Page:EB1911 - Volume 08.djvu/784

of 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

$p_{1}{\dot {q}}_{1}+p_{2}{\dot {q}}_{2}+\ldots =\mathrm {T} +\mathrm {T} ',$

(20)

where $\mathrm {T} \!$ is supposed expressed as in (8), and $\mathrm {T} '\!$ as in (19). Hence if, for the moment, we denote by $\delta \!$ a variation affecting the velocities, and therefore the momenta, but not the configuration, we have

$p_{1}\delta {\dot {q}}_{1}+{\dot {q}}_{1}\delta p+p_{2}\delta {\dot {q}}_{2}+{\dot {q}}_{2}\delta p_{2}+\ldots =\delta \mathrm {T} +\delta \mathrm {T} '={\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{1}}}\delta {\dot {q}}_{1}+{\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{2}}}\delta {\dot {q}}_{2}+\ldots +{\frac {\partial \mathrm {T} '}{\partial p_{1}}}\delta p_{1}+{\frac {\partial \mathrm {T} '}{\partial p_{2}}}\delta p_{2}+\ldots$

(21)

In virtue of (13) this reduces to

${\dot {q}}_{1}\delta p_{1}+{\dot {q}}_{2}\delta p_{2}+\ldots ={\frac {\partial \mathrm {T} '}{\partial p_{1}}}\delta p_{1}+{\frac {\partial \mathrm {T} '}{\partial p_{2}}}\delta p_{2}+\ldots$

(22)

Since $\delta p_{1},\delta p_{2},\ldots \!$ may be taken to be independent, we infer that

${\dot {q}}_{1}={\frac {\partial \mathrm {T} '}{\partial p_{1}}},\qquad \qquad {\dot {q}}_{2}={\frac {\partial \mathrm {T} '}{\partial p_{2}}},\ldots$

(23)

In the very remarkable exposition of the matter given by James Clerk Maxwell in his Electricity 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 in terms of the velocities corresponding to some of the co-ordinates, say $q_{1},q_{2},\ldots q_{m}\!$, and Routh’s modification.of the momenta corresponding to the remaining co-ordinates, which (for the sake of distinction) we may denote by $\chi ,\chi ',\chi '',\ldots \!$. Thus, $\mathrm {T} \!$ being expressed as a homogeneous quadratic function of ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m},{\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\ldots \!$, the momenta corresponding to the co-ordinates $\chi ,\chi ',\chi '',\ldots \!$ may be written

$\kappa ={\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}}},\qquad \kappa '={\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}'}},\qquad \kappa ''={\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}''}},\ldots$

(24)

These equations, when written out in full, determine ${\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\ldots \!$ as linear functions of ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m},\kappa ,\kappa ',\kappa '',\ldots \!$ We now consider the function

$R=\mathrm {T} -\kappa {\dot {\chi }}-\kappa '{\dot {\chi }}'-\kappa ''{\dot {\chi }}''-\ldots ,$

(25)

supposed expressed, by means of the above relations in terms of ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m},\kappa ,\kappa ',\kappa '',\ldots \!$. Performing the operation $\delta \!$ on both sides of (25), we have

${\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{1}}}\delta {\dot {q}}_{1}+\ldots +{\frac {\partial \mathrm {R} }{\partial \kappa }}\delta \kappa +\ldots ={\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{1}}}\delta {\dot {q}}_{1}+\ldots +{\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}}}\delta {\dot {\chi }}+\ldots -\kappa \partial {\dot {\chi }}-{\dot {\chi }}\delta \kappa -\ldots ,$

(26)

where, for brevity, only one term of each type has been exhibited. Omitting the terms which cancel in virtue of (24), we have

${\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{1}}}\delta {\dot {q}}_{1}+\ldots +{\frac {\partial \mathrm {R} }{\partial \kappa }}\delta \kappa +\ldots ={\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{1}}}\delta {\dot {q}}_{1}+\ldots -{\dot {\chi }}\delta \kappa -\ldots$

(27)

Since the variations $\delta q_{1},\delta q_{2},\ldots \delta q_{m},\delta \kappa ,\delta \kappa ',\delta \kappa '',\ldots \!$ may be taken to be independent, we have

$p_{1}={\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{1}}}={\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{1}}},\qquad p_{2}={\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{2}}}={\frac {\partial \mathrm {R} }{\partial {\dot {q}}_{2}}},\ldots$

(28)

and

${\dot {\chi }}=-{\frac {\partial \mathrm {R} }{\partial \kappa }},\qquad {\dot {\chi }}'=-{\frac {\partial \mathrm {R} }{\partial \kappa '}},\qquad {\dot {\chi }}''=-{\frac {\partial \mathrm {R} }{\partial \kappa ''}},\ldots$

(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

$\mathrm {T} ={\mathfrak {T}}+\mathrm {K} ,$

(30)

where ${\mathfrak {T}}$ involves the velocities ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m}\!$ alone, and $\mathrm {K} \!$ the momenta $\kappa ,\kappa ',\kappa '',\ldots \!$ alone. For in virtue of (29) we have, from (25),

$\mathrm {T} =\mathrm {R} -\left(\kappa {\frac {\partial \mathrm {R} }{\partial \kappa }}+\kappa '{\frac {\partial \mathrm {R} }{\partial \kappa '}}+\kappa ''{\frac {\partial \mathrm {R} }{\partial \kappa ''}}+\ldots \right),$

(31)

and it is evident that the terms in $\mathrm {R} \!$ which are bilinear in respect of the two sets of variables ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m}\!$ and $\kappa ,\kappa ',\kappa '',\ldots \!$ 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, but is otherwise free. J. L. F. Bertrand’s theorem is to the effect that the kinetic Maximum and minimum energy.energy is greater than 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 ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m}\!$, whilst the given impulses are $\kappa ,\kappa ',\kappa '',\ldots \!$. Hence the energy in the actual motion is greater than in the constrained motion by the amount ${\mathfrak {T}}$.

Again, suppose that the system is started with prescribed velocity components ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m}\!$, by means of proper impulses of the corresponding types, but is otherwise free, so that in the motion actually generated we have $\kappa =0,\kappa '=0,\kappa ''=0,\ldots \!$ and therefore \mathrm{K} = 0. The kinetic energy is therefore less than in any other motion consistent with the prescribed velocity-conditions by the value which $\mathrm {K} \!$ assumes when $\kappa ,\kappa ',\kappa '',\ldots \!$ 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. The equations of motion of any particle of the system are of the formLagrange’s equations.

$m{\ddot {x}}=\mathrm {X} ,\qquad m{\ddot {y}}=\mathrm {Y} ,\qquad m{\ddot {z}}=\mathrm {Z}$

(1)

Now let $x+\delta x,y+\delta y,z+\delta z\!$ 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

$\Sigma m({\ddot {x}}\delta x+{\ddot {y}}\delta y+{\ddot {z}}\delta z)=\Sigma (\mathrm {X} \delta x+\mathrm {Y} \delta y+\mathrm {Z} \delta z)$

(2)

Lagrange’s investigation consists in the transformation of (2) into an equation involving the independent variations $\delta q_{1},\delta q_{2},\ldots \delta q_{n}\!$.

It is important to notice that the symbols $\delta \!$ and $d/dt\!$ are commutative, since

$\delta {\dot {x}}={\frac {d}{dt}}(x+\delta x)-{\frac {dx}{dt}}={\frac {d}{dt}}\delta x,\And \!\!\!{\text{c.}}$

(3)

Hence

$\Sigma m({\ddot {x}}\delta x+{\ddot {y}}\delta y+{\ddot {z}}\delta z)={\frac {d}{dt}}\Sigma m({\dot {x}}\delta x+{\dot {y}}\delta y+{\dot {z}}\delta z)-\Sigma m({\dot {x}}\delta {\dot {x}}+{\dot {y}}\delta {\dot {y}}+{\dot {z}}\delta {\dot {z}})$

$={\frac {d}{dt}}(p_{1}\delta q_{1}+p_{2}\delta q_{2}+\ldots )-\delta \mathrm {T} ,$

(4)

by § 1 (14). The last member may be written

${\dot {p}}_{1}\delta q_{1}+p_{1}\delta {\dot {q}}_{1}+{\dot {p}}_{2}\delta q_{2}+p_{2}\delta {\dot {q}}_{2}+\ldots -{\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{1}}}\delta {\dot {q}}_{1}-{\frac {\partial \mathrm {T} }{\partial q_{1}}}\delta q_{1}-{\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{2}}}\delta {\dot {q}}_{2}-{\frac {\partial \mathrm {T} }{\partial q_{2}}}\delta q_{2}-\ldots$

(5)

Hence, omitting the terms which cancel in virtue of § 1 (13), we find

$\Sigma m({\ddot {x}}\delta x+{\ddot {y}}\delta y+{\ddot {z}}\delta z)=\left({\dot {p}}_{1}-{\frac {\partial \mathrm {T} }{\partial q_{1}}}\right)\delta q_{1}+\left({\dot {p}}_{2}-{\frac {\partial \mathrm {T} }{\partial q_{2}}}\right)\delta q_{2}+\ldots$

(6)

For the right-hand side of (2) we have

$\Sigma (\mathrm {X} \delta x+\mathrm {Y} \delta y+\mathrm {Z} \delta z)=\mathrm {Q} _{1}\delta q_{1}+\mathrm {Q} _{2}\delta q_{2}+\ldots ,$

(7)

where

$\mathrm {Q} _{r}=\Sigma \left(\mathrm {X} {\frac {\partial x}{\partial q_{r}}}+\mathrm {Y} {\frac {\partial y}{\partial q_{r}}}+\mathrm {Z} {\frac {\partial z}{\partial q_{r}}}\right).$

(8)

The quantities $\mathrm {Q} _{1},\mathrm {Q} _{2},\ldots \!$ are called the generalized components of force acting on the system.

Comparing (6) and (7) we find

${\dot {p}}_{1}-{\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{1}}}=\mathrm {Q} _{1},\qquad {\dot {p}}_{2}-{\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{2}}}=\mathrm {Q} _{2},\ldots ,$

(9)

or, restoring the values of $p_{1},p_{2},\ldots \!$,

${\frac {d}{dt}}\left({\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{1}}}\right)-{\frac {\partial \mathrm {T} }{\partial q_{1}}}=\mathrm {Q} _{1},\qquad {\frac {d}{dt}}\left({\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{2}}}\right)-{\frac {\partial \mathrm {T} }{\partial q_{2}}}=\mathrm {Q} _{2},\ldots$

(10)

These are Lagrange’s general equations of motion. Their number is of course equal to that of the co-ordinates $q_{1},q_{2},\ldots \!$ 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 (see Mechanics), 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 $(q_{1},q_{2},\ldots q_{n})\!$ is independent of the path, and may therefore be regarded as a definite function of $q_{1},q_{2},\ldots q_{n}\!$. Denoting this function (the potential energy) by $\mathrm {V} \!$, we have, if there be no extraneous force on the system,

$\Sigma (\mathrm {X} \delta x+\mathrm {Y} \delta y+\mathrm {Z} \delta z)=-\delta \mathrm {V} ,$

(11)

and therefore

$\mathrm {Q} _{1}=-{\frac {\partial \mathrm {V} }{\partial q_{1}}},\qquad \mathrm {Q} _{2}=-{\frac {\partial \mathrm {V} }{\partial q_{2}}},\ldots .$

(12)