Page:EB1911 - Volume 08.djvu/786

This must be equal to the rate at which the forces acting on the system do work, viz. to

$\omega \Sigma (x\mathrm {Y} -y\mathrm {X} )+\mathrm {Q} _{1}{\dot {q}}_{1}+\mathrm {Q} _{2}{\dot {q}}_{2}+\ldots +\mathrm {Q} _{n}{\dot {q}}_{n},$

where the first term represents the work done in virtue of the rotation.

We have still to notice the modifications which Lagrange’s equations undergo when the co-ordinates $q_{1},q_{2},\ldots q_{n}\!$ are not all independently variable. In the first place, we may suppose them connected by a number $m(<n)\!$ of relations of the typeConstrained systems.

$\mathrm {A} (t,q_{1},q_{2},\ldots q_{n})=0,\qquad \mathrm {B} (t,q_{1},q_{2},\ldots q_{n})=0,\And {\text{c.}}$

(28)

These may be interpreted as introducing partial constraints into a previously free system. The variations $\delta q_{1},\delta q_{2},\ldots \delta q_{n}\!$ in the expressions (6) and (7) of § 2 which are to be equated are no longer independent, but are subject to the relations

${\frac {\partial \mathrm {A} }{\partial q_{1}}}\delta q_{1}+{\frac {\partial \mathrm {A} }{\partial q_{2}}}\delta q_{2}+\ldots =0,\qquad {\frac {\partial \mathrm {B} }{\partial q_{1}}}\delta q_{1}+{\frac {\partial \mathrm {B} }{\partial q_{2}}}\delta q_{2}+\ldots =0,\And {\text{c.}}$

(29)

Introducing indeterminate multipliers $\lambda ,\mu ,\ldots \!$, one for each of these equations, we obtain in the usual manner $n\!$ equations of the type

${\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{r}}}-{\frac {\partial \mathrm {T} }{\partial q_{r}}}=\mathrm {Q} _{r}+\lambda {\frac {\partial \mathrm {A} }{\partial q_{r}}}+\mu {\frac {\partial \mathrm {B} }{\partial q_{r}}}+\ldots ,$

(30)

in place of § 2 (10). These equations, together with (28), serve to determine the $n\!$ co-ordinates $q_{1},q_{2},\ldots q_{n}\!$ and the $m\!$ multipliers $\lambda ,\mu ,\ldots \!$.

When $t\!$ does not occur explicitly in the relations (28) the system is said to be holonomic. The term connotes the existence of integral (as opposed to differential) relations between the co-ordinates, independent of the time.

Again, it may happen that although there are no prescribed relations between the co-ordinates $q_{1},q_{2},\ldots q_{n}\!$, yet from the circumstances of the problem certain geometrical conditions are imposed on their variations, thus

$\mathrm {A} _{1}\delta q_{1}+\mathrm {A} _{2}\delta q_{2}+\ldots =0,\qquad \mathrm {B} _{1}\delta q_{1}+\mathrm {B} _{2}\delta q_{2}+\ldots =0,\And {\text{c.}},$

(31)

where the coefficients are functions of $q_{1},q_{2},\ldots q_{n}\!$ and (possibly) of $t\!$. It is assumed that these equations are not integrable as regards the variables $q_{1},q_{2},\ldots q_{n}\!$; otherwise, we fall back on the previous conditions. Cases of the present type arise, for instance, in ordinary dynamics when we have a solid rolling on a (fixed or moving) surface. The six co-ordinates which serve to specify the position of the solid at any instant are not subject to any necessary relation, but the conditions to be satisfied at the point of contact impose three conditions of the form (31). The general equations of motion are obtained, as before, by the method of indeterminate multipliers, thus

${\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{r}}}-{\frac {\partial \mathrm {T} }{\partial q_{r}}}=\mathrm {Q} _{r}+\lambda \mathrm {A} _{r}+\mu \mathrm {B} _{r}+\ldots$

(32)

The co-ordinates $q_{1},q_{2},\ldots q_{n}$, and the indeterminate multipliers $\lambda ,\mu ,\ldots \!$, are determined by these equations and by the velocity-conditions corresponding to (31). When $t$ does not appear explicitly in the coefficients, these velocity-conditions take the forms

$\mathrm {A} _{1}{\dot {q}}_{1}+\mathrm {A} _{2}{\dot {q}}_{2}+\ldots =0,\qquad \mathrm {B} _{1}{\dot {q}}_{1}+\mathrm {B} _{2}{\dot {q}}_{2}+\ldots =0,\And \!\!\!{\text{c.}}$

(33)

Systems of this kind, where the relations (31) are not integrable, are called non-holonomic.

4. Hamiltonian Equations of Motion.

In the Hamiltonian form of the equations of motion of a conservative system with unvarying relations, the kinetic energy is supposed expressed in terms of the momenta $p_{1},p_{2},\ldots \!$ and the co-ordinates $q_{1},q_{2},\ldots \!$, as in § 1 (19). Since the symbol $\delta \!$ now denotes a variation extending to the co-ordinates as well as to the momenta, we must add to the last member of § 1 (21) terms of the types

${\frac {\partial \mathrm {T} }{\partial q_{1}}}\delta q_{1}+{\frac {\partial \mathrm {T} '}{\partial q_{2}}}\delta q_{2}+\ldots$

(1)

Since the variations $\delta p_{1},\delta p_{2},\ldots \delta q_{1},\delta q_{2},\ldots \!$ may be taken to be independent, we infer the equations § 1 (23) as before, together with

${\frac {\partial \mathrm {T} }{\partial q_{1}}}=-{\frac {\partial \mathrm {T} '}{\partial q_{1}}},\qquad {\frac {\partial \mathrm {T} }{\partial q_{2}}}=-{\frac {\partial \mathrm {T} '}{\partial q_{2}}},\ldots ,$

(2)

Hence the Lagrangian equations § 2 (14) transform into

${\dot {p}}_{1}=-{\frac {\partial }{\partial q_{1}}}(\mathrm {T} '+\mathrm {V} ),\qquad {\dot {p}}_{2}=-{\frac {\partial }{\partial q_{2}}}(\mathrm {T} '+\mathrm {V} ),\ldots$

(3)

If we write

$\mathrm {H} =\mathrm {T} '+\mathrm {V} ,$

(4)

so that $\mathrm {H} \!$ denotes the total energy of the system, supposed expressed in terms of the new variables, we get

${\dot {p}}_{1}=-{\frac {\partial \mathrm {H} }{\partial q_{1}}},\qquad {\dot {p}}_{2}=-{\frac {\partial \mathrm {H} }{\partial q_{2}}},\ldots$

(5)

If to these we join the equations

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

(6)

which follow at once from § 1 (23), since $\mathrm {V} \!$ does not involve $p_{1},p_{2},\ldots \!$, we obtain a complete system of differential equations of the first order for the determination of the motion.

The equation of energy is verified immediately by (5) and (6), since these make

${\frac {d\mathrm {H} }{dt}}={\frac {\partial \mathrm {H} }{\partial p_{1}}}{\dot {p}}_{1}+{\frac {\partial \mathrm {H} }{\partial p_{2}}}{\dot {p}}_{2}+\ldots +{\frac {\partial \mathrm {H} }{\partial q_{1}}}{\dot {q}}_{1}+{\frac {\partial \mathrm {H} }{\partial q_{2}}}{\dot {q}}_{2}+\ldots =0.$

(7)

The Hamiltonian transformation is extended to the case of varying relations as follows. Instead of (4) we write

$\mathrm {H} =p_{1}{\dot {q}}_{1}+p_{2}{\dot {q}}_{2}+\ldots -\mathrm {T} +\mathrm {V} ,$

(8)

and imagine $\mathrm {H}$ to be expressed in terms of the momenta $p_{1},p_{2},\ldots \!$, the co-ordinates $q_{1},q_{2},\ldots \!$, and the time. The internal forces of the system are assumed to be conservative, with the potential energy $\mathrm {V} \!$. Performing the variation $\delta \!$ on both sides, we find

$\delta \mathrm {H} ={\dot {q}}_{1}\delta p_{1}+\ldots -{\frac {\partial \mathrm {T} }{\partial q_{1}}}\delta q_{1}+{\frac {\partial \mathrm {V} }{\partial q_{1}}}\delta q+\ldots ,$

(9)

terms which cancel in virtue of the definition of $p_{1},p_{2},\ldots \!$ being omitted. Since $\delta p_{1},\delta p_{2},\ldots ,\delta q_{1},\delta q_{2},\ldots \!$ may be taken to be independent, we infer

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

(10)

and

${\frac {\partial }{\partial q_{1}}}(\mathrm {T} -\mathrm {V} )=-{\frac {\partial \mathrm {H} }{\partial q_{1}}},\qquad {\frac {\partial }{\partial q_{2}}}(\mathrm {T} -\mathrm {V} )=-{\frac {\partial \mathrm {H} }{\partial q_{2}}},\ldots .$

(11)

It follows from (11) that

${\dot {p}}_{1}=-{\frac {\partial \mathrm {H} }{\partial q_{1}}},\qquad {\dot {p}}_{2}=-{\frac {\partial \mathrm {H} }{\partial q_{2}}},\ldots .$

(12)

The equations (10) and (12) have the same form as above, but $\mathrm {H}$ is no longer equal to the energy of the system.

5. Cyclic Systems.

A cyclic or gyrostatic system is characterized by the following properties. In the first place, the kinetic energy is not affected if we alter the absolute values of certain of the co-ordinates, which we will denote by $\chi ,\chi ',\chi '',\ldots \!$, provided the remaining co-ordinates $q_{1},q_{2},\ldots q_{m}\!$ and the velocities, including of course the velocities ${\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\ldots \!$, are unaltered. Secondly, there are no forces acting on the system of the types $\chi ,\chi ',\chi '',\ldots \!$. This case arises, for example, when the system includes gyrostats which are free to rotate about their axes, the co-ordinates $\chi ,\chi ',\chi '',\ldots \!$ then being the angular co-ordinates of the gyrostats relatively to their frames. Again, in theoretical hydrodynamics we have the problem of moving solids in a frictionless liquid; the ignored co-ordinates $\chi ,\chi ',\chi '',\ldots \!$ then refer to the fluid, and are infinite in number. The same question presents itself in various physical speculations where certain phenomena are ascribed to the existence of latent motions in the ultimate constituents of matter. The general theory of such systems has been treated by E. J. Routh, Lord Kelvin, and H. L. F. Helmholtz.

If we suppose the kinetic energy $\mathrm {T} \!$ to be expressed, as in Lagrange’s method, in terms of the co-ordinates and the velocities, the equations of motion corresponding to $\chi ,\chi ',\chi '',\ldots \!$ reduce, in virtue of the above hypotheses, to the formsRouth’s equations.

${\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}}}=0,\qquad {\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}'}}=0,\qquad {\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}''}}=0,\ldots ,$

(1)

whence

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

(2)

where $\kappa ,\kappa ',\kappa '',\ldots \!$ are the constant momenta corresponding to the cyclic co-ordinates $\chi ,\chi ',\chi '',\ldots \!$. These equations are linear in ${\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\ldots \!$; solving them with respect to these quantities and substituting in the remaining Lagrangian equations, we obtain $m\!$ differential equations to determine the remaining co-ordinates $q_{1},q_{2},\ldots q_{m}\!$. The object of the present investigation is to ascertain the general form of the resulting equations. The retained co-ordinates $q_{1},q_{2},\ldots q_{m}\!$ may be called (for distinction) the palpable co-ordinates of the system; in many practical questions they are the only co-ordinates directly in evidence.

If, as in § 1 (25), we write

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

(3)

and imagine $\mathrm {R}$ to be expressed by means of (2) as a quadratic function of ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{m},\kappa ,\kappa ',\kappa '',\ldots \!$ with coefficients which are in general functions of the co-ordinates $q_{1},q_{2},\ldots q_{m}\!$, then, performing the operation $\delta \!$ on both sides, we find

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

$+{\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}}}\delta {\dot {\chi }}+\ldots +{\frac {\partial \mathrm {T} }{\partial \chi _{1}}}\delta q_{1}+\ldots -\kappa \delta {\dot {\chi }}-{\dot {\chi }}\delta \kappa -\ldots .$

(4)