(3)and a velocity function of the formφ = xψ,(4)where ψ is a function of λ only, so that ψ is constant over an ellipsoid; and we seek to determine the motion set up, and the form of ψ which will satisfy the equation of continuity.Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane,l =px, m =py, n =pza2+ λb2+ λc2+ λ(5)1 =p2x2+p2y2+p2z2,(a2+ λ)2(b2+ λ)2(c2+ λ)2p2= (a2+ λ) l2+ (b2+ λ) m2+ (c2+ λ) n2,= a2l2+ b2m2+ c2n2+ λ,(7)2pdp=dλ;dsds(8)Thencedφ=dxψ + xdψdsdsds=dxψ + 2 (a2+ λ)dψldp,dsdλds(9)so that the velocity of the liquid may be resolved into a component -ψ parallel to Ox, and −2(a2+ λ)l dψ/dλ along the normal of the ellipsoid; and the liquid flows over an ellipsoid along a line of slope with respect to Ox, treated as the vertical.Along the normal itselfdφ{ψ + 2(a2+ λ)dψ}l,dsdλ(10)so that over the surface of an ellipsoid where λ and ψ are constant, the normal velocity is the same as that of the ellipsoid itself, moving as a solid with velocity parallel to OxU = −ψ − 2 (a2+ λ)dψ,dλ(11)and so the boundary condition is satisfied; moreover, any ellipsoidal surface λ may be supposed moving as if rigid with the velocity in (11), without disturbing the liquid motion for the moment.The continuity is secured if the liquid between two ellipsoids λ and λ1, moving with the velocity U and U1of equation (11), is squeezed out or sucked in across the plane x = 0 at a rate equal to the integral flow of the velocity ψ across the annular area α1− α of the two ellipsoids made by x = 0; or ifαU − α1U1=∫λ1λψdαdλ,dλ(12)α = π√ (b2+ λ.c2+ λ).(13)Expressed as a differential relation, with the value of U from (11),d[αψ + 2 (a2+ λ) αdψ]− ψdα= 0,dλdλdλ(14)3αdψ+ 2 (a2+ λ)d(αdψ)= 0,dλdλdλ(15)and integrating(a2+ λ)3/2αdψ= a constant,dλ(16)so that we may putψ =∫M dλ,(a2+ λ) P(17)P2= 4 (a2+ λ) (b2+ λ) (c2+ λ),(18)where M denotes a constant; so that ψ is an elliptic integral of the second kind.The quiescent ellipsoidal surface, over which the motion is entirely tangential, is the one for which2 (a2+ λ)dψ+ ψ = 0,dλ(19)and this is the infinite boundary ellipsoid if we make the upper limit λ1= ∞.The velocity of the ellipsoid defined by λ = 0 is thenU = −2a2dψ0− ψ0dλ=M−∫∞0M dλabc(a2+ λ)P=M(1 − A0),abc(20)with the notationA or Aλ=∫∞λabc dλ(a2+ λ) P= −2abcd∫∞λdλ,da2P(21)so that in (4)φ =MxA =UxA, φ1=xAλ,abc1 − A01 − A0(22)in (1) for an ellipsoid.The impulse required to set up the motion in liquid of density ρ is the resultant of an impulsive pressure ρφ over the surface S of the ellipsoid, and is therefore∫ ∫ ρφl dS = ρψ0∫ ∫ xl dS = ρψ0(volume of the ellipsoid) = ψ0W′,(23)where W′ denotes the weight of liquid displaced.Denoting the effective inertia of the liquid parallel to Ox by αW′. the momentumαW′U = ψ0W′(24)α =ψ0=A0;U1 − A0(25)in this way the air drag was calculated by Green for an ellipsoidal pendulum.Similarly, the inertia parallel to Oy and Oz isβW′ =B0W′, γW′ =C0W′,1 − B01 − C0(26)Bλ, Cλ=∫∞λabc dλ;(b2+ λ, c2+ λ) P(27)andA + B + C = abc / ½P, A0+ B0+ C0= 1.(28)For a spherea = b = c, A0= B0= C0=1⁄3, α = β = γ = ½,(29)so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical accelerationW − W′g.W + ½W′(30)Thus a spherical air bubble, in which W/W′ is insensible, will begin to rise in water with acceleration 2g.45. When the liquid is bounded externally by the fixed ellipsoid λ = λ1, a slight extension will give the velocity function φ of the liquid in the interspace as the ellipsoid λ = 0 is passing with velocity U through the confocal position; φ must now take the form x(ψ + N), and will satisfy the conditions in the shapeφ = UxA + B1+ C1= Uxabc+∫λ1λabcdλa1b1c1(a2+ λ) P,B0+ C0− B1− C11 −abc−∫λ10abcdλa1b1c1(a2+ λ) P(1)and any confocal ellipsoid defined by λ, internal or external to λ = λ1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along OxUBλ+ Cλ− B1− C1.B0+ C0− B1− C1Since − Ux is the velocity function for the liquid W′ filling the ellipsoid λ = 0, and moving bodily with it, the effective inertia of the liquid in the interspace isA0+ B1+ C1W′.B0+ C0− B1− C1(2)If the ellipsoid is of revolution, with b = c,φ = ½UxA + 2B1,B0− B1(3)and the Stokes’ current function ψ can be written downψ = − ½ Uy2B − B1;B0− B1(4)reducing, when the liquid extends to infinity and B1= 0, toφ = ½ UxA, ψ = − ½ Uy2B;B0B0(5)so that in the relative motion past the body, as when fixed in the current U parallel to xO,φ′ = ½Ux(1 +A), ψ′ = ½Uy2(1 −B).B0B0(6)Changing the origin from the centre to the focus of a prolate spheroid, then putting b2= pa, λ = λ′a, and proceeding to the limit where a = ∞, we find for a paraboloid of revolutionB = ½p,B=p,p + λ′B0p + λ′(7)y2= p + λ′ − 2x,p + λ′(8)with λ′ = 0 over the surface of the paraboloid; and thenψ′ = ½ U [ y2− p √ (x2+ y2) + px ];(9)ψ = −½ Up [ √ (x2+ y2) − x ];(10)φ = −½ Up log [ √ (x2+ y2) + x ].(11)The relative path of a liquid particle is along a stream lineψ′ = ½ Uc2, a constant,(12)x =p2y2− (y2− c2)2, √ (x2+ y2) =p2y2− (y2− c2)22p (y2− c2)2p (y2− c2)(13)a C4; while the absolute path of a particle in space will be given bydy= −r − x=y2− c2,dxy2py(14)y2− c2= a2e−x/p.(15)46. Between two concentric spheres, witha2+ λ = r2, a2+ λ1= a12,(1)A = B = C = a3/ 3r3,φ = ½ Uxa3/r3+ 2 a3/a13, ψ = ½ Uy2a3/r3− a3/a13;1 − a4/a121 − a3/a13(2)and the effective inertia of the liquid in the interspace isA0+ 2A1W′ = ½a13+ 2a3W′.2A0− 2A1a13− a3(3)When the spheres are not concentric, an expression for the effective inertia can be found by the method of images (W. M. Hicks,Phil. Trans., 1880).The image of a source of strength μ at S outside a sphere of radius a is a source of strength μa/ƒ at H, where OS = ƒ, OH = a2/ƒ, and a line sink reaching from the image H to the centre O of line strength −μ/a; this combination will be found to produce no flow across the surface of the sphere.Taking Ox along OS, the Stokes’ function at P for the source S is μ cos PSx, and of the source H and line sink OH is μ(a/ƒ) cos PHx and −(μ/a)(PO − PH); so thatψ = μ(cos PSx +acos PHx −PO − PH),ƒa(4)and ψ = −μ, a constant, over the surface of the sphere, so that there is no flow across.When the source S is inside the sphere and H outside, the line sink must extend from H to infinity in the image system; to realize physically the condition of zero flow across the sphere, an equal sink must be introduced at some other internal point S′.When S and S′ lie on the same radius, taken along Ox, the Stokes’ function can be written down; and when S and S′ coalesce a doublet is produced, with a doublet image at H.For a doublet at S, of moment m, the Stokes’ function ismdcos PSx = −my2;dƒPS3(5)and for its image at H the Stokes’ function ismdcos PHx = −ma3y2;dƒƒ3PH3(6)so that for the combinationψ = my2(a31−1)= my2(a3−ƒ3),ƒ3PH3PS3ƒ3PH3PS3(7)and this vanishes over the surface of the sphere.There is ao Stokes’ function when the axis of the doublet at S does not pass through O; the image system will consist of an inclined doublet at H, making an equal angle with OS as the doublet S, and of a parallel negative line doublet, extending from H to O, of moment varying as the distance from O.A distribution of sources and doublets over a moving surface will enable an expression to be obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it.The method of electrical images will enable the stream function ψ′ to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle π/m, where m is an integer (R. A. Herman,Quart. Jour. of Math.xxii.).Thus for m = 2, the spheres are orthogonal, and it can be verified thatψ′ = ½ Uy2(1 −a13−a23+a3),r13r23r3(8)where a1, a2, a = a1a2/√ (a12+ a22) is the radius of the spheres and their circle of intersection, and r1, r2, r the distances of a point from their centres.The corresponding expression for two orthogonal cylinders will beψ′ = Uy(1 −a12−a22+a2).r12r22r2(8)With a2= ∞, these reduce toψ′ = ½Uy2(1 −a5)x, or Uy(1 −a4)x,r5ar4a(10)for a sphere or cylinder, and a diametral plane.Two equal spheres, intersecting at 120°, will requireψ′ = ½Uy2[x−a3+a4(a − 2x)+a3−a4(a + 2x)],a2r132r152r232r25(11)with a similar expression for cylinders; so that the plane x = 0 may be introduced as a boundary, cutting the surface at 60°. The motion of these cylinders across the line of centres is the equivalent of a line doublet along each axis.47. The extension of Green’s solution to a rotation of the ellipsoid was made by A. Clebsch, by taking a velocity functionφ = xyχ(1)for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface λ may be in rotation about Oz without disturbing the motion ifR = −[ 1/ (a2+ λ) + 1/ (b2+ λ) ] χ + 2 dx/dλ,1 / (b2+ λ) − 1 / (a2+ λ)(2)and that the continuity of the liquid is secured if(a2+ λ)3/2(b2+ λ)3/2(c2+ λ) ½dχ= constant,dλ(3)χ =∫∞λN dλ=N·Bλ− Aλ;(a2+ λ) (b2+ λ) Pabca2− b2(4)and at the surface λ = 0,R = −[ (1/a2+ 1/b2) · N/abc · (B0− A0)/(a2− b2) ] − N/abc · 1/a2b2,1/b2− 1/a2(5)N= R1/b2− 1/a2,abc1/a2b2− [ (1/a2+ 1/b2) · (B0− A0) / (a2− b2) ](6)= R(a2− b2)2/ (a2+ b2)(a2− b2) / (a2+ b2) − (B0− A0)The velocity function of the liquid inside the ellipsoid λ = 0 due to the same angular velocity will beφ1= Rxy (a2− b2) / (a2+ b2),(7)and on the surface outsideφ0= xyχ0= xyNB0− A0,abca2− b2(8)so that the ratio of the exterior and interior value of φ at the surface isφ0=B0− A0,φ1(a2− b2) / (a2+ b2) − (B0− A0)(9)and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid λ = 0.The extension to the case where the liquid is bounded externally by a fixed ellipsoid λ = λ1is made in a similar manner, by puttingφ = xy (χ + M),(10)and the ratio of the effective angular inertia in (9) is changed to(B0− A0) − (B1− A1) +a12− b12abca12+ b12a1b1c1.a2− b2−a12− b12abc− (B0− A0) + (B1− A1)a2+ b2a12+ b12a1b1c1(11)Make c = ∞ for confocal elliptic cylinders; and thenAλ =∫∞λab=ab(1 −√b2+ λ),(a2+ λ) √ (4a2+ λb2+ λ)a2− b2a2+ λ(12)Bλ =ab( √a2+ λ− 1), Cλ = 0;a2− b2b2+ λand then as above in § 31, witha = c ch α, b = c sh α, a1= √ (a2+ λ) = c ch α1, b1= c sh α1(13)the ratio in (11) agrees with § 31 (6).As before in § 31, the rotation may be resolved into a shear-pair, in planes perpendicular to Ox and Oy.A torsion of the ellipsoidal surface will give rise to a velocity function of the form φ = xyzΩ, where Ω can be expressed by the elliptic integrals Aλ, Bλ, Cλ, in a similar manner, sinceΩ = L∫∞λdλ / P3.48. The determination of the φ’s and χ’s is a kinematical problem, solved as yet only for a few cases, such as those discussed above.But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R. The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.Conversely, if the kinetic energy T is expressed as a quadratic function of x1, x2, x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding arex1=dT, x2=dT, x3=dT,dUdVdW(1)y1=dT, y2=dT, y3=dT;dPdQdRbut when it is expressed as a quadratic function of x1, x2, x3, y1, y2, y3,U =dT, V =dT, W =dT,dx1dx2dx3(2)P =dT, Q =dT, R =dT.dy1dy2dy3The second system of expression was chosen by Clebsch and adopted by Halphen in hisFonctions elliptiques; and thence the dynamical equations followX =dx1− x2dT+ x3dT, Y = ..., Z = ...,dtdy3dy2(3)L =dy1− y2dT+ y3dT− x2dT+ x3dT, M = ..., N = ...,dtdy3dy2dx3dx2
(3)
and a velocity function of the form
φ = xψ,
(4)
where ψ is a function of λ only, so that ψ is constant over an ellipsoid; and we seek to determine the motion set up, and the form of ψ which will satisfy the equation of continuity.
Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane,
(5)
p2= (a2+ λ) l2+ (b2+ λ) m2+ (c2+ λ) n2,= a2l2+ b2m2+ c2n2+ λ,
p2= (a2+ λ) l2+ (b2+ λ) m2+ (c2+ λ) n2,
= a2l2+ b2m2+ c2n2+ λ,
(7)
(8)
Thence
(9)
so that the velocity of the liquid may be resolved into a component -ψ parallel to Ox, and −2(a2+ λ)l dψ/dλ along the normal of the ellipsoid; and the liquid flows over an ellipsoid along a line of slope with respect to Ox, treated as the vertical.
Along the normal itself
(10)
so that over the surface of an ellipsoid where λ and ψ are constant, the normal velocity is the same as that of the ellipsoid itself, moving as a solid with velocity parallel to Ox
(11)
and so the boundary condition is satisfied; moreover, any ellipsoidal surface λ may be supposed moving as if rigid with the velocity in (11), without disturbing the liquid motion for the moment.
The continuity is secured if the liquid between two ellipsoids λ and λ1, moving with the velocity U and U1of equation (11), is squeezed out or sucked in across the plane x = 0 at a rate equal to the integral flow of the velocity ψ across the annular area α1− α of the two ellipsoids made by x = 0; or if
(12)
α = π√ (b2+ λ.c2+ λ).
(13)
Expressed as a differential relation, with the value of U from (11),
(14)
(15)
and integrating
(16)
so that we may put
(17)
P2= 4 (a2+ λ) (b2+ λ) (c2+ λ),
(18)
where M denotes a constant; so that ψ is an elliptic integral of the second kind.
The quiescent ellipsoidal surface, over which the motion is entirely tangential, is the one for which
(19)
and this is the infinite boundary ellipsoid if we make the upper limit λ1= ∞.
The velocity of the ellipsoid defined by λ = 0 is then
(20)
with the notation
(21)
so that in (4)
(22)
in (1) for an ellipsoid.
The impulse required to set up the motion in liquid of density ρ is the resultant of an impulsive pressure ρφ over the surface S of the ellipsoid, and is therefore
∫ ∫ ρφl dS = ρψ0∫ ∫ xl dS = ρψ0(volume of the ellipsoid) = ψ0W′,
(23)
where W′ denotes the weight of liquid displaced.
Denoting the effective inertia of the liquid parallel to Ox by αW′. the momentum
αW′U = ψ0W′
(24)
(25)
in this way the air drag was calculated by Green for an ellipsoidal pendulum.
Similarly, the inertia parallel to Oy and Oz is
(26)
(27)
and
A + B + C = abc / ½P, A0+ B0+ C0= 1.
(28)
For a sphere
a = b = c, A0= B0= C0=1⁄3, α = β = γ = ½,
(29)
so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration
(30)
Thus a spherical air bubble, in which W/W′ is insensible, will begin to rise in water with acceleration 2g.
45. When the liquid is bounded externally by the fixed ellipsoid λ = λ1, a slight extension will give the velocity function φ of the liquid in the interspace as the ellipsoid λ = 0 is passing with velocity U through the confocal position; φ must now take the form x(ψ + N), and will satisfy the conditions in the shape
(1)
and any confocal ellipsoid defined by λ, internal or external to λ = λ1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox
Since − Ux is the velocity function for the liquid W′ filling the ellipsoid λ = 0, and moving bodily with it, the effective inertia of the liquid in the interspace is
(2)
If the ellipsoid is of revolution, with b = c,
(3)
and the Stokes’ current function ψ can be written down
(4)
reducing, when the liquid extends to infinity and B1= 0, to
(5)
so that in the relative motion past the body, as when fixed in the current U parallel to xO,
(6)
Changing the origin from the centre to the focus of a prolate spheroid, then putting b2= pa, λ = λ′a, and proceeding to the limit where a = ∞, we find for a paraboloid of revolution
(7)
(8)
with λ′ = 0 over the surface of the paraboloid; and then
ψ′ = ½ U [ y2− p √ (x2+ y2) + px ];
(9)
ψ = −½ Up [ √ (x2+ y2) − x ];
(10)
φ = −½ Up log [ √ (x2+ y2) + x ].
(11)
The relative path of a liquid particle is along a stream line
ψ′ = ½ Uc2, a constant,
(12)
(13)
a C4; while the absolute path of a particle in space will be given by
(14)
y2− c2= a2e−x/p.
(15)
46. Between two concentric spheres, with
a2+ λ = r2, a2+ λ1= a12,
(1)
A = B = C = a3/ 3r3,
(2)
and the effective inertia of the liquid in the interspace is
(3)
When the spheres are not concentric, an expression for the effective inertia can be found by the method of images (W. M. Hicks,Phil. Trans., 1880).
The image of a source of strength μ at S outside a sphere of radius a is a source of strength μa/ƒ at H, where OS = ƒ, OH = a2/ƒ, and a line sink reaching from the image H to the centre O of line strength −μ/a; this combination will be found to produce no flow across the surface of the sphere.
Taking Ox along OS, the Stokes’ function at P for the source S is μ cos PSx, and of the source H and line sink OH is μ(a/ƒ) cos PHx and −(μ/a)(PO − PH); so that
(4)
and ψ = −μ, a constant, over the surface of the sphere, so that there is no flow across.
When the source S is inside the sphere and H outside, the line sink must extend from H to infinity in the image system; to realize physically the condition of zero flow across the sphere, an equal sink must be introduced at some other internal point S′.
When S and S′ lie on the same radius, taken along Ox, the Stokes’ function can be written down; and when S and S′ coalesce a doublet is produced, with a doublet image at H.
For a doublet at S, of moment m, the Stokes’ function is
(5)
and for its image at H the Stokes’ function is
(6)
so that for the combination
(7)
and this vanishes over the surface of the sphere.
There is ao Stokes’ function when the axis of the doublet at S does not pass through O; the image system will consist of an inclined doublet at H, making an equal angle with OS as the doublet S, and of a parallel negative line doublet, extending from H to O, of moment varying as the distance from O.
A distribution of sources and doublets over a moving surface will enable an expression to be obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it.
The method of electrical images will enable the stream function ψ′ to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle π/m, where m is an integer (R. A. Herman,Quart. Jour. of Math.xxii.).
Thus for m = 2, the spheres are orthogonal, and it can be verified that
(8)
where a1, a2, a = a1a2/√ (a12+ a22) is the radius of the spheres and their circle of intersection, and r1, r2, r the distances of a point from their centres.
The corresponding expression for two orthogonal cylinders will be
(8)
With a2= ∞, these reduce to
(10)
for a sphere or cylinder, and a diametral plane.
Two equal spheres, intersecting at 120°, will require
(11)
with a similar expression for cylinders; so that the plane x = 0 may be introduced as a boundary, cutting the surface at 60°. The motion of these cylinders across the line of centres is the equivalent of a line doublet along each axis.
47. The extension of Green’s solution to a rotation of the ellipsoid was made by A. Clebsch, by taking a velocity function
φ = xyχ
(1)
for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface λ may be in rotation about Oz without disturbing the motion if
(2)
and that the continuity of the liquid is secured if
(3)
(4)
and at the surface λ = 0,
(5)
(6)
The velocity function of the liquid inside the ellipsoid λ = 0 due to the same angular velocity will be
φ1= Rxy (a2− b2) / (a2+ b2),
(7)
and on the surface outside
(8)
so that the ratio of the exterior and interior value of φ at the surface is
(9)
and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid λ = 0.
The extension to the case where the liquid is bounded externally by a fixed ellipsoid λ = λ1is made in a similar manner, by putting
φ = xy (χ + M),
(10)
and the ratio of the effective angular inertia in (9) is changed to
(11)
Make c = ∞ for confocal elliptic cylinders; and then
(12)
and then as above in § 31, with
a = c ch α, b = c sh α, a1= √ (a2+ λ) = c ch α1, b1= c sh α1
(13)
the ratio in (11) agrees with § 31 (6).
As before in § 31, the rotation may be resolved into a shear-pair, in planes perpendicular to Ox and Oy.
A torsion of the ellipsoidal surface will give rise to a velocity function of the form φ = xyzΩ, where Ω can be expressed by the elliptic integrals Aλ, Bλ, Cλ, in a similar manner, since
Ω = L∫∞λdλ / P3.
48. The determination of the φ’s and χ’s is a kinematical problem, solved as yet only for a few cases, such as those discussed above.
But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R. The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.
Conversely, if the kinetic energy T is expressed as a quadratic function of x1, x2, x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.
These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.
Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are
(1)
but when it is expressed as a quadratic function of x1, x2, x3, y1, y2, y3,
(2)
The second system of expression was chosen by Clebsch and adopted by Halphen in hisFonctions elliptiques; and thence the dynamical equations follow
(3)