Page:EB1911 - Volume 08.djvu/785

Hence the typical Lagrange’s equation may be now written in the form

${\frac {d}{dt}}\left({\frac {\partial \mathrm {T} }{\partial {\dot {q}}_{r}}}\right)-{\frac {\partial \mathrm {T} }{\partial q_{r}}}=-{\frac {\partial \mathrm {V} }{\partial q_{r}}},$

(13)

or, again,

${\dot {p}}_{r}=-{\frac {\partial }{\partial q_{r}}}(\mathrm {V} -\mathrm {T} ).$

(14)

It has been proposed by Helmholtz to give the name kinetic potential to the combination $\mathrm {V} -\mathrm {T}$.

As shown under Mechanics, § 22, we derive from (10)

${\frac {d\mathrm {T} }{dt}}=\mathrm {Q} _{1}{\dot {q}}_{1}+\mathrm {Q} _{2}{\dot {q}}_{2}+\ldots ,$

(15)

and therefore in the case of a conservative system free from extraneous force,

${\frac {d}{dt}}(\mathrm {T} +\mathrm {V} )=0{\text{ or }}\mathrm {T} +\mathrm {V} ={\text{const.}},$

(16)

which is the equation of energy. For examples of the application of the formula (13) see Mechanics, § 22.

3. Constrained Systems.

It has so far been assumed that the geometrical relations, if any, which exist between the various parts of the system 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 typeCase of varying relations.

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

(1)

was made by J. M. L. Vieille. We now have

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

(2)

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

(3)

so that the expression § 1 (8) for the kinetic energy is to be replaced by

$2\mathrm {T} =\alpha _{0}+2\alpha _{1}{\dot {q}}_{1}+2\alpha _{2}{\dot {q}}_{2}+\ldots +\mathrm {A} _{11}{\dot {q}}_{1}^{2}+\mathrm {A} _{22}{\dot {q}}_{2}^{2}+\ldots +\mathrm {A} _{12}{\dot {q}}_{1}{\dot {q}}_{2}+\ldots ,$

(4)

where

$\left.{\begin{aligned}\alpha _{0}&=\Sigma m\left\{\left({\frac {\partial x}{\partial t}}\right)^{2}+\left({\frac {\partial y}{\partial t}}\right)^{2}+\left({\frac {\partial z}{\partial t}}\right)^{2}\right\},\\\alpha _{r}&=\Sigma m\left\{{\frac {\partial x}{\partial t}}\,{\frac {\partial x}{\partial q_{r}}}+{\frac {\partial y}{\partial t}}\,{\frac {\partial y}{\partial q_{r}}}+{\frac {\partial z}{\partial t}}\,{\frac {\partial z}{\partial q_{r}}}\right\},\end{aligned}}\right\}$

(5)

and the forms of $\mathrm {A} _{rr},\mathrm {A} _{rs}\!$ are as given by § 1 (7). It is to be remembered that the coefficients $\alpha _{0},\alpha _{1},\alpha _{2},\ldots \mathrm {A} _{11},\mathrm {A} _{22},\ldots \mathrm {A} _{12}\ldots \!$ will in general involve $t\!$ explicitly as well as implicitly through the co-ordinates $q_{1},q_{2},\ldots \!$. Again, we find

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

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

(6)

where $p_{r}\!$ is 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 $\mathrm {T} \!$ is a homogeneous quadratic function of the velocities ${\dot {q}}_{1},{\dot {q}}_{2}\ldots \!$.

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 ($\chi \!$) in place of $t\!$, so far as it appears explicitly in the relations (1). We have then

$2\mathrm {T} =\alpha _{0}{\dot {\chi }}^{2}+2(\alpha _{1}{\dot {q}}_{1}+\alpha _{2}{\dot {q}}_{2}+\ldots ){\dot {\chi }}+\mathrm {A} _{11}{\dot {q}}_{1}^{2}+\mathrm {A} _{22}{\dot {q}}_{2}^{2}+\ldots +2\mathrm {A} _{12}{\dot {q}}_{1}{\dot {q}}_{2}+\ldots .$

(7)

The equations of motion will be as in § 2 (10), with the additional equation

${\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {\chi }}}}-{\frac {\partial \mathrm {T} }{\partial \chi }}=\mathrm {X} ,$

(8)

where $\mathrm {X} \!$ is the force corresponding to the co-ordinate $\chi$. We may suppose $\mathrm {X}$ to be adjusted so as to make ${\ddot {\chi }}=0$, and in the remaining equations nothing is altered if we write $t$ for $\chi$ 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 $\mathrm {X} {\dot {\chi }}$ on the right-hand side; this represents the rate at which work is being done by the constraining forces required to keep ${\dot {\chi }}$ 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 $\phi \!$ be the angular co-ordinate of the solid, we find without difficulty

$2\mathrm {T} =m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})+2{\dot {\phi }}m(x{\dot {y}}-y{\dot {x}})+{\mathrm {I} +m(x^{2}+y^{2})}{\dot {\phi }}^{2},$

(9)

where $\mathrm {I} \!$ is the moment of inertia of the solid. The equations of motion, viz.

${\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {x}}}}-{\frac {\partial \mathrm {T} }{\partial x}}=\mathrm {X} ,\qquad {\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {y}}}}-{\frac {\partial \mathrm {T} }{\partial y}}=\mathrm {Y} ,\qquad {\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {z}}}}-{\frac {\partial \mathrm {T} }{\partial z}}=\mathrm {Z} ,$

(10)

and

${\frac {d}{dt}}\,{\frac {\partial \mathrm {T} }{\partial {\dot {\phi }}}}-{\frac {\partial \mathrm {T} }{\partial \phi }}=\Phi ,$

(11)

become

$m({\ddot {x}}-2{\dot {\phi }}{\dot {y}}-x{\dot {\phi }}^{2}-y{\ddot {\phi }})=\mathrm {X} ,m({\ddot {y}}+2{\dot {\phi }}{\dot {x}}-y{\dot {\phi }}^{2}+x{\ddot {\phi }})=\mathrm {Y} ,m{\ddot {z}}=\mathrm {Z} ,$

(12)

and

${\frac {d}{dt}}[{\mathrm {I} +m(x^{2}+y^{2})}{\dot {\phi }}+m(x{\dot {y}}-y{\dot {x}})]=\Phi .$

(13)

If we suppose $\Phi \!$ adjusted so as to maintain ${\ddot {\phi }}=0\!$, or (again) if we suppose the moment of inertia $\mathrm {I} \!$ to be infinitely great, we obtain the familiar equations of motion relative to moving axes, viz.

$m({\ddot {x}}-2\omega {\dot {y}}-\omega ^{2}x)=\mathrm {X} ,m({\ddot {y}}+2\omega {\dot {x}}-\omega ^{2}y)=\mathrm {Y} ,m{\ddot {z}}=\mathrm {Z} ,$

(14)

where $\omega \!$ has been written for $\phi \!$. These are the equations which we should have obtained by applying Lagrange’s rule at once to the formula

$2\mathrm {T} =m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})+2m\omega (x{\dot {y}}-y{\dot {x}})+m\omega ^{2}(x^{2}+y^{2}),$

(15)

which gives the kinetic energy of the particle referred to axes rotating with the constant angular velocity $\omega \!$. (See Mechanics, § 13.)

More generally, let us suppose that we have a certain group of co-ordinates $\chi ,\chi ',\chi '',\ldots \!$ whose absolute values do not affect the expression for the kinetic energy, and that by suitable forces of the corresponding types the velocity-components ${\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\ldots \!$ are maintained constant. The remaining co-ordinates being denoted by $q_{1},q_{2},\ldots q_{n}\!$, we may write

$2\mathrm {T} ={\mathfrak {T}}+\mathrm {T} _{0}+2(\alpha _{1}{\dot {q}}_{1}+\alpha _{2}{\dot {q}}_{2}+\ldots ){\dot {\chi }}+2(\alpha '_{1}{\dot {q}}_{1}+\alpha '_{2}{\dot {q}}_{2}+\ldots ){\dot {\chi }}'+\ldots ,$

(16)

where ${\mathfrak {T}}$ is a homogeneous quadratic function of the velocities ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{n}\!$ of the type § 1 (8), whilst $\mathrm {T} _{0}\!$ is a homogeneous quadratic function of the velocities ${\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\ldots \!$ alone. The remaining terms, which are bilinear in respect of the two sets of velocities, are indicated more fully. The formulae (10) of § 2 give $n$ equations of the type

${\frac {d}{dt}}\left({\frac {\partial {\mathfrak {T}}}{\partial q_{r}}}\right)-{\frac {\partial {\mathfrak {T}}}{\partial q_{r}}}+(r,1){\dot {q}}_{1}+(r,2){\dot {q}}_{2}+\ldots -{\frac {\partial \mathrm {T} _{0}}{\partial q_{r}}}=\mathrm {Q} _{r}$

(17)

where

$(r,s)=\left({\frac {\partial \alpha _{r}}{\partial q_{s}}}-{\frac {\partial \alpha _{s}}{\partial q_{r}}}\right){\dot {\chi }}+\left({\frac {\partial \alpha '_{r}}{\partial q_{s}}}-{\frac {\partial \alpha '_{s}}{\partial q_{r}}}\right){\dot {\chi }}'+\ldots .$

(18)

These quantities $(r,s)\!$ are subject to the relations

$(r,s)=-(s,r),(r,r)=0$

(19)

The remaining dynamical equations, equal in number to the co-ordinates $\chi ,\chi ',\chi '',\ldots \!$, yield expressions for the forces which must be applied in order to maintain the velocities ${\dot {\chi }},{\dot {\chi }}',{\dot {\chi }}'',\ldots \!$ constant; they need not be written down. If we follow the method by which the equation of energy was established in § 2, the equations (17) lead, on taking account of the relations (19), to

${\frac {d}{dt}}({\mathfrak {T}}-\mathrm {T} _{0})=\mathrm {Q} _{1}{\dot {q}}_{1}+\mathrm {Q} _{2}{\dot {q}}_{2}+\ldots +\mathrm {Q} _{n}{\dot {q}}_{n},$

(20)

or, in case the forces $\mathrm {Q} _{r}\!$ depend only on the co-ordinates $q_{1},q_{2},\ldots q_{n}\!$ and are conservative,

${\mathfrak {T}}+\mathrm {V} -\mathrm {T} _{0}={\text{const.}}$

(21)

The conditions that the equations (17) should be satisfied by zero values of the velocities ${\dot {q}}_{1},{\dot {q}}_{2},\ldots {\dot {q}}_{n}\!$ are

$\mathrm {Q} _{r}=-{\frac {\partial \mathrm {T} _{0}}{\partial q_{r}}},$

(22)

or in the case of conservative forces

${\frac {\partial }{\partial q_{r}}}(\mathrm {V} -\mathrm {T} _{0})=0,$

(23)

i.e. the value of $\mathrm {V} -\mathrm {T} _{0}\!$ must be stationary.

We may apply this to the case of a system whose configuration relative to axes rotating with constant angular velocity ($\omega \!$) is defined by means of the $n$ co-ordinates $q_{1},q_{2},\ldots q_{n}\!$. This is important on account of its bearing on the kinetic theory of the tides. Since the Cartesian co-ordinates $x,y,z\!$ of any particle $m$ of Rotating axes.the system relative to the moving axes are functions of $q_{1},q_{2},\ldots q_{n}\!$, of the form § 1 (1), we have, by (15)

$2{\mathfrak {T}}=\Sigma m({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}),\qquad 2\mathrm {T} _{0}=\omega ^{2}\Sigma m(x^{2}+y^{2}),$

(24)

$\alpha _{r}=\Sigma m\left(x{\frac {\partial y}{\partial q_{r}}}-y{\frac {\partial x}{\partial q_{r}}}\right),$

(25)

whence

$(r,s)=2\omega \cdot \Sigma m{\frac {\partial (x,y)}{\partial (q_{s},q_{r})}}.$

(26)

The conditions of relative equilibrium are given by (23).

It will be noticed that this expression $\mathrm {V} -\mathrm {T} _{0}\!$, which is to be stationary, differs from the true potential energy by a term which represents the potential energy of the system in relation to fictitious “centrifugal forces.” The question of stability of relative equilibrium will be noticed later (§ 6).

It should be observed that the remarkable formula (20) may in the present case be obtained directly as follows. From (15) and (14) we find

${\frac {d\mathrm {T} }{dt}}={\frac {d}{dt}}({\mathfrak {T}}+\mathrm {T} _{0})+\omega \cdot \Sigma m(x{\ddot {y}}-y{\ddot {x}})={\frac {d}{dt}}({\mathfrak {T}}-\mathrm {T} _{0})+\omega \cdot \Sigma (x\mathrm {Y} -y\mathrm {X} ).$

(27)