Applying the principle of angular momentum to constrained systems of point masses

With the addition of the collinearity (or “centrality”) condition, Newton's third law is said to hold in the “strong” form.

For background material see, e.g.,

H.

 

Goldstein

, (; , ), Sections 1-1 and 1-2; 3rd edition, 2001, co-authored by C. P. Poole and J. L. Safko; J. L. Synge, “Classical Dynamics,” in Handbuch der Physik (Encyclopedia of Physics), Vol. III/1: Principles of Classical Mechanics and Field Theory, edited by S. Flügge, pp. 1–225 (Springer-Verlag, Berlin, 1960), Section 44. Similar treatments can be found in standard textbooks on mechanics.

W.

 

Stadler

, “,”  , – ().

In the case of a simple pendulum, if the constraint is replaced by two collinear internal forces, the principle of angular momentum then holds for it as a theorem.

Related aspects of Stadler's point mass model were considered by

J.

Casey

and

W.

Stadler

, “,”  , – ()

and by

W.

 

Stadler

, “Controllability implications of Newton's third law,”  , – ().

S.

Felszeghy

, “,”  , – (), challenged Stadler's conclusions in Ref. 4. Felszeghy pointed out that the removal of the rigid rod resulted in missing information, but he did not seem to realize that some new assumption must be made; see Sec. II of the present discussion. Felszeghy also proposed a 4-particle pendulum, where an additional massless particle braces the double pendulum sideways, resulting in a truss structure. Actually, infinitely many different constraining systems are imaginable that produce the same motion of the two point masses but have different internal force systems.

W.

 

Stadler

, in his “Rebuttal to ‘On the adequacy of Newtonian particle mechanics for solving the rigid double pendulum,’”  , – (), defended his views on the 3-particle model. In my opinion, a single rigid rod is the simplest constraint system for the double pendulum and ought to be dealt with directly.

Each such action-reaction pair acts on two particles, one being a point mass and the other belonging to the continuum; the two particles are coincident. The action-reaction pair therefore produces no resultant torque.

How to deal with unknown constraint reactions was recognized as a thorny problem early in the history of mechanics. During the half century following the publication of Newton's Principia in 1687, it was gradually recognized that some statement should be made about the effect of the internal forces as a whole. Reference may be made to the following excellent older books:

L. A.

 

Pars

, (, ), Sections 19.5 and 19.7; H. Lamb, Dynamics (U.P., Cambridge, 1942), Sections 52-53; E. J. Routh, The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies (Macmillan and Co., Limited, London and New York, 1905; reprinted by Dover Publications, New York, 1960), Sections 66-70. A common suggestion is to regard a two- or three-dimensional rigid body as a finite number of point masses set in a rigid massless frame—“an imponderable skeleton, carrying ponderable particles of matter” (Routh, Section 69, contributed by G. Airy). The skeleton consists of linked massless rigid rods and the point masses are attached at the joints. The rods are two-force members, transmitting tensions and thrusts but no shearing forces or bending moments. Each rod exerts equal and opposite forces on the particles at its two ends. Consequently, considering the forces exerted by all of the rods on all of the point masses, one has an equilibrated system of forces. For the rigid double pendulum, the constraining massless rigid rod is one-dimensional and it is not a two-force member; I am making the assumption that the forces exerted on the rod by the pivot and the two point masses form by themselves an equilibrated force system.

The principle of linear momentum also holds for the complete pendulum $B$⁠, as can be readily deduced with the aid of Eqs. (1) and (4).

In the century following the publication of Newton's Principia in 1687, the leading mathematicians and natural philosophers struggled to extend the principles of mechanics to cover constrained systems (and especially rigid bodies), elastic solids, and incompressible inviscid fluids. Leonhard Euler (1707–1783), throughout his long career, made major contributions to mechanics, both by establishing basic equations and by solving difficult problems. After returning again and again to the foundations of mechanics, Euler, when already blind, presented a paper in Latin to the St. Petersburg Academy of Sciences on 16 October, 1775, in which he proposed that both the principles of linear momentum and angular momentum should hold for all types of motions of all kinds of (classical) bodies. An informative history of the principle of angular momentum may be found in the fifth essay in

C.

Truesdell

, (, , ).

See also

J.

 

Casey

, “Areal velocity and angular momentum for non-planar problems in particle mechanics,”  , – ().

See Section 196 of

C.

Truesdell

and

R. A.

Toupin

, “,” in , edited by

S.

Flügge

(, , ), pp. –.

The reader is also referred to the illuminating discussion by

S. S.

 

Antman

, “The simple pendulum is not so simple,”  , – ().

For particle systems, Lagrange's equations (including constraint forces) may be derived from Newton's second law; see

J.

 

Casey

, “,”  , – (). For a rigid body, Lagrange's equations follow from Euler's laws; see J. Casey, “On the advantages of a geometrical viewpoint in the derivation of Lagrange's equations for a rigid continuum,” Z. Angew. Math Phys. 46, S805–S847 (1995).