynamics of the unconstrained physical system is now represented by the dynamics of a single abstract particle P of mass m moving through configuration space. In the presence of L holonomic constraints L<3N, which are allowed to be time-dependent, P will be confined to a moving manifold M, called the configuration space of the constrained system, or simply the constraint manifold, of dimension n=3N−L.7 Let qα α=1,2,…,n be convected coordinates on M; these are our “generalized coordinates.” The position vector of P can now be written as a function r=rqα,t. The covariant basis vectors in M are defined by (4)aα=∂r∂qαα=1,2,…,n.The inner products aαβ=[aα,aβ],(α=1,2,…,n;β=1,2,…,n) furnish a Riemannian metric on M. The velocity v of the particle P has the decomposition (5a)v=v′+v″,(5b)v′=∑α=1nq˙αaα,(5c)v″=∂r∂t.The component v″=v″qα,t is the velocity of the point A of M that P instantaneously occupies at time t; the component v′=v′qα,q˙α,t is the velocity of P relative to A, and it lies in the tangent space to M at A. In general, these two components are not orthogonal to one another. The kinetic energy of the system can be expressed as (6)T=12m[v,v]=T2+T1+T0,where8(7a)T2=12m[v′,v′]=12m∑α=1n∑β=1naαβq˙αq˙β,(7b)T1=m[v′,v″]=m∑α=1nbαq˙α,bα=[v″,aα],α=1,2,…,n(7c)T0=12m[v″,v″].The Lagrange’s equations for the system can be written in the general form (8)ddt ∂T∂q˙γ−∂T∂qγ=Qγ,γ=1,2,…,nwhere (9)Qγ=[Φ,aγ]γ=1,2,…,nare the covariant components of Φ in the manifold M.9 These “generalized forces” need not be derivable from a potential, and no assumption whatsoever is being made regarding the nature of the constraint forces (which may, for example, be dissipative). Recalling the relationship that the abstract vectors r, Φ bear to physical position and force vectors, and making use of (4), we obtain (10)Qγ=∑i=1NFis˙∂ri∂qγγ=1,2,…,n.Further, we note that the velocity of Pi i=1,2,…,N can be expressed as (11a)vi=r˙i=vi′+vi″,i=1,2,…,Nwith (11b)vi′=∑α=1n ∂ri∂qα q˙α,vi″=∂ri∂ti=1,2,…,N.Let P be the power of all of the forces acting on the system, i.e., (12)P=∑i=1NFis˙r˙i=[Φ,v].It is obvious from (12), (3), and (6) that (13)P=m[v˙,v]=T˙.Further, it is evident from (12), (5), and (11a11b) that P can always be decomposed as (14a)P=P′+P″,(14b)P′=[Φ,v′]=∑γ=1nQγq˙γ=∑i=1NFis˙vi′,P″=[Φ,v″]=∑i=1NFis˙vi″.One can now establish the following results:10(15a)P′=ddt T2−T0+∂T∂t,(15b)P″=ddt T1+2T0−∂T∂t.To prove (15a), note that by virtue of (14b)1 and (8), (16)P′=∑γ=1nddt ∂T∂q˙γ−∂T∂qγq˙γ=∑γ=1nddt ∂T∂q˙γ q˙γ−∂T∂q˙γ q¨γ−∂T∂qγ q˙γ.But, in view of (6) and (7a∂T∂q˙γ=12m∑β=1naγβq˙β+12m∑β=1naβγq˙β+mbγ,(17)γ=1,2,…,n.Hence, (18)∑γ=1n ∂T∂q˙γ q˙γ=2T2+T1.Also, the total time derivative of T is given by (19)T˙=∂T∂t+∑γ=1n∂T∂qγ q˙γ+∂T∂q˙γ q¨γ.From (16) and (19), it is clear that (20)P′=ddt ∑γ=1n ∂T∂q˙γ q˙γ+∂T∂t−T˙.Equation (15a) follows immediately from (20), (18), and (6). Equation (15b) may be readily deduced from (13), (14a), (6), and (20). We mention two important consequences of (15a): (I) Suppose that: (a) the constraint forces do not contribute to the power P′; and (b) the remaining forces are derivable from a potential function V. Let L=T−V be the Lagrangian function. Then, (21)ddt T2−T0+V+∂L∂t=0.To prove this, note that in view of assumption (b), the covariant components of the nonconstraint forces will be11(22)Qγ*=−∂V∂qγγ=1,2,…,n.The contribution of the components Qγ* to the power P′ is (23)∑γ=1nQγ*q˙γ=−∑γ=1n=1∂V∂qγ q˙γ=∂V∂t−V˙.By assumption (a), the constraint forces contribute nothing to P′. With the help of (14b)1 and (15a), it then follows that (24)∂V∂t−V˙=ddt T2−T0+∂L+V∂t,from which (21) can be concluded at once. (II) (Jacobi Integral). In addition to the conditions (a) and (b) assumed in (I), suppose that the Lagrangian does not depend explicitly on t. Then, (21) immediately yields the integral: (25)T2−T0+V=const.=E′say.Let us take some illustrative examples involving a single particle P of mass m. The configuration space now coincides with the physical three-dimensional space having the ordinary Euclidean metric. Example 1. Suppose that P moves on a horizontal plane (elevator floor) that is being driven vertically upwards in a Newtonian frame. The floor is the constraint manifold. Using a fixed rectangular Cartesian coordinate system, we may write the velocity components in (5b5c) as v′=xi˙+yj˙,v″=hk˙, where h=ht is a prescribed function. The force acting on P is F=Fxi+Fyj+N−mgk, where N is the force supplied by the floor. The kinetic energy of P comprises (26)T2=12mx˙2+y˙2,T1=0,T0=12mh˙2.Lagrange’s equations yield mx¨=Fx,my¨=Fy. The two portions of the power that appear in (14b) are P′=Fxx˙+Fyy˙,P″=N−mgh˙. Equations (15a,b) reduce to (27)P′=dT2dt,P″=dT0dt=mhh¨˙.Example 2. Suppose that P is constrained to move on a frictionless horizontal circle fixed at the origin of a Newtonian frame, and having a prescribed radius lt. Suppose that no other forces are applied to P. Here, the constraint manifold is a time-dependent circle centered at the origin and we may take the polar coordinate θ as our generalized coordinate. Let er,eθ be the usual orthonormal basis of polar coordinates. The covariant basis vector is a1=∂r/∂θ=leθ. The velocity components in (5b5c) are v′=θ˙leθ,v″=le˙r. The kinetic energy of P comprises (28)T2=12ml2θ˙2,T1=0,T0=12ml˙2.The only force acting on P is that of the constraint and it points in the radial direction. The Lagrange’s equation is d/dt∂T/∂θ˙=0, which yields the angular momentum integral ml2θ˙=const. The power decomposes as P′=0=ddt T2−T0+∂T∂t,(29)P=P″=2 dT0dt−∂T∂t=ml−¨lθ˙2l˙.Example 3. Suppose that P is constrained to move on a frictionless rigid circle that is rotating with constant angular velocity Ω about a fixed vertical axis under the influence of gravity. Let us use spherical coordinates r, θ, φ, the angle φ being measured from the positive z-axis. Taking φ as our generalized coordinate, the position vector of P is r=rφ,t, and the velocity components in (5b5c) are v′=φ˙leφ,v″=lΩ sin φeθ. The kinetic and potential energies are T2=12ml2φ˙2,T1=0,T0=12ml2Ω2 sin2φ,(30)V=mgl cos φ.The constraint force has the form N=Nrer+Nθeθ. The Lagrange’s equation yields φ−¨g/l+Ω2 cos φsin φ=0. The Lagrangian does not depend explicitly on time. We therefore have a Jacobi integral (25). The portion P″ of the power is (31)P″=2 dT0dt=ml2Ω2φ ˙sin 2φ,i.e., the power supplied by Nθ.Example 4. Suppose that P is confined to move on a fixed plane z=0 under the action of potential forces. If fixed Cartesian coordinates are used, obviously P″=0 and T+V=const. Instead, take another frame of reference, also with z=0, but which rotates with constant angular velocity Ω about the z-axis. Let b1,b2 be an orthonormal basis fixed to the rotating frame and let the position vector of P be written as r=ρ1b1+ρ2b2. The constraint manifold M coincides with the rotating frame. Choosing ρ1 and ρ2 as generalized coordinates, we see that the covariant basis on M is b1,b2. The velocity components in (5b5c) are (32)v′=ρ˙1b1+ρ˙2b2,v″=Ω−ρ2b1+ρ1b2.The kinetic energy of P has the three portions T2=12mρ˙12+ρ˙22,T1=mΩρ1ρ˙2−ρ˙1ρ2,(33)T0=12mΩ2ρ12+ρ22,and V=Vρ1,ρ2,t. The Lagrange’s equations yield −∂V∂ρ1=mρ¨1−2Ωρ˙2−Ω2ρ1,(34)−∂V∂ρ2=mρ¨2+2Ωρ˙1−Ω2ρ2.In view of the definitions (14a14b), (35a)P′=−gradVs˙ρ˙1b1+ρ˙2b2=−∂V∂ρ1 ρ˙1−∂V∂ρ2 ρ˙2=∂V∂t−V˙,(35b)P″=−gradVs˙Ω−ρ2b1+ρ1b2=Ωρ2 ∂V∂ρ1−ρ1 ∂V∂ρ2.Applying the relations (15a15b) to (33), we find that (36a)P′=ddt T2−T0=mρ˙1ρ¨1+ρ˙2ρ¨2−mΩ2ρ1ρ˙1+ρ2ρ˙2,(36b)P″=ddt T1+2T0=mΩρ1ρ¨2−ρ2ρ¨1+2mΩ2ρ1ρ˙1+ρ2ρ˙2.With the help of (34), Eqs. (35a,b) are seen to be equivalent to (36a36b). Example 5. Suppose that P is confined to move on a fixed horizontal plane, with rectangular Cartesian coordinates x and y, under the action of potential forces and subject to the nonholonomic constraint −sin ωtx˙+cos ωty˙=0, where ω=const. Assume that the constraint force is parallel to the vector having the components −sin ωt,cos ωt. The kinetic energy is (37)T=T2=12mx˙2+y˙2.The Lagrange’s equations are (38)mx¨+∂V∂x=−λ sin ωt,my¨+∂V∂y=λ cos ωt,where the multiplier λ is plus or minus times the magnitude of the constraint force. The power P″ is zero and the constraint force makes no contribution to P′; (II) yields the energy integral T+V=const." @default.
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• W1993472591 title "A Basic Power Decomposition in Lagrangian Mechanics" @default.
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