Page:EB1911 - Volume 08.djvu/783

issue in 1875, when the baronetcy became extinct, the estate passing to a collateral branch of the family. After the coronation of George IV. the ceremony was allowed to lapse, but at the coronation of King Edward VII. H. S. Dymoke bore the standard of England in Westminster Abbey.

DYNAMICS (from Gr. δύναμις, strength), the name of a branch of the science of Mechanics (q.v.). The term was at one time restricted to the treatment of motion as affected by force, being thus opposed to Statics, which investigated equilibrium or conditions of rest. In more recent times the word has been applied comprehensively to the action of force on bodies either at rest or in motion, thus including “dynamics” (now termed kinetics) in the restricted sense and “statics.”

Analytical Dynamics.—The fundamental principles of dynamics, and their application to special problems, are explained in the articles Mechanics and Motion, Laws of, where brief indications are also given of the more general methods of investigating the properties of a dynamical system, independently of the accidents of its particular constitution, which were inaugurated by J. L. Lagrange. These methods, in addition to the unity and breadth which they have introduced into the treatment of pure dynamics, have a peculiar interest in relation to modern physical speculation, which finds itself confronted in various directions with the problem of explaining on dynamical principles the properties of systems whose ultimate mechanism can at present only be vaguely conjectured. In determining the properties of such systems the methods of analytical geometry and of the infinitesimal calculus (or, more generally, of mathematical analysis) are necessarily employed; for this reason the subject has been named Analytical Dynamics. The following article is devoted to an outline of such portions of general dynamical theory as seem to be most important from the physical point of view.

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 $q_{1},q_{2},\ldots q_{n}$, 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

$x=f(q_{1},q_{2},\ldots q_{n}),y=\And \!\!\!{\text{c.}},z=\!\And \!\!\!{\text{c.}},$

(1)

where the functions $f\!$ differ (of course) from particle to particle. In modern language, the variables $q_{1},q_{2},\ldots q_{n}$ are generalized co-ordinates serving to specify the configuration of the system; their derivatives with respect to the time are denoted by ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{n}$, and are called the generalized 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 the path.

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 of impulsive motion. Whatever the actual motion of the system at any instant, we may conceive it to be generated instantaneously from rest by Impulsive motion.the application of proper impulses. On this view we have, if $x,y,z\!$ be the rectangular co-ordinates of any particle $m\!$,

$m{\dot {x}}=\mathrm {X} ',m{\dot {y}}=\mathrm {Y} ',m{\dot {z}}=\mathrm {Z} ',$

(2)

where $\mathrm {X} ',\mathrm {Y} ',\mathrm {Z} '\!$ are the components of the impulse on $m\!$. Now let $\delta x,\delta y,\delta z\!$ be any infinitesimal variations of $x,y,z\!$ which are consistent with the connexions of the system, and let us form the equation

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

(3)

where the sign $\Sigma$ indicates (as throughout this article) a summation extending over all the particles of the system. To transform (3) into an equation involving the variations $\delta q_{1},\delta q_{2},\ldots$ of the generalized co-ordinates, we have

${\dot {x}}={\frac {\partial x}{\partial q_{1}}}{\dot {q}}_{1}+{\frac {\partial x}{\partial q_{2}}}{\dot {q}}_{2}+\ldots ,\And \!\!\!{\text{c.}},\And \!\!\!{\text{c.}}$

(4)

$\delta x={\frac {\partial x}{\partial q_{1}}}\delta q_{1}+{\frac {\partial x}{\partial q_{2}}}\delta q_{2}+\ldots ,\And \!\!\!{\text{c.}},\And \!\!\!{\text{c.}}$

(5)

and therefore

$\Sigma m({\dot {x}}\delta x+{\dot {y}}\delta y+{\dot {z}}\delta z)=\mathrm {A} _{11}{\dot {q}}_{1}+\mathrm {A} _{12}{\dot {q}}_{2}+\ldots )\delta q_{1}+(\mathrm {A} _{21}{\dot {q}}_{1}+\mathrm {A} _{22}{\dot {q}}_{2}+\ldots )\delta q_{2}+\ldots ,$

(6)

where

$\left.{\begin{aligned}\mathrm {A} _{rr}=\Sigma m\left\{\left({\frac {\partial x}{\partial q_{r}}}\right)^{2}+\left({\frac {\partial y}{\partial q_{r}}}\right)^{2}+\left({\frac {\partial z}{\partial q_{r}}}\right)^{2}\right\},\\\mathrm {A} _{rs}=\Sigma m\left\{{\frac {\partial x}{\partial q_{r}}}{\frac {\partial x}{\partial q_{s}}}+{\frac {\partial y}{\partial q_{r}}}{\frac {\partial y}{\partial q_{s}}}+{\frac {\partial z}{\partial q_{r}}}{\frac {\partial z}{\partial q_{s}}}\right\}=\mathrm {A} _{sr}.\end{aligned}}\qquad \qquad \right\}$

(7)

If we form the expression for the kinetic energy $\mathrm {T} \!$ of the system, we find

$2\mathrm {T} =\Sigma m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})=\mathrm {A} _{11}{\dot {q}}_{1}^{2}+\mathrm {A} _{22}{\dot {q}}_{2}^{2}\ldots 2\mathrm {A} _{12}{\dot {q}}_{1}{\dot {q}}_{2}+\ldots$

(8)

The coefficients $\mathrm {A} _{11},\mathrm {A} _{22},\ldots \mathrm {A} _{12},\ldots$ are by an obvious analogy called the coefficients of inertia of the system; they are in general functions of the co-ordinates $q_{1},q_{2},\ldots$. The equation (6) may now be written

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

(9)

This maybe regarded as the cardinal formula in Lagrange’s method. For the right-hand side of (3) we may write

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

(10)

where

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

(11)

The quantities $\mathrm {Q} _{1},\mathrm {Q} _{2},\ldots$ are called the generalized components of impulse. Comparing (9) and (10), we have, since the variations $\delta q_{1},\delta q_{2},\ldots$ are independent,

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

(12)

These are the general equations of impulsive motion.

It is now usual to write

$p_{r}={\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{r}}}.$

(13)

The quantities $p_{1},p_{2},\ldots$ represent the effects of the several component impulses on the system, and are therefore called the generalized components of momentum. In terms of them we have

$\Sigma m({\dot {x}}\delta x+{\dot {y}}\delta y+{\dot {z}}\delta z)=p_{1}\delta q_{1}+p_{2}\delta q_{2}+\ldots$

(14)

Also, since $\mathrm {T} \!$ is a homogeneous quadratic function of the velocities ${\dot {q}}_{1},{\dot {q}}_{2}\ldots$,

$2\mathrm {T} =p_{1}{\dot {q}}_{1}+p_{2}{\dot {q}}_{2}+\ldots$

(15)

This follows independently from (14), assuming the special variations $\delta x={\dot {x}}dt$, &c., and therefore $\delta q_{1}={\dot {q}}_{1}dt,\delta q_{2}={\dot {q}}_{2}dt,\ldots$

Again, if the values of the velocities and the momenta in any other motion of the system through the same configuration be distinguished by accents, we have the identityReciprocal theorems.

$p_{1}{\dot {q}}'_{1}+p_{2}{\dot {q}}'_{2}+\ldots =p'_{1}{\dot {q}}_{1}+p'_{2}{\dot {q}}_{2}+\ldots ,$

(16)

each side being equal to the symmetrical expression

$\mathrm {A} _{11}{\dot {q}}_{1}{\dot {q}}'_{1}+\mathrm {A} _{22}{\dot {q}}_{2}{\dot {q}}'_{2}+\ldots +\mathrm {A} _{12}({\dot {q}}_{1}{\dot {q}}'_{2}+{\dot {q}}'_{1}{\dot {q}}_{2})+\ldots$

(17)

The theorem (16) leads to some important reciprocal relations. Thus, let us suppose that the momenta $p_{1},p_{2},\ldots$ all vanish with the exception of $p_{1}\!$, and similarly that the momenta $p'_{1},p'_{2},\ldots$ all vanish except $p'_{2}\!$. We have then $p_{1}{\dot {q}}'_{1}=p'_{2}{\dot {q}}_{2}$, or

${\dot {q}}_{2}:p_{1}={\dot {q}}'_{1}:p'_{2}$

(18)

The interpretation is simplest when the co-ordinates $q_{1},q_{2}\!$ are both of the same kind, e.g. both lines or both angles. We may then conveniently put $p_{1}=p'_{2}\!$, 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 $\mathrm {P} \!$, will produce a certain velocity at any other point $\mathrm {Q} \!$; the theorem asserts that an equal velocity will be produced at $\mathrm {P} \!$ by an equal blow at $\mathrm {Q} \!$. Again, an impulsive couple acting on any link $\mathrm {A} \!$ will produce a certain angular velocity in any other link $\mathrm {B}$; an equal couple applied to $\mathrm {B} \!$ will produce an equal angular velocity in $\mathrm {A} \!$. Also if an impulse $\mathrm {F} \!$ applied at $\mathrm {P} \!$ produce an angular velocity $\omega \!$ in a link $\mathrm {A} \!$, a couple $\mathrm {F} a$ applied to $\mathrm {A} \!$ will produce a linear velocity $\omega a$ at $\mathrm {P} \!$. 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 $p_{1},p_{2},\ldots$ as linear functions of the velocities ${\dot {q}}_{1},{\dot {q}}_{2},\ldots$ Solving these, we can express ${\dot {q}}_{1},{\dot {q}}_{2}\ldots$ as linear functions of $p_{1},p_{2},\ldots$ The resulting equations give us the velocities produced by any given system of impulses. Further, by substitution in (8), we can express the kinetic Velocities in terms of momenta.energy as a homogeneous quadratic function of the momenta $p_{1},p_{2},\ldots$ The kinetic energy, as so expressed, will be denoted by $\mathrm {T} '\!$; thus

$2\mathrm {T} '=\mathrm {A} '_{11}p_{1}^{2}+\mathrm {A} '_{22}p_{2}^{2}+\ldots +2\mathrm {A} '_{12}p-p_{2}+\ldots$

(19)

where $\mathrm {A} '_{11},\mathrm {A} '_{22},\ldots \mathrm {A} '_{12},\ldots$ are certain coefficients depending on the configuration. They have been called by Maxwell the coefficients of mobility of the system. When the form (19) is given, the values