1.General Equations of Impulsive Motion.The systems contemplated by Lagrange are composed of discrete particles, or of rigid bodies, in finite number, connected (it may be) in various ways by invariable geometrical relations, the fundamental postulate being that the position of every particle of the system at any time can be completely specified by means of the instantaneous values of a finite number of independent variables q1, q2, ... qn, each of which admits of continuous variation over a certain range, so that if x, y, z be the Cartesian co-ordinates of any one particle, we have for examplex = ƒ(q1, q2, ... qn), y = &c., z = &c.,(1)where the functions ƒ differ (of course) from particle to particle. In modern language, the variables q1, q2, ... qnaregeneralized co-ordinatesserving to specify theconfigurationof the system; their derivatives with respect to the time are denoted by q˙1, q˙2, ... q˙n, and are called thegeneralized components of velocity. The continuous sequence of configurations assumed by the system in any actual or imagined motion (subject to the given connexions) is called thepath.For the purposes of a connected outline of the whole subject it is convenient to deviate somewhat from the historical order of development, and to begin with the consideration ofImpulsive motion.impulsivemotion. Whatever the actual motion of the system at any instant, we may conceive it to be generated instantaneously from rest by the application of proper impulses. On this view we have, if x, y, z be the rectangular co-ordinates of any particle m,mẋ = X′, mẏ = Y′, mz˙ = Z′,(2)where X′, Y′, Z′ are the components of the impulse on m. Now let δx, δy, δz be any infinitesimal variations of x, y, z which are consistent with the connexions of the system, and let us form the equationΣm(ẋδx + ẏδy + z˙δz) = Σ(X′δx + Y′δy + Z′δz),(3)where the sign Σ indicates (as throughout this article) a summation extending over all the particles of the system. To transform (3) into an equation involving the variations δq1, δq2, ... of the generalized co-ordinates, we haveẋ =∂xq˙1+∂xq˙2+ ..., &c., &c.∂q1∂q2(4)δx =∂xδq1+∂xδq2+ ..., &c., &c.∂q1∂q2(5)and thereforeΣm(ẋδx + ẏδy + z˙δz) = A11q˙1+ A12q˙2+ ...)δq1+ (A21q˙1+ A22q˙2+ ...)δq2+ ...,(6)whereArr= Σm{ (∂x)²+(∂y)²+(∂z)²},∂qr∂qr∂qr(7)Ars= Σm{∂x∂x+∂y∂y+∂z∂z}= Asr.∂qr∂qs∂qr∂qs∂qr∂qsIf we form the expression for the kinetic energy Τ of the system, we find2Τ = Σm(ẋ² + ẏ² + z˙²) = A11q˙1² + A22q˙2² ... 2A12q˙1q˙2+ ...(8)The coefficients A11, A22, ... A12, ... are by an obvious analogy called thecoefficients of inertiaof the system; they are in general functions of the co-ordinates q1, q2,.... The equation (6) may now be writtenΣm(ẋδx + ẏδy + z˙δz) =∂Τδq1+∂Τδq2+ ...∂q˙1∂q˙2(9)This maybe regarded as the cardinal formula in Lagrange’s method. For the right-hand side of (3) we may writeΣ(X′δx + Y′δy + Z′δz) = Q′1δq1+ Q′2δq2+ ... ,(10)whereQ′r= Σ(X′∂x+ Y′∂y+ Z′∂z).∂qr∂qr∂qr(11)The quantities Q1, Q2, ... are called thegeneralized components of impulse. Comparing (9) and (10), we have, since the variations δq1, δq2,... are independent,∂Τ= Q′1,∂Τ= Q′2, ...∂q˙1∂q˙2(12)These are the general equations of impulsive motion.It is now usual to writepr=∂Τ.∂q˙r(13)The quantities p1, p2, ... represent the effects of the several component impulses on the system, and are therefore called thegeneralized components of momentum. In terms of them we haveΣm(ẋδx + ẏδy + z˙δz) = p1δq1+ p2δq2+ ...(14)Also, since Τ is a homogeneous quadratic function of the velocities q˙1, q˙2...,2Τ = p1q˙1+ p2q˙2+ ...(15)This follows independently from (14), assuming the special variations δx = ẋdt, &c., and therefore δq1= q˙1dt, δq2= q˙2dt, ...Again, if the values of the velocities and the momentaReciprocal theorems.in any other motion of the system through the same configuration be distinguished by accents, we have the identityp1q˙′1+ p2q˙′2+ ... = p′1q˙1+ p′2q˙2+ ...,(16)each side being equal to the symmetrical expressionA11q˙1q˙′1+ A22q˙2q˙′2+ ... + A12(q˙1q˙′2+ q˙′1q˙2) + ...(17)The theorem (16) leads to some important reciprocal relations. Thus, let us suppose that the momenta p1, p2, ... all vanish with the exception of p1, and similarly that the momenta p′1, p′2, ... all vanish except p′2. We have then p1q˙′1= p′2q˙2, orq˙2: p1= q˙′1: p′2(18)The interpretation is simplest when the co-ordinates q1, q2are both of the same kind,e.g.both lines or both angles. We may then conveniently put p1= p′2, and assert that the velocity of the first type due to an impulse of the second type is equal to the velocity of the second type due to an equal impulse of the first type. As an example, suppose we have a chain of straight links hinged each to the next, extended in a straight line, and free to move. A blow at right angles to the chain, at any point P, will produce a certain velocity at any other point Q; the theorem asserts that an equal velocity will be produced at P by an equal blow at Q. Again, an impulsive couple acting on any link A will produce a certain angular velocity in any other link B; an equal couple applied to B will produce an equal angular velocity in A. Also if an impulse F applied at P produce an angular velocity ω in a link A, a couple Fa applied to A will produce a linear velocity ωa at P. Historically, we may note that reciprocal relations in dynamics were first recognized by H.L.F. Helmholtz in the domain of acoustics; their use has been greatly extended by Lord Rayleigh.The equations (13) determine the momenta p1, p2,... as linear functions of the velocities q˙1, q˙2,... Solving these, we can express q˙1, q˙2... as linear functions of p1, p2,... The resulting equations give us the velocities produced by any givenVelocities in terms of momenta.system of impulses. Further, by substitution in (8), we can express the kinetic energy as a homogeneous quadratic function of the momenta p1, p2,... The kinetic energy,as so expressed, will be denoted by Τ`; thus2Τ` = A`11p1² + A`22p2² + ... + 2A`12p-p2+ ...(19)where A`11, A`22,... A`12,... are certain coefficients depending on the configuration. They have been called by Maxwell thecoefficients of mobilityof the system. When the form (19) is given, the valuesof the velocities in terms of the momenta can be expressed in a remarkable form due to Sir W.R. Hamilton. The formula (15) may be writtenp1q˙1+ p2q˙2+ ... = Τ + Τ`,(20)where Τ is supposed expressed as in (8), and Τ` as in (19). Hence if, for the moment, we denote by δ a variation affecting the velocities, and therefore the momenta, but not the configuration, we havep1δq˙1+ q˙1δp + p2δq˙2+ q˙2δp2+ ... = δΤ + δΤ`=∂Τδq˙1+∂Τδq˙2+ ... +∂Τ`δp1+∂Τ`δp2+ ...∂q˙1∂q˙2∂p1∂p2(21)In virtue of (13) this reduces toq˙1δp1+ q˙2δp2+ ... =∂Τ`δp1+∂Τ`δp2+ ...∂p1∂p2(22)Since δp1, δp2, ... may be taken to be independent, we infer thatq˙1=∂Τ`, q˙2=∂Τ`, ...∂p1∂p2(23)In the very remarkable exposition of the matter given by James Clerk Maxwell in hisElectricity and Magnetism, the Hamiltonian expressions (23) for the velocities in terms of the impulses are obtained directly from first principles, and the formulae (13) are then deduced by an inversion of the above argument.An important modification of the above process was introduced by E.J. Routh and Lord Kelvin and P.G. Tait. Instead of expressing the kinetic energy in terms of the velocities alone, or in terms of the momenta alone, we may express it inRouth’s modification.terms of the velocities corresponding to some of the co-ordinates, say q1, q2, ... qm, and of the momenta corresponding to the remaining co-ordinates, which (for the sake of distinction) we may denote by χ, χ′, χ″, .... Thus, Τ being expressed as a homogeneous quadratic function of q˙1, q˙2, ... q˙m, χ˙, χ˙′, χ˙″, ..., the momenta corresponding to the co-ordinates χ, χ′, χ″, ... may be writtenκ =∂Τ, κ′ =∂Τ, κ″ =∂Τ, ...∂χ˙∂χ˙′∂χ˙″(24)These equations, when written out in full, determine χ˙, χ˙′, χ˙″, ... as linear functions of q˙1, q˙2, ... q˙m, κ, κ′, κ″,... We now consider the functionR = Τ − κχ˙ − κ′χ˙′ − κ″χ˙″ − ... ,(25)supposed expressed, by means of the above relations in terms of q˙1, q˙2, ... q˙m, κ, κ′, κ″, ... Performing the operation δ on both sides of (25), we have∂Rδq˙1+ ... +∂Rδκ + ... =∂Τδq˙1+ ... +∂Τδχ˙ + ... − κ∂χ˙ − χ˙δκ − ... ,∂q˙1∂κ∂q˙1∂χ˙(26)where, for brevity, only one term of each type has been exhibited. Omitting the terms which cancel in virtue of (24), we have∂Rδq˙1+ ... +∂Rδκ + ... =∂Τδq˙1+ ... − χ˙δκ − ...∂q˙1∂κ∂q˙1(27)Since the variations δq1, δq2, ... δqm, δκ, δκ′, δκ″, ... may be taken to be independent, we havep1=∂Τ=∂R, p2=∂Τ=∂R, ...∂q˙1∂q˙1∂q˙2∂q˙2(28)andχ˙ = −∂R, χ˙′ = −∂R, χ˙″ = −∂R, ...∂κ∂κ′∂κ″(29)An important property of the present transformation is that, when expressed in terms of the new variables, the kinetic energy is the sum of two homogeneous quadratic functions, thusΤ = ⅋ + K,(30)where ⅋ involves the velocities q˙1, q˙2, ... q˙malone, and K the momenta κ, κ′, κ″, ... alone. For in virtue of (29) we have, from (25),Τ = R −(κ∂R+ κ′∂R+ κ″∂R+ ...),∂κ∂κ′∂κ″(31)and it is evident that the terms in R which are bilinear in respect of the two sets of variables q˙1, q˙2, ... q˙mand κ, κ′, κ″, ... will disappear from the right-hand side.It may be noted that the formula (30) gives immediate proof of two important theorems due to Bertrand and to Lord Kelvin respectively. Let us suppose, in the first place, that the system is started by given impulses of certain types,Maximum and minimum energy.but is otherwise free. J.L.F. Bertrand’s theorem is to the effect that the kinetic energy isgreaterthan if by impulses of the remaining types the system were constrained to take any other course. We may suppose the co-ordinates to be so chosen that the constraint is expressed by the vanishing of the velocities q˙1, q˙2, ... q˙m, whilst the given impulses are κ, κ′, κ″,... Hence the energy in the actual motion is greater than in the constrained motion by the amount ⅋.Again, suppose that the system is started with prescribed velocity components q˙1, q˙2, ... q˙m, by means of proper impulses of the corresponding types, but is otherwise free, so that in the motion actually generated we have κ = 0, κ′ = 0, κ″ = 0, ... and therefore K = 0. The kinetic energy is thereforelessthan in any other motion consistent with the prescribed velocity-conditions by the value which K assumes when κ, κ′, κ″, ... represent the impulses due to the constraints.Simple illustrations of these theorems are afforded by the chain of straight links already employed. Thus if a point of the chain be held fixed, or if one or more of the joints be made rigid, the energy generated by any given impulses is less than if the chain had possessed its former freedom.2.Continuous Motion of a System.We may proceed to the continuous motion of a system. TheLagrange’s equations.equations of motion of any particle of the system are of the formmẍ = X, mÿ = Y, mz¨ = Z(1)Now let x + δx, y + δy, z + δz be the co-ordinates of m in any arbitrary motion of the system differing infinitely little from the actual motion, and let us form the equationΣm (ẍδx + ÿδy + z¨δz) = Σ (Xδx + Yδy + Zδz)(2)Lagrange’s investigation consists in the transformation of (2) into an equation involving the independent variations δq1, δq2, ... δqn.It is important to notice that the symbols δ and d/dt are commutative, sinceδẋ =d(x + δx) −dx=dδx, &c.dtdtdt(3)HenceΣm(ẍδx + ÿδy + z¨δz) =dΣm (ẋδx + ẏδy + z˙δz) − Σm (ẋδẋ + ẏδẏ + z˙δz˙)dt=d(p1δq1+ p2δq2+ ...) − δΤ,dt(4)by § 1 (14). The last member may be writtenṗ1δq1+ p1δq˙1+ ṗ2δq2+ p2δq˙2+ ... −∂Τδq˙1−∂Τδq1−∂Τδq˙2−∂Τδq2− ...∂q˙1∂q1∂q˙2∂q2(5)Hence, omitting the terms which cancel in virtue of § 1 (13), we findΣm(ẍδx + ÿδy + z¨δz) =(ṗ1−∂Τ)δq1+(ṗ2−∂Τ)δq2+ ...∂q1∂q2(6)For the right-hand side of (2) we haveΣ(Xδx + Yδy + Zδz) = Q1δq1+ Q2δq2+ ... ,(7)whereQr= Σ(X∂x+ Y∂y+ Z∂z).∂qr∂qr∂qr(8)The quantities Q1, Q2, ... are called thegeneralized components of forceacting on the system.Comparing (6) and (7) we findṗ1−∂Τ= Q1, ṗ2−∂Τ= Q2, ... ,∂q˙1∂q˙2(9)or, restoring the values of p1, p2, ...,d(∂Τ)−∂Τ= Q1,d(∂Τ)−∂Τ= Q2, ...dt∂q˙1∂q1dt∂q˙2∂q2(10)These are Lagrange’s general equations of motion. Their number is of course equal to that of the co-ordinates q1, q2, ... to be determined.Analytically, the above proof is that given by Lagrange, but the terminology employed is of much more recent date, having been first introduced by Lord Kelvin and P.G. Tait; it has greatly promoted the physical application of the subject. Another proof of the equations (10), by direct transformation of co-ordinates, has been given by Hamilton and independently by other writers (seeMechanics), but the variational method of Lagrange is that which stands in closest relation to the subsequent developments of the subject. The chapter of Maxwell, already referred to, is a most instructive commentary on the subject from the physical point of view, although the proof there attempted of the equations (10) is fallacious.In a “conservative system” the work which would have to be done by extraneous forces to bring the system from rest in some standard configuration to rest in the configuration (q1, q2, ... qn) is independent of the path, and may therefore be regarded as a definite function of q1, q2, ... qn. Denoting this function (thepotential energy) by V, we have, if there be no extraneous force on the system,Σ (Xδx + Yδy + Zδz) = − δV,(11)and thereforeQ1= −∂V, Q2= −∂V, ....∂q1∂q2(12)Hence the typical Lagrange’s equation may be now written in the formd(∂Τ)−∂Τ= −∂V,dt∂q˙r∂qr∂qr(13)or, again,ṗr= −∂(V − Τ).∂qr(14)It has been proposed by Helmholtz to give the namekinetic potentialto the combination V − Τ.As shown underMechanics, § 22, we derive from (10)dΤ= Q1q˙1+ Q2q˙2+ ... ,dt(15)and therefore in the case of a conservative system free from extraneous force,d(Τ + V) = 0 or Τ + V = const.,dt(16)which is the equation of energy. For examples of the application of the formula (13) seeMechanics, § 22.3.Constrained Systems.It has so far been assumed that the geometrical relations, if any, which exist between the various parts of the systemCase of varying relations.are of the type § 1 (1), and so do not contain t explicitly. The extension of Lagrange’s equations to the case of “varying relations” of the typex = ƒ(t, q1, q2, ... qn), y = &c., z = &c.,(1)was made by J.M.L. Vieille. We now haveẋ =∂x+∂xq˙1+∂xq˙2+ ..., &c., &c.,∂t∂q1∂q2(2)∂x =∂xδq1+∂xδq2+ ..., &c., &c.,∂q1∂q2(3)so that the expression § 1 (8) for the kinetic energy is to be replaced by2Τ = α0+ 2α1q˙1+ 2α2q˙2+ ... + A11q˙1² + A22q˙2² + ... + A12q˙1q˙2+ ...,(4)whereα0= Σm{ (∂x)²+(∂y)²+(∂z)²},∂t∂t∂t(5)αr= Σm{∂x∂x+∂y∂y+∂z∂z},∂t∂qr∂t∂qr∂t∂qrand the forms of Arr, Arsare as given by § 1 (7). It is to be remembered that the coefficients α0, α1, α2, ... A11, A22, ... A12... will in general involve t explicitly as well as implicitly through the co-ordinates q1, q2,.... Again, we findΣm (ẋδx + ẏδy + z˙δz) = (α1+ A11q˙1+ A12q˙2+ ...) δq1+ (α2+ A21q˙1+ A22q˙2+ ...) ∂q2+ ...=∂Τδq1+∂Τδq2+ ... = p1δq1+ p2δq2+ ...,∂q˙1∂q˙2(6)where pris defined as in § 1 (13). The derivation of Lagrange’s equations then follows exactly as before. It is to be noted that the equation § 2 (15) does not as a rule now hold. The proof involved the assumption that Τ is a homogeneous quadratic function of the velocities q˙1, q˙2....It has been pointed out by R.B. Hayward that Vieille’s case can be brought under Lagrange’s by introducing a new co-ordinate (χ) in place of t, so far as it appears explicitly in the relations (1). We have then2Τ = α0χ˙² + 2(α1q˙1+ α2q˙2+ ...) χ˙ + A11q˙1² + A22q˙2² + ... + 2A12q˙1q˙2+ ....(7)The equations of motion will be as in § 2 (10), with the additional equationd∂Τ−∂Τ= X,dt∂χ˙∂χ(8)where X is the force corresponding to the co-ordinate χ. We may suppose X to be adjusted so as to make χ¨ = 0, and in the remaining equations nothing is altered if we write t for χ before, instead of after, the differentiations. The reason why the equation § 2 (15) no longer holds is that we should require to add a term Xχ˙ on the right-hand side; this represents the rate at which work is being done by the constraining forces required to keep χ˙ constant.As an example, let x, y, z be the co-ordinates of a particle relative to axes fixed in a solid which is free to rotate about the axis of z. If φ be the angular co-ordinate of the solid, we find without difficulty2Τ = m (ẋ² + ẏ² +z˙²) + 2φ˙m (xẏ − yẋ) + {I + m (x² + y²)} φ˙²,(9)where I is the moment of inertia of the solid. The equations of motion, viz.d∂Τ−∂Τ= X,d∂Τ−∂Τ= Y,d∂Τ−∂Τ= Z,dt∂ẋ∂xdt∂ẏ∂ydt∂z˙∂z(10)andd∂Τ−∂Τ= Φ,dt∂φ˙∂φ(11)becomem (ẍ − 2φ˙ẏ − xφ˙² − yφ¨) = X, m (ÿ + 2φ˙ẋ − yφ˙² + xφ¨) = Y, mz¨ = Z,(12)andd[{I + m (x² + y²)} φ˙ + m (xẏ − yẋ)] = Φ.dt(13)If we suppose Φ adjusted so as to maintain φ¨ = 0, or (again) if we suppose the moment of inertia I to be infinitely great, we obtain the familiar equations of motion relative to moving axes, viz.m (ẍ − 2ωẏ − ω²x) = X, m (ÿ + 2ωẋ − ω²y) = Y, mz¨ = Z,(14)where ω has been written for φ. These are the equations which we should have obtained by applying Lagrange’s rule at once to the formula2Τ = m (ẋ² + ẏ² + z˙²) + 2mω (xẏ − yẋ) + mω² (x² + y²),
1.General Equations of Impulsive Motion.
The systems contemplated by Lagrange are composed of discrete particles, or of rigid bodies, in finite number, connected (it may be) in various ways by invariable geometrical relations, the fundamental postulate being that the position of every particle of the system at any time can be completely specified by means of the instantaneous values of a finite number of independent variables q1, q2, ... qn, each of which admits of continuous variation over a certain range, so that if x, y, z be the Cartesian co-ordinates of any one particle, we have for example
x = ƒ(q1, q2, ... qn), y = &c., z = &c.,
(1)
where the functions ƒ differ (of course) from particle to particle. In modern language, the variables q1, q2, ... qnaregeneralized co-ordinatesserving to specify theconfigurationof the system; their derivatives with respect to the time are denoted by q˙1, q˙2, ... q˙n, and are called thegeneralized components of velocity. The continuous sequence of configurations assumed by the system in any actual or imagined motion (subject to the given connexions) is called thepath.
For the purposes of a connected outline of the whole subject it is convenient to deviate somewhat from the historical order of development, and to begin with the consideration ofImpulsive motion.impulsivemotion. Whatever the actual motion of the system at any instant, we may conceive it to be generated instantaneously from rest by the application of proper impulses. On this view we have, if x, y, z be the rectangular co-ordinates of any particle m,
mẋ = X′, mẏ = Y′, mz˙ = Z′,
(2)
where X′, Y′, Z′ are the components of the impulse on m. Now let δx, δy, δz be any infinitesimal variations of x, y, z which are consistent with the connexions of the system, and let us form the equation
Σm(ẋδx + ẏδy + z˙δz) = Σ(X′δx + Y′δy + Z′δz),
(3)
where the sign Σ indicates (as throughout this article) a summation extending over all the particles of the system. To transform (3) into an equation involving the variations δq1, δq2, ... of the generalized co-ordinates, we have
(4)
(5)
and therefore
Σm(ẋδx + ẏδy + z˙δz) = A11q˙1+ A12q˙2+ ...)δq1+ (A21q˙1+ A22q˙2+ ...)δq2+ ...,
(6)
where
(7)
If we form the expression for the kinetic energy Τ of the system, we find
2Τ = Σm(ẋ² + ẏ² + z˙²) = A11q˙1² + A22q˙2² ... 2A12q˙1q˙2+ ...
(8)
The coefficients A11, A22, ... A12, ... are by an obvious analogy called thecoefficients of inertiaof the system; they are in general functions of the co-ordinates q1, q2,.... The equation (6) may now be written
(9)
This maybe regarded as the cardinal formula in Lagrange’s method. For the right-hand side of (3) we may write
Σ(X′δx + Y′δy + Z′δz) = Q′1δq1+ Q′2δq2+ ... ,
(10)
where
(11)
The quantities Q1, Q2, ... are called thegeneralized components of impulse. Comparing (9) and (10), we have, since the variations δq1, δq2,... are independent,
(12)
These are the general equations of impulsive motion.
It is now usual to write
(13)
The quantities p1, p2, ... represent the effects of the several component impulses on the system, and are therefore called thegeneralized components of momentum. In terms of them we have
Σm(ẋδx + ẏδy + z˙δz) = p1δq1+ p2δq2+ ...
(14)
Also, since Τ is a homogeneous quadratic function of the velocities q˙1, q˙2...,
2Τ = p1q˙1+ p2q˙2+ ...
(15)
This follows independently from (14), assuming the special variations δx = ẋdt, &c., and therefore δq1= q˙1dt, δq2= q˙2dt, ...
Again, if the values of the velocities and the momentaReciprocal theorems.in any other motion of the system through the same configuration be distinguished by accents, we have the identity
p1q˙′1+ p2q˙′2+ ... = p′1q˙1+ p′2q˙2+ ...,
(16)
each side being equal to the symmetrical expression
A11q˙1q˙′1+ A22q˙2q˙′2+ ... + A12(q˙1q˙′2+ q˙′1q˙2) + ...
(17)
The theorem (16) leads to some important reciprocal relations. Thus, let us suppose that the momenta p1, p2, ... all vanish with the exception of p1, and similarly that the momenta p′1, p′2, ... all vanish except p′2. We have then p1q˙′1= p′2q˙2, or
q˙2: p1= q˙′1: p′2
(18)
The interpretation is simplest when the co-ordinates q1, q2are both of the same kind,e.g.both lines or both angles. We may then conveniently put p1= p′2, and assert that the velocity of the first type due to an impulse of the second type is equal to the velocity of the second type due to an equal impulse of the first type. As an example, suppose we have a chain of straight links hinged each to the next, extended in a straight line, and free to move. A blow at right angles to the chain, at any point P, will produce a certain velocity at any other point Q; the theorem asserts that an equal velocity will be produced at P by an equal blow at Q. Again, an impulsive couple acting on any link A will produce a certain angular velocity in any other link B; an equal couple applied to B will produce an equal angular velocity in A. Also if an impulse F applied at P produce an angular velocity ω in a link A, a couple Fa applied to A will produce a linear velocity ωa at P. Historically, we may note that reciprocal relations in dynamics were first recognized by H.L.F. Helmholtz in the domain of acoustics; their use has been greatly extended by Lord Rayleigh.
The equations (13) determine the momenta p1, p2,... as linear functions of the velocities q˙1, q˙2,... Solving these, we can express q˙1, q˙2... as linear functions of p1, p2,... The resulting equations give us the velocities produced by any givenVelocities in terms of momenta.system of impulses. Further, by substitution in (8), we can express the kinetic energy as a homogeneous quadratic function of the momenta p1, p2,... The kinetic energy,as so expressed, will be denoted by Τ`; thus
2Τ` = A`11p1² + A`22p2² + ... + 2A`12p-p2+ ...
(19)
where A`11, A`22,... A`12,... are certain coefficients depending on the configuration. They have been called by Maxwell thecoefficients of mobilityof the system. When the form (19) is given, the valuesof the velocities in terms of the momenta can be expressed in a remarkable form due to Sir W.R. Hamilton. The formula (15) may be written
p1q˙1+ p2q˙2+ ... = Τ + Τ`,
(20)
where Τ is supposed expressed as in (8), and Τ` as in (19). Hence if, for the moment, we denote by δ a variation affecting the velocities, and therefore the momenta, but not the configuration, we have
p1δq˙1+ q˙1δp + p2δq˙2+ q˙2δp2+ ... = δΤ + δΤ`
(21)
In virtue of (13) this reduces to
(22)
Since δp1, δp2, ... may be taken to be independent, we infer that
(23)
In the very remarkable exposition of the matter given by James Clerk Maxwell in hisElectricity and Magnetism, the Hamiltonian expressions (23) for the velocities in terms of the impulses are obtained directly from first principles, and the formulae (13) are then deduced by an inversion of the above argument.
An important modification of the above process was introduced by E.J. Routh and Lord Kelvin and P.G. Tait. Instead of expressing the kinetic energy in terms of the velocities alone, or in terms of the momenta alone, we may express it inRouth’s modification.terms of the velocities corresponding to some of the co-ordinates, say q1, q2, ... qm, and of the momenta corresponding to the remaining co-ordinates, which (for the sake of distinction) we may denote by χ, χ′, χ″, .... Thus, Τ being expressed as a homogeneous quadratic function of q˙1, q˙2, ... q˙m, χ˙, χ˙′, χ˙″, ..., the momenta corresponding to the co-ordinates χ, χ′, χ″, ... may be written
(24)
These equations, when written out in full, determine χ˙, χ˙′, χ˙″, ... as linear functions of q˙1, q˙2, ... q˙m, κ, κ′, κ″,... We now consider the function
R = Τ − κχ˙ − κ′χ˙′ − κ″χ˙″ − ... ,
(25)
supposed expressed, by means of the above relations in terms of q˙1, q˙2, ... q˙m, κ, κ′, κ″, ... Performing the operation δ on both sides of (25), we have
(26)
where, for brevity, only one term of each type has been exhibited. Omitting the terms which cancel in virtue of (24), we have
(27)
Since the variations δq1, δq2, ... δqm, δκ, δκ′, δκ″, ... may be taken to be independent, we have
(28)
and
(29)
An important property of the present transformation is that, when expressed in terms of the new variables, the kinetic energy is the sum of two homogeneous quadratic functions, thus
Τ = ⅋ + K,
(30)
where ⅋ involves the velocities q˙1, q˙2, ... q˙malone, and K the momenta κ, κ′, κ″, ... alone. For in virtue of (29) we have, from (25),
(31)
and it is evident that the terms in R which are bilinear in respect of the two sets of variables q˙1, q˙2, ... q˙mand κ, κ′, κ″, ... will disappear from the right-hand side.
It may be noted that the formula (30) gives immediate proof of two important theorems due to Bertrand and to Lord Kelvin respectively. Let us suppose, in the first place, that the system is started by given impulses of certain types,Maximum and minimum energy.but is otherwise free. J.L.F. Bertrand’s theorem is to the effect that the kinetic energy isgreaterthan if by impulses of the remaining types the system were constrained to take any other course. We may suppose the co-ordinates to be so chosen that the constraint is expressed by the vanishing of the velocities q˙1, q˙2, ... q˙m, whilst the given impulses are κ, κ′, κ″,... Hence the energy in the actual motion is greater than in the constrained motion by the amount ⅋.
Again, suppose that the system is started with prescribed velocity components q˙1, q˙2, ... q˙m, by means of proper impulses of the corresponding types, but is otherwise free, so that in the motion actually generated we have κ = 0, κ′ = 0, κ″ = 0, ... and therefore K = 0. The kinetic energy is thereforelessthan in any other motion consistent with the prescribed velocity-conditions by the value which K assumes when κ, κ′, κ″, ... represent the impulses due to the constraints.
Simple illustrations of these theorems are afforded by the chain of straight links already employed. Thus if a point of the chain be held fixed, or if one or more of the joints be made rigid, the energy generated by any given impulses is less than if the chain had possessed its former freedom.
2.Continuous Motion of a System.
We may proceed to the continuous motion of a system. TheLagrange’s equations.equations of motion of any particle of the system are of the form
mẍ = X, mÿ = Y, mz¨ = Z
(1)
Now let x + δx, y + δy, z + δz be the co-ordinates of m in any arbitrary motion of the system differing infinitely little from the actual motion, and let us form the equation
Σm (ẍδx + ÿδy + z¨δz) = Σ (Xδx + Yδy + Zδz)
(2)
Lagrange’s investigation consists in the transformation of (2) into an equation involving the independent variations δq1, δq2, ... δqn.
It is important to notice that the symbols δ and d/dt are commutative, since
(3)
Hence
(4)
by § 1 (14). The last member may be written
(5)
Hence, omitting the terms which cancel in virtue of § 1 (13), we find
(6)
For the right-hand side of (2) we have
Σ(Xδx + Yδy + Zδz) = Q1δq1+ Q2δq2+ ... ,
(7)
where
(8)
The quantities Q1, Q2, ... are called thegeneralized components of forceacting on the system.
Comparing (6) and (7) we find
(9)
or, restoring the values of p1, p2, ...,
(10)
These are Lagrange’s general equations of motion. Their number is of course equal to that of the co-ordinates q1, q2, ... to be determined.
Analytically, the above proof is that given by Lagrange, but the terminology employed is of much more recent date, having been first introduced by Lord Kelvin and P.G. Tait; it has greatly promoted the physical application of the subject. Another proof of the equations (10), by direct transformation of co-ordinates, has been given by Hamilton and independently by other writers (seeMechanics), but the variational method of Lagrange is that which stands in closest relation to the subsequent developments of the subject. The chapter of Maxwell, already referred to, is a most instructive commentary on the subject from the physical point of view, although the proof there attempted of the equations (10) is fallacious.
In a “conservative system” the work which would have to be done by extraneous forces to bring the system from rest in some standard configuration to rest in the configuration (q1, q2, ... qn) is independent of the path, and may therefore be regarded as a definite function of q1, q2, ... qn. Denoting this function (thepotential energy) by V, we have, if there be no extraneous force on the system,
Σ (Xδx + Yδy + Zδz) = − δV,
(11)
and therefore
(12)
Hence the typical Lagrange’s equation may be now written in the form
(13)
or, again,
(14)
It has been proposed by Helmholtz to give the namekinetic potentialto the combination V − Τ.
As shown underMechanics, § 22, we derive from (10)
(15)
and therefore in the case of a conservative system free from extraneous force,
(16)
which is the equation of energy. For examples of the application of the formula (13) seeMechanics, § 22.
3.Constrained Systems.
It has so far been assumed that the geometrical relations, if any, which exist between the various parts of the systemCase of varying relations.are of the type § 1 (1), and so do not contain t explicitly. The extension of Lagrange’s equations to the case of “varying relations” of the type
x = ƒ(t, q1, q2, ... qn), y = &c., z = &c.,
(1)
was made by J.M.L. Vieille. We now have
(2)
(3)
so that the expression § 1 (8) for the kinetic energy is to be replaced by
2Τ = α0+ 2α1q˙1+ 2α2q˙2+ ... + A11q˙1² + A22q˙2² + ... + A12q˙1q˙2+ ...,
(4)
where
(5)
and the forms of Arr, Arsare as given by § 1 (7). It is to be remembered that the coefficients α0, α1, α2, ... A11, A22, ... A12... will in general involve t explicitly as well as implicitly through the co-ordinates q1, q2,.... Again, we find
Σm (ẋδx + ẏδy + z˙δz) = (α1+ A11q˙1+ A12q˙2+ ...) δq1+ (α2+ A21q˙1+ A22q˙2+ ...) ∂q2+ ...
(6)
where pris defined as in § 1 (13). The derivation of Lagrange’s equations then follows exactly as before. It is to be noted that the equation § 2 (15) does not as a rule now hold. The proof involved the assumption that Τ is a homogeneous quadratic function of the velocities q˙1, q˙2....
It has been pointed out by R.B. Hayward that Vieille’s case can be brought under Lagrange’s by introducing a new co-ordinate (χ) in place of t, so far as it appears explicitly in the relations (1). We have then
2Τ = α0χ˙² + 2(α1q˙1+ α2q˙2+ ...) χ˙ + A11q˙1² + A22q˙2² + ... + 2A12q˙1q˙2+ ....
(7)
The equations of motion will be as in § 2 (10), with the additional equation
(8)
where X is the force corresponding to the co-ordinate χ. We may suppose X to be adjusted so as to make χ¨ = 0, and in the remaining equations nothing is altered if we write t for χ before, instead of after, the differentiations. The reason why the equation § 2 (15) no longer holds is that we should require to add a term Xχ˙ on the right-hand side; this represents the rate at which work is being done by the constraining forces required to keep χ˙ constant.
As an example, let x, y, z be the co-ordinates of a particle relative to axes fixed in a solid which is free to rotate about the axis of z. If φ be the angular co-ordinate of the solid, we find without difficulty
2Τ = m (ẋ² + ẏ² +z˙²) + 2φ˙m (xẏ − yẋ) + {I + m (x² + y²)} φ˙²,
(9)
where I is the moment of inertia of the solid. The equations of motion, viz.
(10)
and
(11)
become
m (ẍ − 2φ˙ẏ − xφ˙² − yφ¨) = X, m (ÿ + 2φ˙ẋ − yφ˙² + xφ¨) = Y, mz¨ = Z,
(12)
and
(13)
If we suppose Φ adjusted so as to maintain φ¨ = 0, or (again) if we suppose the moment of inertia I to be infinitely great, we obtain the familiar equations of motion relative to moving axes, viz.
m (ẍ − 2ωẏ − ω²x) = X, m (ÿ + 2ωẋ − ω²y) = Y, mz¨ = Z,
(14)
where ω has been written for φ. These are the equations which we should have obtained by applying Lagrange’s rule at once to the formula
2Τ = m (ẋ² + ẏ² + z˙²) + 2mω (xẏ − yẋ) + mω² (x² + y²),