Trajectories in the Conservative Field of Force, Part 1,
by
RINOSUKE OGURA in Sendai.
CONTENTS. Pares. Introduction...... 124 OHAPTERI. General properties of natural families. 1-2. Natural families in space...... 125 3. Determination of SPby the extremals...... 128 9-5. Natural families on a surface...... 130 6. Particular case where the extremals are geodesics...... 132 7. Other formulas (for reactions)...... 133 CHAPTERII. Asymptotic curves and free orbits on a surface. 8. Asymptotic curves as orbits:...... 135 9. Free orbits on a surface...... 136 10. Orbits in a plane...... 138 11. Determination of the force by which one family of asymptotic curves can be described...... 139 12. Minimal surfaces...... 144 13. Determination of the force by which two families of asymptotic curves can be described...... 146 14. Minimal surfaces...... 147 15. Pseudospherical surfaces...... 149 16. Ruled surfaces...... 150 17. Ruled surfaces of the second degree...... 153 18. The ordinary helicoid...... 155 19. The differential equation of the surfaces whose asymptotic curves can be the orbits in the given field of force...... 156 20. Discussion of some differential equations...... 159 21. Application of Goursat's transformation of minimal surfaces...... 162 CHAPTERIII. Application of Joukovsky's method, especially to the orthogonal System. 22. Joukovsky's method: a general ease...... 164 23. The orthogonal system...... 167 M. Some examples: asymptotic curves, geodesic ellipses and hyperbolas, etc.. 168 24 RINNOSUEE OGURA:
25, Lines of curvature...... 169 20, Particular ease whore one family of orbits are geodesics...... 171 27. Gteodesics and geodesic parallels...... 173 28-29. Some examples: Surface of revolution, -surface of constant total curvature, etc...... 174
OUAPTER IV. General method for the determination of So by two families of extremals on a surface, and the differential equation of all extremals. 30. General method for the determination of P...... 177 31. Particular oases...... 180 32; Equidistantial system...... 181 33. The differential equation of all extremals on a surface...... 183 34. Examples: Surface of revolution, Uidistantial system, etc...... 184
Introduction. The main object of this paper is to treat' some: properties of the trajectories. in the conservative field of force, from the standpoint of differential geometry. (1) During any motion of a particle in the field, the sum of the kinetic and potential energies is constant: where q is the velocity of the particle with unit mass and U is the force-function. Thus each motion corresponds to a definite value of the constant h representing the_ total energy., Throughout Part I of this paper we will confine ourselves to the case in which h. has a def nice value. Besides we will proceed to consider a more general case of the so-called natural family, that is, the curves which can be regarded as the totality of extremals in a variation-problem of the form
∫s1sgφ(x, y, z)ds=min where So is any point-function and s is the curve-length. . Some ex- amples of such families are:
(i) Bee the classical works by Routh, Appel, etc. also Stitakel, Elementare Dynamik (Eneyklopidle der mathematisohen Wissensohaften, Bd. 4, Teilband 1.), p. 493- 506; and especially E. Kasnor, Differential-geometric aspects of dynamics (The PrWoeton Colloquium 1913); Kasner, Transactions of the American Math. Society, 8 (1907), p. 135 and Mid, 10 (1909), p. 201; Ph. Frank, Journal fair d. refine u. angew. 11athematik, 134 (1908), p. 156. TRATEOTORIFSIN THE CONSERVATIEVFIELD OF FORCE, PART I. 125
(i) For the orbits in the conservativefield of force, by the principle of least action,
∫qs=min, and hence ψ=q=√2(U+h)
(ii) For the brachistochronaesin the conservativefield of force,
and hence
(iii) For the positions of equilibrivzmof a uniform chord in the conservativefield of force,
∫Uds=min, and hence φ=-U (iv) For the paths of light in the medium with index of refraction u, by Fermat's principle,
∫ μds=min,hence and φ=μ The contents which precede will 'show in full to the reader the matter we are treating. Among others we hope to notify here that in the first chapter we have proved some already known theorems together with new ones, and in the first article (Art. 22) of the third chapter we have reproduced Joukovsky's method with a slight, modification.
CHAPTER I. General Properties of Natural Families.
Natural families in space. 1. Let us begin with a consideration of a natural family, that is, the totality of extremals with an integral of the form
(1) where a denotes the arc measured along the curve-from a fixed point end 126 KINNOSUKE OGUBA:
The differential equations: of, the extremals are then given by the rules of the calculus of variations
(2)
where
Multiplying equations (2) by x", J", z" respectively and adding, we have
(3) p being the radius of curvature of an extremal at the Point (x, y, z); for x'2+y'2+z'2=1,
x'x"+y'y"+z'z'=0
Again multiplying equations (2) by y'z"-z'y',z'x"-x'z"Ir. x'y"-y'x0 respectively and adding, we obtain
=0. (4) x'y'z' x" y" z"
Now aso, aT, aSP are the, components of -the gradient vector of p referred to the coordinate axes. Hquce it follows, from (3), that 4-is equal to the componentof the gradient vectorof cp in the direction of the principal normal to its extremal; and from (4), that TRAIEOTORIES IN THE OONSERVATIVEFIELD OF FORCE, PART I. 127 the gradientt vector of o lies in the osculating plane of its extremal. Two equations (3) and (4) are called the intrinsic equations of the extremals. For example, for the-orbits of a particle in the conservative field of force,
φ=q=2(u+h) and so
Fx, Fy, F, being the components of force, referred to the coordinate axes consequently we arrive at the well known result that the com- ponents of force are q2/p and zero along the principal normal, and the binormal respectively. 2. If (E, n, C) be. the centre of curvature Of an extrernal & the point F(cv, y, x), then
ξ=x+ρ2x", η=y+ρ2y", ζ=z+ρ2z" Hence equation (3) becomes
(5)
which may be considered as the polar plane of the point
y+a0+a} with respect to the sphere
(ξ-x)2+(η-y)2+(ζ-z)2=ψ Therefore the centres of curvature of the o extremuls passing through ann point P lie in the polar plane of the end point of the gradient vector of SPat the point P With respect to the sphere having the point P as its centre and VT as its radius.(1) For convenience, hereafter we will onll thin plane the molar Mane at the point P with respect to (o. For example, for the orbit of a particle in the conservative field of force, equation (5) may be written
(ξ-x)FN+(η-y)Fy+(ζ-z)Fn=q2;
(1) Prof. Kasner confirmed that the centre of curvature of any extremal at a point lies in a fixed plane; but he did not state the property of this plane here obtained, explicitly. 128 I TNOSUHE OGUREA and for the braohistcohrone in the same, field
(ξ-x)Fx+(η-y)Fy+(ξ-z)Fn=-q2 hence these two, planes are symnietria with respect to the point (x, y, z). Conversely, if the centre of curvature of a curve at any paint lie in the polar plane at the point with respect to (p and the gradient vector of So lie in the oseulatinrd plane at the point, then thie-curve is neeessarlti an extremal of the integral ∫s1sqφ(x,y,z)√x2+y2+z2ds. For, these assumptions are equivalent to (5) and (4); and from (5) we can derive equation (3). So the curve satisfies the intrinsic equa- tions (3) and (4) of the extremals. 3, If two functions V (x, y, z), and (x, y, z) have the same polar plane at any point, it follows that
φ(x, y, z)=k,φ(x, z), k being an arbitrarj constant independent of x, y Z. For, in order that the two plates
coincide with each other at any point (x, y, z), it should be
from which we find
φ=kφ Differentiating with respect to x
On the other hand, since TRAJECTORIES IN THE CONSERVATIVEFIELD OF FOROE, PART I. 129
it follows that
Similarly we have
Henceif Sp a=0, mustk be constant independentof x, y, z: Taut the polar plane at a point are determined by the three centres of curvature of anv three extremals passing through the point. Therefore if we know, three extromals which pass through every point in space and whose principal norinals at every point are not in the same plane, the function (p(x,y, z) is determined up to a constant multiple. Here we add an analytical proof: If. (a1, b1, c1) (Es, r3, a3) be the centres of curvature at the point (x, y, z) of three extremals a1, A2 r3, we haveeby, (5)
Since the three principal, normals to these extremals are not in the same plane,
ξ1-x η1-y ζ1-z
ξ1-x η2-y ζ2-z ≠0. ε3-x η3-y ζ3-ｚ
So we have 130 KINNOSUKEOGURA:
from which d(log Sp)=Pdx+Qdy-Rdo. Integrating alone the arc of extremalak we obtain the result
ρ=const.epdz+Qdy+Rdx Thus thefunction (p is in general determined, and can be constructed, if we know 3002 out of the totality of booextremale, each of three systems of 0 curves passing through each point of mace A more detailed discussion in these lines will be given in Part II of this paper Natural families on a surface. 4 We now consider the natural family constrained to a smooth Btu-face. If the eiven surface be (6) f(x, J)z)=O, the differential equations of the extremals of the integral
(1) ∫s1snφ(x,y,z)x'2+y'2+z'2ds are
(7)
where k denotes an indeterminate multiplier. Multiplying these equations (7) by
respectively and adding, TRAJECTORIES IN THE. CONSERVATIVEFIELD OF FORCE, PART 1. 131
or
which may be written
A being the radius of curvature. If we denote by ca the angle between the principal normal to the extremal and the normal to the surface at the point P (x, y, z), and by 2. u. v the direction cosines of the normal to the extremal at the point P in the tangent plane to the surface at this point, the last equation becomes
(8)
By equations (6) and (8) the extremals are determined, 6. Now the geodesic curvature
of an extremal at any point P(x. y. z) is equal to the curvature of the nrojection of the extremaal upon the tangent plane at this point P. Let G (e, v, C) be the centre of curvature of the projection of the extremal, which may be called the centre of geodesic curvature. Then
In consequence of these equations, (8) becomes
(9) 182 KINNOSUKE OGURA:
which is nothing but the polar plane at the point P(x, y, z), with respect to V obtained in Art 2 Hence the centres of geodesic.curvature G of the ooi extremals. pass- ing through any point P lie on the straight line p which in the intereec- Lion of the polar plane (9) and the tangent plane at the point P
Conversoly, ifthe centre of geodesiccurvature G at any point P of a curve lie on the straight line p which is the intersection of the polar plane with respect to V at the point P and the tangent plane to the surface at the same point, then the curve is necessarily an extremal of the integral ∫s1s0ρ(x,y,z)x2+y'2+z'2dds Lastly it can be shown by a process similar to that in Art 3 that (i) if twofunctions SPand have tw same straight line p at. any point, Own
ψ(x,y,z)=k(X,y,Z), (k, const.) and (ii) the function Sois in general determined, and can be constructed, if we know, 2 s out of the totality of s2 extremals, each of two systems of s' curves covering the surface simply, that is, one passing through each point of the surface. In the following chapters we will give a detailed investigation of this matter. 6. Particularly, when cp is constant, equation (8) gives
(I) In the particular case of a uniform catenary in the field of gravity, we can take
-U=g z-c, (g, c, coast.); hence (9) becomes
g(C-z)=gz-c, or When be the point Q whore P is the middle point of the segment GO2 we get
so the point Q lies on a fixed ' horizontal plane See Lindolof-Moigno , Calcul des variations (1861), p. 313; Bolza, Vorlesungen fiber Variationsreohnung (1909), p. 586. TRAJECTORIES IN THE CONSERVATIVE FIELD OF FORGE, PART I. 133
so the extremals are geodesics on the surface. In this, case the straight line P is at infinity, for- the centres of geodesic-curvature are at infinity. Next, when an extremal is a geodesic,
so that the extremal is an orthogonal trajectory of the ool curves
gyp=coast. on the surface; or along the orthogonal trajectories of the extremal SP must be constant Conversely when SP is constant along the orthogonal trajectories of any x extremals, the extremals are geodesics. Let IV=x(u, v), Y=y (u, v), z=z (u, v). be the parametric equations of a surface S, and let vi(u, v)=const. be the orthogonal trcjectories of any given oo' extremals which are geo- desics, then the function (p is given by (10) SP=1P(vl) where (P denotes an arbitrary function. (1) When u=const. and v=const,
are any two different systems of k geodesics on the surface S, there is no function, to within a constant, under. which both systems it=cowl. and v=coast. are extremals. (3) 7, Lastly we will derive some formulas (for reactions in the orbits in the conservative field) which are. sometimes useful.
Firstly, multiplying equations (7) by ax of of respec-
tively and adding
(1), (2) For another proof, see Art. 31, II. 134 KINNOSURE OGURA: so that
(11),
For example, for the orbits of a particle in the conservative field of force, since
kq reaction of the surface, we arrive at the well known result that the acceleration along the normal to the surface is equal to the sum of the force along the normal to the surface and the reaction of the surface. Secondly, multiplying equations (7) by y'z"-z'y", z'x'-x'z", x'y"-y'x" respectively and adding
whence
(12) sin a=0.
For the orbits of a particle in the conservative field of force, we get the well known result that the sum of the components of the force and the reaction along the binormal is equal to zero. TRAJECTORMS IN THE CONSERVATIVE FIELD OF FORCE, PART I. 135
CHAPTER II.
Asymptotic Curves and Free Orbits on a Surface.
Asymptotic curves as orbits.
$. In this chapter Ave, vill confine ourselves to consider, for con- venience. only the orbits in the conservative field of force; but the methods of proof are applicable to any natural family, and the cor- responding results for the general case can be stated at once. Let the equation of a fixed smooth surface S be z=f(x, y) and the orbits on the surface in the conservative field be
(12) where co denotes the angle between the normal to the-surface and the principal normal to the orbit, and A, p, v denote the direction-cosines of the normal to the orbit on the tangent plane. Now consider the particular orbit I' which is defined by
(13)
In order that the curve I' be an orbit, it should be
But since
Σ(ρx')2=1 and Σ λ2=1, we must have
14) sin (0=1 or and
(15)ρx"=± λ, ρy"=± μ, ρz"=±v If follows from (14) that the binormal to the curve I' at any point is the normal to the surface at the point; and, from (15), that the normal to the curve I' on the tangent plane is the principal normal to the curve I'. Hence along the curve I' the tangent plane 136 KINNOSUKE OGURV: at any point coincides with the osoulatinix plane at the point; that is, the curve r is an asymptotic curve on the surface S. Conversely, if any orbit of a particle on the surface S in the con- servative field be an asymptotic curve, then it must be the curve r, For, since the tangent plane coincides with the osculating plane along an asymptotic curve,
ρx"=± λ, ρy"=±12, ρz"=±v, sinω=±1. so equation (12) of the orbit takes the form (13). In consequence of Art. 5, a nece8sarzj and sufllcient condition duct an asymptotic curve on the surface S should be an orbit in the con- servative field having the force function U+h, is that the centre off curvature of the asymptotic curve lies on the polar plane with respect to 1/2(U+h). Analytically, the above condition may ba written
(16)
along the asymptotic curve
(17)
on the surface (18) zf (a, J) Lastly we add a remark. Along the asymptotic curve, since sin w=+1, the acceleration along the normal to the surface must be zero so from (H) or (12) it follows that the sum of the components of the force and the reaction in the direction of the binormal to the orbit, which is an asymptotic curve, is equal to zero.
Free orbits on a surface.
9. If the component of force be zero for the direction of any binormal to an orbit which is an asymptotic curve, then the two equations
(1) In the general case of natural families this equation becomes TRAJECTORIES IN THE CONSERVATIVEFIELD OF FORCE, PART 1. 137
(13)
Fx Fy Fz (19) x' y' z' =0 x" y" z" ehould be satisfied : hence by equations (3) and (4) in Art. 1, the assvmntotic curve is a free orbit on the surface. Conversely, if a particle describe a free orbit on the surface, by Art. 1. we have the above two equations (13) and (19); and then the orbit is an asymptotic curve and the component of force is zero for the direction of the binormal to the orbit. But the binormal to the asymptotic curve at any point coincides with the normal to the surface at this point; so the force-vector touches the surface at the point of impression. (1) Hence we obtain the theorem: A necessary and sufficient condition that a particle should describe a free orbit on tke surface is that the orbit be an asymptotic curve and the force touch the suiface at any point of the orbit. Analytically this condition may be written
(16)
(20)
along the curve
(17)
on the surface (18) z=f (x, y) If the surface be defined in the parametric form x=x (u, v), y=y (u,v), x=z (u, v),
we have
(1) In the ruled surface the force-vector may lie in the surface. 188 INNOSUKE OGURA:
and also
so that equation (20) becomes
(21)
10. Particularly, consider the case. of a plane, for example, x=0. From equation (20) we see that the force must be contained in thus plane; but since any curve on the plane is an asymptotic curve, the above condition for free orbits is reduced to only one equation
or
Hence in this case any orbit is a free one; and the centres of curvature of o orbits at any point (x, y) lie on the straight line
(ξ-x)Fx+(η-y)Fy=q3,
(1) For an example, see Art, 18. TRAJECTORIES IN THI. CONSERVATIVE FIELD OF FORCE, PART 1. 139 which may be called the polar line at the point (x, y) with respect to q. (1), Let us assume that any two orbits passing through any point (x, y) and having different tangents at this point are drawn. Then the polar line at this point is known; and so if a particle arrive at this point by any given path, we can construct graphically the path, in the vicinity of this point, after the particle has passed through this point.
Determination of the conservative force under which one system of asymptotic curves can be described.
11. Now we will determine the most general conservative field of force under which one system of asymptotic curves on the given surface can be described. On the developablesurface two systems of asymptotic curves coin- cide with each other and they form a family of geodesics; hence if v1=const. be the orthogonal trajectories of generators of the surface, it follows by (10) that (22) U+h=di (v), where 0 is an arbitrary positive function. (2) We now proceed to consider the general case. Let the linear ele- ment of a surface be as usual
(23) ds2=Edu2+2Fdudv+Gdv2, where
(24)
and let asymptotic curves be taken for parametric curves.
(1) Pref. E. Kasner did not state this property explicitly, although ho confirmed the existence of this straight line, See also 0. Sohoffels, Leipziger Ierichte 50 (1898), p. 261. (2) From q=2(U+14), we see that U+h must be positive. 140 RTNNOSURE OGUBA
It is well known that for any curve on the surface(1)
where
Btit in our case, since asymptotic curves are taken for parametric curves
(1) G. S chef fors, Anwendung der Differential-und Integralrechnang nuf Geometrie, 13d. 2 (1902), p. 481. TRAJECTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART I. 141
L=O, N=O.
Now along the family of curves
u-const. since
(25) by (23) or the equation
(26) we obtain
(27)
Differentiating (26) with respect to s and remembering (25) -and (27),
Consequently along the curves u=const.
(28)
where
and the curvature of these curves is given by 142 KINNOSUKE OGTJRA:
=A2uE+2A uB%F+BuG,
or which is a well known formula. Therefore along the curves is=const., it follows that
where
(29)
or, introducing Christoffel's symbols, equations (29) map be written
(30)
Now since
equation (16) becomes TRAJECTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART I. 143
Alone any curve on the surface passing through the point (u, v), we have
and
so that
or
While a it ax and 1-d2c dx are the definite functions
of the point (u, v), the direction of the curve and hence dv is quite
arbitrary; therefore it must be
and
Similarly 144 KINNOSUKE OGURA:
Hence if Uu, be the force function under which u=const. can be described, it must satisfy the linear partial differential equation
or
(31)
By a quite similar way, the forcefunction Uv, under which the other family of asymptotic curves
V=const. can be described, satisfies
(32)
where
(33)
or
12. In the particular case where
F=0. that is, for a minimal su, face, our differential equations (31) and (32) become very simple. It is well known that for a minimal surface we can take families of asymptotic curves u=const. v=const. such that
E= G, F=O; L=N=O, M=+1;
(1) In the general case of natural families, equations (31) and (32) are replaced by TRAJEOTORIES IN THE CONSERVATIVEFIELD OF FORGE, PART I. 145
S being the total curvature of the surface. In this case
Now consider the non-ruled minimal surface where
then equation (31) becomes
from which we obtain
01 being an arbitrary positive function. Similarly from (32) or
we get
0, being an arbitrary positive function. But by a theorem due to Lie, the asymptotic curves of the minimal surface can be found by quadratures; hence in the non- ruled minimal surface our problem can be solved by gnactratures ouiy. Conversely, in the surface de=Eclue+2Fdudv+Gdv2,
where u=coast. v=const are families of asymptotic curves, if 146 KINNOSUIC OGURA:
(34)'
then the surface is a non-ruled minimal surface. For, comparing (31) and (32) with the following equations, which are derived from two equations (34)',
it follows that
Du=O and Cv=O; hence
F=0.
It is remarkable that the force-function U+h under which both families of asymptotic curves u and v can be described on the non- ruled minimal surface is equal to a constant multiple of the absolute value of the principal curvature; for in this case
Ic being a positive constant.
Determination of the conservative force under which both systems of asymptotic curves can be described.
13. Now we will determine the force-function U under which both families of asymptotic curves u=const. and v=const. can be described on the surface. In this case U+la must satisfy the two equations
(35)
Solving them with respect to TRAJECTORIES IN THE CONSERVA CIVE FIELD OF FORCE, PART I. 147
(36)
Consequently, in order that such a function U may exist, the con- dition of integrability
(37) must be satisfied and f the condition be fuflled, the required function is given by
or
(38)
k being a positive constant. Thus we have the theorem: When two systems of asymptotic curves on a surface are known, the conservative force, if it exist, under which such two systems may be the orbits on the surface, can be found bu quadratures only. 14. Particularly for the minimal surface ds2=Edu2+Gde, (E=G), we have
so that the condition of, integrability is satisfied, and
or 148 KINNOSUKE OGURA:
k being an arbitrary positive constant. For example, let us take the catenoid x=cosh (u+v)cos(u-v), y=cosh (u+v) sin (u-v), x=u+v; in which case
and E=G=2 cosh2(u+v), F=O,
Also since
and
it can be verified that TRAJECTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART I, 149
Now we find that oil=-coth (it+v), Du=0, Cv=0, Dv=-coth (u+v); whence
(k, a positive const.).
But since C61+y=cosha (2t+V), the force-function is inversely proportional to the square of distance from the axis to the point of the surface. 15. In the pseudosplterieal surface whose total curvature is a negative constant K, we can take
(0 being any solution of the differential equation
The condition of integrability may be written
that is,
or by integration (39)' cos cu=0 (it+v)+tF (u-v) 0, 3P being arbitrary functions. If the condition be satisfied, the force function is given by
For an example, consider the surface where w takes the form w (u+v). (1)
(1) For the geometrical meaning of this form, see Art. 32. 150 KINNOSUKE OGUEA:
In order that the surface may have, a constant negative total curvature, it should be
whore E=u+v; so if we substitute cosf=12,
a, b being arbitrary constants. The condition of integrability is satis- fled, of course, in this case. On the contrary, if we consider the pseudospherical surface where (o takes the form (wf=(uv), (1) the function f is given by
where E=vv.
In this case it is easy to show that the function f does not satisfy the condition of integrability.
Ruled Surfaces.
16. When
(40) the surface is a ruled surface (besides the ordinary helicoid) having the generators u=const., and we have
so that it is required to give a separate discussion in this case. Let an orthogonal trajectory of the generators be the directrix a0=x0(u), y0=y0(u), z0=z0(u), expressed in terms of the arc u measured from a point on it. If 1, m, n be the direction-cosines of the generator through a point (u) of
(1) For this surfaco, see L. Bianchi, Vorlesungen uber Digerentialgeometrie, 2 Aug. (1910), p. 451. TRAJEOTORIFS IN TEE CONSERVATIVE FIELD OF FORGE, PART I. 151
the direotrix, and v, be the distance from the point (u) to a point (u, v,) on the generator through (u), the equations of the ruled surface, are x=xo(u)+v1Z(u), y=Jo(u)+v1in(u), z=z0(u)+v1(u),
where
The linear element of the surface is dsz=(P2v2+2Qu+1)du1+dv
where
Now u=const. is a family of asymptotic curves and the other family is determined by Riccati's equation
a, b, c being known functions of u. It is well known that the solution of this equation takes the form
(as-r*O), where a, b, r, d are known functions of u, and Ic is an arbitrary con- scant. Hence if we put
it follows that v=const. is the other family of asymptotic curves. Thus the equations of the surface an
and we find for the linear element the expression 162 KINNOSUKE.OGURA:
ds2=Edu2+2Fdudv+Gdv2, where
Since v=const. are the orthogonal trajectories of generators u=const., in consequence of Art 6,
Uu+h=0 (v1) or
0 being an arbitrary positive function. But it is well known that the orthogonal trajectories of generators can be found by one quadrature; so the determination of Uu expressed as the function of u and v re- duces to the finding of the solutions of Riccati's equation and to quadratures. Next we will determine the force-function U under which both families of asymptotic curves can be described. If we assume that the surface is not a ruled surface of the second degree, then
must satisfy
that is,
Since the right hand side of this equation should be the total differential, in order that the force-function U exist, it is necessary and sufficient that the condition of integrability TRAJECTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART I, 153
should be satisfied; and if the condition be fulfilled, the force-function is given by
U+h=count. Hence for the ruled surface the force function, if it exist, under which both families of asymptotic curves may be orbits, can be found bra nuadra- tures and by solving R icc a ti's equation. 17. We are now in the position to discuss the ruled surface qt the second degree. I. The hyperboloid of one sheet
can be expressed as follows:
u=const. v=const. being generators. In this case, since
the differential equation of the orthogonal trajectories of generators uc=const, is of Riccati's type
But since. v u is a particular solution, if we substitute
it becomes the linear equation 154 KTh NOSUKE OGURU:
whose solution is
hence
By a similar way we can find Uv+h; and by Art. 6, it is shown that when both systems of generators of a hyperboloid of one sheet can be described as orbits on the surface, the conservative force must be zero at every point of the surface. II. Next the hyperbolic paraboloid
can bo expressed as follows: x=a(u+v), y=b(v-v), z=uv,
u, v being generators. In this case, since
T=a2+b2+4v2, F=a2-b2+4uv, G=a2+b2+4ua, the orthogonal trajectories of u=const. are given by the linear equation
whose solution is
hence TRAJECTORIES IN THE CONSERVATIVE FIELD OF FORGE, PART 1. 155
18. Lastly consider the surface where
(41) and F=O,
i. e.
In this case we find that
and the corresponding surface is an ordinary helicoid having generators v=const., which may be written
x=sinh u cosv, y=sinh u sin v, z=v.
Now we have
1=G=cosh2u, F=O,
L=N=0, M=1,
and also
sink it=ta-i,
from which it is seen that
Then equation (31) becomes 156 KINNO. SUKE OGURA:
whose solution is
0 being an arbitrary positive function. And since u and v form an. orthogonal system, by Art. 6, we get
Uv+h=V(u),
P being an arbitrary positive function. Hence we have arrived at the result
k being an arbitrary positive constant. It is to be noticed that this result is nothing but a particular case of that in Art. 14. This force-function U satisfies equation (21), and consequently under this force, the generators v=cont. and its orthogonal trajectories (helices) u=cont. are, free orbits on the minimal helicoid.
The differential equation of the surfaces whose asymptotic curves are orbits under a given conservative field. (1)
19. We will find the partial differential equation of the surfaces whose asymptotic curves are orbits described by a particle under a given conservative field of force. For which we will hereafter (Art. 19-20) use the following usual notations
If the required surface be (18) P=f(x, y), the differential equation of asymptotic curves is
(17)' rx'2+2sx'y'+ty'2=0,
(1) For some particular cases, see' Art. 6, 12 and 17. TRATEOTORIES IN THE CONSERVATIVE FIELD OF FOROE, PART 1. 157 and the condition that asymptotic curves can be described by a particle under the given force-function U is
(16),
From (17)' it follows that
Differentiating (18) with respect to the curve-length (42) z'=px'+qy" of which
(43)
But from the equation (44) x'2+y'2+z'2=1, we get
(45)
Differentiating equation (44) with respect to the curve-length
(46) x'x"+y'y"+z'z"=0, and again differentiating (42) and by recalling (17)', (47) z"=rx'+2sxy+ty'2+px"+qy"=2px"-qy', thus we have from (43), (45), (46) and (47)
(48)
Differentiating (17)' with respect to the curve-length
+2x'x"r+2(y'x"+x'y")s+2y'y"t=0 158 KINNOSUKEOGURA consequently by (48)
from which it is seen that
(49)
where we have put for the sake of brevity
(60) W3=(1+1p2)t1+2pqt(-s+s1+rt) +(1+q2)(-s+1+rt)2,
Wb=tqp+{1+p1)/ert+t(s+Vsue-rt) XIT(1+q2)-p(-s+s2-t)}-h(-s+1/s-rt)2(1+82)8
On the other hand, by (16)', (47) and (48)
(51)
from which it follows that
(52) TRAJECTORI IN THE CONSERVATIVE FIELD OF FORCE, PART I. 159 where Wj-tt(1+p2)(1-I-22-q)-i-21(-s+s1-2t)(1-1-p24g2+pg2-p2) +(841/7-rt)2(1+q2)(1+p2+q2), (53)
Hence, by comparing (49) with (52), we obtain the partial dif- ferential equation of the third order of the required Surface (54) (U+h)W1W2=W3W5WW6.(1) 20. For the present we consider the five differential equations TVJ=0 (J-1, 2, 3, 4. 5).
I. In the first place, it is well known that
W=0
is the differential equation of the general ruled surface. II. Secondly. since the equation
W2=0
can be resolved into
we obtain two Monge-Ampere's equations of the form (65) r+2f, (p, q)s-f1-f2(p, q)t=0 r-I-2f2(p, q)s+2(p, q)t=0.
flI. Similarly the equation W-1, 0
ist resolved into two Monge-Ampere's equations of the same form as (55); but in this case fl and f, become imaginary, and asymptotic curves of the integral surfaces are also imaginary. Since these two equations
(1) Or, from Art. 20, V, we can show that this equation is reduced to (U+h)W1W2=s1-rtWa2W4 160 KINNOSUK OGURA:
W1=0 and W3=0 are of the form
(at-r't)=s+t. Jf(p,q) (j=1, 2), they include the developable surfaces which are obtained by Poisson's method. (1) IV. The equation IV=O
is a Monge-Amp re's equation of the form (56) r+2f (x,u,p,q,)t+ft-f2t=0, Now the two systems of the characteristics of Mongo-Am pere's equations (55), (56) respectively coincide with each other, and they are given by the simultaneous equations of the form
(52) dy-fdx=0, dp+fdq=0. Eliminating the function f between these two equations, we get
dpdx+dq dy=0,
which is the differential equation of asymptotic curves on the integral surface. And also since
the first equation of (52) rly-fdx=0 takes the form rdc+2sdxdy+tdy2=0.
Consequently two systems of the characteristics of W2=0, Tv3=0, W4=0
coincide with each other respectively, and they lire asymptotic curves of their integral surfaces. (2)
(1) If we ptlt i=00)), 4. (1)*O, the developable surfaces are given by the ordinary differential equation of the first order 1+v)f>{j, 40)1=0. (2) E. Gottrsat, Lecons stir rintegration des equations aux derlveos partielles du second ordre, t. 1 (189a), p. 223. TRAJEOTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART I. 161
In order to discuss asymptotic curves on the integral surface of Wi=0, if we transform by the polar reciprocation with respect to the paraboloid a2+y2-2z=0, which is equivalent to Legendre's transformation
x=p, y=q, p=x, q=-J,
then the equation W2=0 is transformed into
(1+x2)(1-x2-y21+J1-}-xYI- (-1+s-12(1+7)(1+xr+11)=0.
The differential equation Td P+2sdxdy+tdy=0
of asymptotic curves on the new surface is then (1+x2)(1+x2+y2)dy2-2(1+x2+y2+x1y1-x2y2) +(1+y2) (1+x2+J2)d2=0.
Hence if this differential equation be solved, asymptotic curves on the original surface are known: bacause the asymptotic curves of a surface are transformed into those of the transformed surface by the polar reciprocation. V. In the last place, the differential equation
tdy+sdx=+vtdx
of asymptotic curves on the integral surface of
W5=0
becomes dxz-I-{r(1+q2)-t(1+W)c2x-d-q +j(1+q2)s-pqtidyl=0,
which is nothing. but the differential equation of lines of curvature. Therefore the integral surfaces of W=0, whose asymptotic curves are 162 FINNOSUKE OGURA: not imaginary, must be developable. In fact, 116=0 is equivalent to t/s-t2 (1+p)2p 2t (s+s-rt)-1-(1f-7)(-s+s-rt)}=0, or
1s-rt+1V=0; and, as we have seen already, asymptotic curves of W3=0 are imaginary.
Application of toursat's trap eformation of the minimal surface.
21. By a certain transformation, let the surface x=x (vt,v), y=y (ft, v), z=z (u, v) be transformed into X (u, v))=J (t) v), r=r (it, v)) where v, v and it, v are families of asymptotic curves on these two surfaces respectively. Here we will give an example of such a transformation that when Uu Uv are given, Vu, Uu can be found at once. Get S and Svbe a minimal surface and its conjugate respectively: 2 R f(1. zt)F(r) dr, S: u=Rfi(1+r2) F(r)dr, z=1lf2rF(r)dr;
my=Bi (1-r2) F(r) dr,
So: yo=ltf (1+v1) F(r) (dr, xo=ldf2irF (r)dr,
where r is a complex parameter, F (z) is any function of r and B (0) denotes the real part of 0. TRMECTORIES IN THE CONSERVATIVEFIELD OF FORCE, PART I. 163
If we transform S by Goursat's transformaton(1)
x=x, k being a positive constant, then another minimal surface, S is obtained: a=Rf(1-r2) VF (kr) dr,
J=B fi (1+r2)lct F(kr)dT, i=Rf2rVF(ler)dr.
By this transformation asymptotic curves on S are transformed into those on hence u=u, v=v; and if we put fy 2F (r) dr=a+ij' (a,jreal), the equationsof asymptoticcurves may be written
Prof. Gour sat showed that when 1, m, n are the direction-cosines of the normal to the surface S,
from which we find
ro being the conjugate imaginary to r. Hence if we put
Te(u+v)+t (u-v)'re=e(u+v)-t (u-v),
(1) Goursat, Acta mathematica 11 (1888), p. 135 164 KINNOSUR OGURA the linear element of the surfaco S takes the form
Therefore by Art. 14, it follows that
CHAPTER III.
Application of Jonkovsky's Method, especially to Orthogonal Systems.
Joukovskv's Method.
22. Let S be a surface referred to any system of coordinates 2i, v, and let v, (u, v) be a function of a and v. Prof. N. E. Joukovskv discovered a method to determine the force-function Uv, under which a given family of carves v, (u, v)=const=
can be the orbits of a particle constrained to move on the surface. Here we will reproduce his method with some slight modificatiovs(2)
Let the linear element of the surface S be
(23) ds2=Fd'u+2 Fdu dv+Gdv2,
and list ul (v, v)=const.
be the integral of the differential equation
(1) Also we have the remarkable formula
(2) N. E. Joukovsky, lostimmtmg der Krhtftefunktion nach einem gegobonen System von llahnon, Arboiten der phys. Soction dot Iraiserl. Gesellsahaften fur Frenndo der Naturkundo, Moskau, 13d. 3, Soft 2 (1890) [Rnssisch]. I regret very much that I have not been able to consult with his original paper; but the outline of his method will be found in r. T. Wh-ittaker, A-treatise on the-analytical. dynamics (1901), p. 107 and also in Tortschritte der Mathematik, 22; p. 890. TRAJECTORIES IN THE CONSERVATIVE FIELD OF FORCE, PAST I. 165
which is the family of the orthogonal trajectories of the given family v=const. Then we can take (57) de=E, d'u2-G, dv, 2, where
(58)
The kinetic energy T of a particle with unit mass which moves on the surface is given by
where dots denote differentiations with respect to the time t ; and Lagrange's equations of motion are therefore
where Uv, denotes the force-function which it is required to determine. These equations are to be satisfied by v1=0; they then become
(69)
(60)
But since 166 KINNOSUKE OGURA:
equation (59) becomes
(61)
Now we, will consider the case where
(62)
Mat is, v1=canst, is not a, family of geodesics. Eliminating u12between (60) and (61), we have
whose first integral is
(63) where f (v1) is independent of u1. While by (60), equation (63) be- comes
since the left hand side represents the total energy of the particle whose orbits are v1=const., it must be -f(v1)=h where h denotes a definite constaut. (2) Henco from (63) we have
(1) It seems to me that Prof. Joukovsky did not discuss the case where av=0. For this case, see Art. 26-29. (2) See the introduction of this paper. TRA, TECTOIUFS IN THE CONSERVATIVE FIEGD OF FORCE, PAIR I. 167
and therefore
64) or Uv1+1c=A,(ul) 0 (ul), where 0 is an arbitrary positive function. Similarly, in the case where u1=coast. is not a family of geodesics, we obtain UV1+1a=01(vI)T(VI),
where F is an arbitrary positive function.
Families of orbits which form an orthogonal system.
23. Particularly let u=const. and v=coast. form an orthogonal system, none of which is a family of geodesics. Then by (64) we find that
Hence for the existence of the force-function U under which. u=coast, and v=coast. can be orbits, it is necessary and sufficient that
or E=1, (u, v). (P (u), G=I, (v, v). (v)
so that these two families u and v form an isothermal system. Conversely, if u and v form an isothermal system, we can take E=A (u, v) 0 (u), G=A, (u, v) V (v), A(u, v)>0,
and hence the force-function is given by
(65)
lc being a positivo constant. Taut when u and. v1 form, an isothermal system and the equation of one of these two families is known, by the 168 KINNOSUKE OGUIRA: well known theorem due to Lie, the equation of the other family can be found by quadraturs only. Thus we arrive at the theorem: A necessary and suffcient condition that the conservative force under which two orthogonal families of curves (which are not geodesics and geodesic parallels) can be orbits is that these families form an isothermal system; and if the condition be satisfied, the force can be found by quadratures, even when only; the equation of one of these two families is lenown. 24. Here we give some examples. I. In the minimal surface, two families n and v of asymptotic curves form an isothermal system, and the linear element may be
where K denotes the total curvature. Hence the force-function is given by
U+h=-K which coincides with the result of Art. 14. II. It is well known that a system of geodesic ellipses and hyperbolas is orthogonal, and the surface with an isothermal system of geodesic ellipses and hyperbolas is necessarily Liouville's surface. (1) Consequently for the existence of the conservative force under which both families of geodesicellipses and hyperbolascan be orbits, it is necessary and sufficient that the surface is Liouville's type. For this surface referred to thesa families, we find ds2{u(n)+13 (v)}Glut+dv2), and then
III. An orthogonal system of the curves having constant geodesic curvature forms an isothermal system; and hence in a surface referred to these families the linear element has the expression(2)
(1) Dini, Annali di Matematica, (2), 3 (1869), p. 269; Bianchi, loo. cit, p. 171. (2) Bianchi, loo. cit., p. 175. TRAJEOTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART 1. 169
Hence we have U+h=1c4a(u)+13(v)
Particularly in the pseudospherieal surface whose total curvature is K=-(cat+b2), (a, b, const.), the families of curves ra=const. and v=const. having geodesic curvature a and b respectively form an isothermal system, and
hence it follows that U+It=1c (au-F-bv)2. 25. Now consider the case where is and v are lines of curvatures which are not plane curves. In order that there exists a conservative force under which two systems of lines of curvature can be the orbits, it is necessary and sufficient that the surface is an isothermic supface; and in the isothermie surface the forca-function U can be found by quadratures. There are some remarkable surfaces which are isothermie.(1) I. In the minimal surface we can so choose families u, v of lines of curvatures that the linear element has the form
where K denotes the total curvature. Henca we find
Uu+h=V-K.(v), U+h=1-K(1)0c), and U+t=1c1-f. II. More generally, in the surface of constant mean curvature, fi R1, I2 be the principal radii of curvature,
(c=0).
(1) See H. Willgrod, Ueber Flitchen, woloho sioh duroh ihre Ertimmungslinien in unondlich kleine Quadrate theilen lessen, Gattinger Dies. (1883); Darboux, Legons stir ]a theorie generale doe surfaces, t. 2 (1889), p. 239. 170 KINNOSUKSOGURA:
In this ease we can so choose u and v that.
from which we have
III. Quadrics. (i) The equation of the ellipsoid
(a2>b2>c2) can be replaced by
(-c2>u2>-b2>v2>-a2) and the parametriic curves are the lines of curvature. We find for the linear element the expression
hence
In a similar manner, zve can treat the hyperbbloids of one sheet and of two sheets respectively. (ii) The egtmtibh of the paraboloid 2s=ae+byl can be replaced by
and the parametric curves are the lines of curvature. In this case
(1) Willgrod, loc, cit., p. 30. TRAJECTORIES IN THE CONSERVATIVE I IELD OF FORCE, PART. I. 171 from which we have
IV. Lastly we consider the transformation by reciprocal radii with respect to the sphere (x-a)2-(y-b)2+(z-c)2=B2. BY this transformation the isothermic surface 8 is transformed into the other surface f5 which is also isothermic, and the linear elements of these are respectively (42=A (v, 47)10(41) die+(v) dv2f,
and the lines of curvature of S become those of Hence we get
(k, a positive const.), or
where a denotes the distance from the centre (a, b, c) of the fundamental sphere of inversion to the point (x, y, z) of the original surface.
Exceptional case where one family of orbits consists of geodesics.
26. In the previous articles (Art. 22-25) we have confined ourselves to orbits which are not geodesics. We are now in a position to discuss the exceptional case. Let us assume that in Art. 22, instead of (62),
(66)
that is, the curves (67) vL=const. are geodesics. The orthogonal trajectories of the family are given by the equation 172 KTNNOSUBE OGURA:
(68)
Hence if it, (it, v) denote the left band side of this equation, we find by Art. 6, (69) UV+7t=(h (2t1), whore, 0 is an arbitrary positive function. In the particular case where v=const. are geodesics, Uv+h=(h (It), where
Moreover, if v=const. be geodesics and t=const. be their geodesic parallels, the linear element may be written (70) (Is=tht2+G (lt2. Firstly we consider the case where
(71)
that is, u=const. are also geodesics. 'By the change of parameter v=fvrcly,
the linear element is transformed into
(182=cliff-1-clv
In consequence of this form it follows that the surface is developable, and Uu+h=715(v)=(h (v), Uv=T (2t),
and hence U-ft=coast.,
which has been shown in Art. 6.
(1) Darbonx, loc. cit., t. 2 (1881)), p. 421; Bianchi, loc. cit., p. 167. TRAJECTORIES IN THE COLTS RVATTVE M MD OF FORCE, PART 1. 173
27. Secondly, if we assume that
(72) it follows that by Art. 6 or equation (69), Uv+h=(1) (11), and by Art. 22,
Hence for the existence of the force-function U under which V, V call be orbits, it is necessary that G takes the form
Conversely, if the condition be satisfied, the function U exists and is determined by
If the parameter v be transformed by V=YT(v)(IV
the linear element becomes
(731=(lu+01(u)dv1, where
and then we have
(73)
e being an arbitrary constant. By this form of the linear element it follows that the surface is a surface of revolution whose meridians and parallels are u=const. and v1=coast. respectively, or a surface ap- plicable to it. Consequently we have the theorem: A necessary and svicient condition that the conservative force under which a family of geodesics and its geodesic parallels can be orbits is that the surface is a surface of revolution whose meridians are (liven geodesics or a surface applicable to it. It is to be noticed that this theorem is nothing but a particular 174 KINNOSUKE OGURA.: case of that in Art. 23; because by the transformation of the parameter
the linear element takes the form de=f2(u1) (du2+dv12). 28. If the equations of a surface of revolution be x=ucosv, y=usinv, xf(u), the linear element is cls2=11+fl(u)ldu2+w2dv3,
If we put uh= J1+1+P(u) A or u=0(u0, the linear element is transformed into ds=du12+S02 (ui) dv2; from which we obtain the force-function under which u1, v and therefore u, v can be orbits:
k being a positive constant. Hence the force-function at a point on the surface is inversely proportional to the square of the distance from the axis of revolution to the point. Since the surface of revolution is Liouville's type, there is the geodesic representation G other than similar transformations and deformations. In general, any Liouviile's surface S having the linear element
(ls2=If1(vi)-f2 (v1)1. (dull+dvl2) is transformed, -by G, into 9 having the linear element
For example, we consider the pseudospherical surface S of the elliptic type
(1) Dini, loo. cit. Scheffers, loo, oft., p. 420. TRAJEOTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART I. 175
where -zalz denotes the total curvature S. If we put
v=v, the linear element becomes
so that we can put especially
A (vi)=Ra.
Hence by the transformation G it is transformed into the surface 3 whose linear element is
or ds-2=du-u2c1, where we lave put
(74) v=v.
It will be seen that the surface 3 may be considered as a plane referred to the polar coordinates u and v; and equations (74) denote the well known transformation due to B el t ram i. (-1) For the two surfaces S find 8, since
And or
it follows that
(1) Beltrami, Giornale di Miatematicho, 6 (1868), p. 284; Bisnchi, loc. cit., p. 441. 176 KINNOSUKK OGTRA:
where 14k' and k" are arbitrary positive constants. Also in the plane S we see that the force is an attraction inversely proportional to the cube of the distance from the origin of coordinates, which is easily verified. 29. We add now two examples. I. In the surface of constant cntrvatire the linear element may be. written
when and
when hence
find
respectively. II. Let i denote the distance of a point of a general helicoid from the axis and let x=ca(r) be the equation of the meridian. Then the equations of the helicoid are 2=icosV-rsinv=(r)+inv, from which (VI=I1+V/2(r)}c1r242m p(r)dclv-f-(I-1-in)dv
If we pat
(k, arbitrary constant.),
n2=12 (r2+m2), the linear element becpinos (lI1+O12(U)}clu2+u2(IV" or (ls2=(1v12+f2 (2c')(1V12, TIATECTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART I. 177
Where 2c=fl-1-h1(2c)chc and ztf (ul).
Conscauently the force-function under which a family of helices μ1=const (and hence r=coust.) and the orqlogonal trajectories (which are geodesics v1=coast., and hence v+mfS)-(1r=coast.) can be
orbits, is given by
1. being an arbitrary positive constant. In the ordinary helicoid
x=rcosv, I=rsinv, z=mv+const.,
if we put
r=m, sink u,
the force-function has the expression
which coincides with the result in Art. 1S.
CHAPTER IV
A General Method for the Determination of the Function
9 by Two Families of Extrmals on a Surface, and tl to Differential Equation of All Extremals.
The third method for the determination of s
30. In the second and third cliapters we have shown two different methods for the determination of the function cp under which a given family or two given families of curves on a surface can bo extremals. Nevertheless, Prof. Joukovsky's-method in the third chapter is applicable only to the orbits in the conservative field of force where the differential ea-tation of the orthogonal trajectories of one family of orbits is integ- rrible. Oil the other hand, the method in the second chapter where 178 IINNOSUKE OGUBA: the two families of extremals are asymptoticcurves may be also applicable to any families of doves; but the process is very tedious. Therefore in this chapter we will give the third method which is of the mosb simple and general nature. If f (u, v)=coast. be a family of the extremals with (1)" JS-SP (u, v) ds=min. on the surface S having the linear element referred to any. system of coordinates u, v,
(23)" ds2=E the+2F du dv+Gdv=1 then f (u, v)=const= is a family of the extremals with (75) J'd8=min. on the surface S having the linear element (76) ds2=(u, v) (E due+2 Fdu dv+G dv2). Hence by the conformal transformation (77) (14-2=[n2c141, or
(78) the extremals with (1)" on the surfaces S are transformed into the geodesics on the surface, 9, and conversely. Consequently we have the theorem: A necessary and sufficient condition that the function qO, under which the two families u= const. and v=const. can be extremals on the surface, may exist is that the families u and v become geodesics on the transformed surface 3 by a conformal transformation; and if the condition be fulfilled, the function p is given by
Now the necessary and sufficient condition that the curves v=const. be geodesics on the surface, b7 is that
(79) TRAM, CTORIES IN THE CONSERVATIVE FIELD OF FORCE, PART I. 179 but in consequence of the identity
the above condition becomes
(80) which is nothing but the condition that v=const. may be extremals, for tx f action s' on the surface S. (Compare with equation (32) in Art. 11). Similarly the necessary and sufficient condition that the curves u=coast. may be extremals for the function Sp on the surface S is that
(81) or
(82)
Hence if u=const. and v=const. be the extremals for the func- tion Sp, two equations (80) and (82) should be satisfied simultaneously; and then
(83)
The condition of integrabity of equations (83) is MY KINNOSUK EOGURA: and f the condition befulfilled, the function Sois cliven by the equation
(85) where 1c denotes an arbitrary constant. 31. Now we consider the two particular cases. I. When u=coast. mud v=con st. form an orthogonal system, equation (82) becomes
whose integral is
ρ=G1/2(v),
0 being an arbitrary positive function. And also equation (80) becomes
from which
T=L"-1qr (u), ! being an arbitrary positive function. Hence the condition of integrability is
of which it follows that the families ii=const. and v=const. form an isothermal eystem (see Art. 23). And if the condition be satisfied, the function SP is given by
where lc denotes an arbitrary constant. II. If a=cnst. be geodesics on the surface 8, equation (82) takes the form
hence when the integral of the differential equation Z+'clu+Gdv=0(1)
(1) This is the differential equation of the orthogonal trajectories of u=const. TRAJECTORIES IN THE CONSERVATIVE FIELD OF FOIIOE, PART I. 181 be v, (u, v)=coast., the function is given by
(P (P (v, l where (h denotes an arbitrary function (see Art. 6). Lastly, since the quantity I. G-F2 is not zero, the simultaneous equations
have only one solution (p=eonst. This is an alternate proof of the last theorem in Art. 6. 32. The linear element of a surface referred to the cquidistantictl system has the form
(7,82=t1'u2+2 cos cv (lit dv+dv2.
Consequently, as in Art. 15, the condition of integrability is cos m=((u+v)+(u-v), where (h and T denote arbitrary functions; and if the condition be satisfied, the function sp is given by
k being an arbitrary constant. Since the linear element may be written
t=at+V, -v=v-v, if to be a function of v+v (or v-v), u-v=coast. (or zt+v=coast.) are geodesics, and the surface is deformable to the surface of revolu- tion. (1) Conversely, if the diagonal curves of equidistantial system be geodesics and geodesic parallels, the surface is deformable to the surface of revolution; and when the surface is referred to the equidistantial system, the linear element takes the form ds2=(lie+20 (u+v) du dv+dv2 or cis2=du+2 (P (v-v) du dv+dv2
(1) A. Voss, Mathematische Annalen, 19 (1882), p. 3; A. Voss, Dyck's Katalog math. u. math. -phys. Modelle (1892). p. 16. 182 KINNOSUKE OGURA
Hence in the surface whose diagonal curves of equidistantial system are geodesics and geodesic parallels, the two families of equidis- tantial system may be the natural family. Already, in Art.: 15, we have dealt with the pseudospherical surface, in which case the equidistantial system consists of asymptotic curves. Here we consider the surface whose equidistantial sytem is a conjugate system. Let the equations of any two curves be x1=X1(u), J1=Jl (u), z1=z1 (u) and x2=a(v), Jl=1/2 (v), z2=z2(v) respectively, where u and v denote the lengths of these curves respectively. Then the linear element of the surface of translation is of the form ds2-du+2 cos to clu dv+dv2, where w is the angle between the curves u=cont. and v=const., which form a conjugate system. For example, in the paraboloid
x=u, I=v, z=au1+bv1 we have xl=u, 1=0, z1=au2 and x1=0, y2=v, P2=v2; from which cos w=2 au+2 by=(cc-1-b) (u+v)+(a-b) (u-v), and therefore
In the ordinary helicoid x=a (cosu+cosv), y=a (sinu+sinv), z=b (u+v), we have x1=acosu, y1=asinu, z1=bu and x1=acosv, v2=asinv, z=bv; from which COsCa=a2 Cos (u-v)+b2, and hence TRAJEOTORIES IN THE CONSERVATIVEFIELD OF FOROE, PART I. 183
Differential equation of all extremals.
33, Suppose that the function Sp has been found, under which u=coast. and v=const. are extremals. Then we can find the differential equation of all extremals on the surface S for this function Sp. For, since these extremals on the surface S-are geodesics on the surface S (Art. 30), the differential equation of them is of the form(1)
where
by the assumption, and(2)
(1) Bianahi, loo, cit., p. 153.
(2) 184 KINNOSUKE OGUIRA:
Henco the differential equation of the extremalsfor Sois
(86)
Here we add a remark: The so-called general equation of orbits in the conservative field of force denotes all curves for all values of the energy constant h on the contrary, we have considered the case where 11has a definite value.(2) 34. Lastly we will give some examples. I. In the surface ds2=A (it) (d2+dv2), equation (86) becomes
whose solution is given by v=au4
a, b being arbitrary constants; and then
For example, in the surface of revolution,
dsl=due q-r2 (2t) de,
the function (p is Ic; and the extremals are
(1) If u and v be geodesics, that is, SP=const., then the equation becomes the dif- ferontial equation of geodesics. (2) For an example, see the foot-note in Art. 31, 1 (p. 185j. TRATEOTORIE. S IN THE CONSERVATIVE FIELD OF FORCE, PART 1, 185 a, b being constants. If 8 be the angle between the extreinal and v=const
from which it follows that the extremals make a constant angle with the meridians. Consequently the extremals are loxodromic curves on the surface. Particularly in the plane referred to polar coordinates, the linear element is cs=-chi=-1.2h dv", and the extremals are
V=(logn+b, which are logarithmic spirals. (1) II. In the surface referred to the equidistantial system, the linear element is
ds2=du1+2cos co dudv+dv1 and the condition of integrability has the form COS(0(1)(u+v)+ 0(u-v); and the differential equation (86) becomes
(1) In the conservative field of force 2(U+1i)=u1, (Art. 28),
the extremals of the integral
(lv=miil.
are given by
(86)' or
where, a and c are arbitrary constants. Substituting 2L+ta) for u-1 in equation (86)'
Now in order to eliminate h, differentiating with respect to v and dividing by 2 d/du/v wo
obtain
which is the so-called general equation of orbits. X86 KINNOSURE OQURA: PIIAJEOTORIES-IN THE CONSERVATIVE FIELD.
Particularly in the case where
cu=ft+v). the above equation is transformed into
where s=11-4-1
lay changing the independent variable from is to,
Again substituting
we obtain
whose solution is
a being the constant of integration. Hence the general solution of the original equation is given by
where b denotes the constant of integration.