The Relative vis viva Equation

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The principle of vis viva (living forces) can be applied to the relative arbitrary motions in coordinate planes.

This can be done by adding to the given forces other forces that are opposed to those capable of forcing the material points to remain invariably linked to the moving planes to which the relative motions are referred.

The proposition which is its object cannot in general be applied to other equations of motion than those of the vis viva.

But I had not then examined whether:

  • the method could be used on certain equations of motion
  • in cases where it does not apply, one could give a simple expression for the new correction terms.

I answer this today by giving this general proposition: To establish any equation of relative motion for a system of bodies or for any machine, it is sufficient to add to the existing forces 2 kinds of supplementary forces.

  1. Those forces which must be taken into account for the vis viva equation. [centripetal]

These are forces opposed to those forces capable of maintaining the material points invariably linked to the moving planes.

  1. Those forces directed perpendicularly to [centrifugal]:
  • the relative speeds and
  • the axis of rotation of the moving planes

These are equal to twice the product of the angular velocity of the moving planes multiplied by the quantity of relative motion projected onto a plane perpendicular to this axis.

These latter forces have the greatest analogy with ordinary centrifugal forces.

The centrifugal force:

  • is equal to the quantity of motion multiplied by the angular velocity of the tangent to the curve described
  • is directed perpendicularly to the velocity and in the osculating plane — perpendicularly also to the axis of rotation of the tangent.

Thus, to pass from ordinary centrifugal forces to the second forces whose doubles enter into the preceding statement, it is only necessary to replace the angular velocity of the tangent by that of the moving planes, and to substitute for the direction of the axis of rotation of this tangent, the direction of the axis of rotation of these same moving planes.

In other words, it suffices to substitute for everything that relates to the magnitude and direction of the rotation of the tangent, that which relates to the moving planes, and to take the double of the forces thus obtained.

This is why I call them composite centrifugal forces.

They participate in:

  • the relative motion by the quantity of motion
  • the motion of the moving planes by the use of their axis of rotation and of their angular velocity.

An equation of non-vis viva relative motion needs the double of the composite centrifugal forces in addition to what is required for this equation.

The directions of these second supplementary forces being perpendicular to the relative velocities.

And so it follows immediately that they disappear in the equation of vis viva for relative motion, since this equation does not employ the components of the forces in the direction of the relative velocities.

The difference in the equations of relative motion is from this disappearance of these composite centrifugal forces.

Relative motions have forces with two supplementary terms:

  1. —Xₑ, —Yₑ, —Zₑ

These are forces opposed to those which would be capable of forcing the moving points to remain invariably linked to the moving coordinate planes.

  1. The others are expressed by
2(rmdydt−qmdzdt),2(pmdzdt−rmdxdt),2(qmdxdt−pmdydt). \begin{align*} 2 \left( rm \frac{dy}{dt} - qm \frac{dz}{dt} \right), \\ 2 \left( pm \frac{dz}{dt} - rm \frac{dx}{dt} \right), \\ 2 \left( qm \frac{dx}{dt} - pm \frac{dy}{dt} \right). \end{align*}

We remark that if x, y, z, x’, y’, z’, are the projections of two lengths r and r’; the parallelogram constructed on r and r’ and whose expression is rr’ sin(rr’), has for projections on the coordinate planes:

(xz′−zx′),(xy′−yz′),(yx′−xy′). \begin{align*} (xz' - zx'), \\ (xy' - yz'), \\ (yx' - xy'). \end{align*}

The above expressions can also be the projections on the coordinate axes of a length equal to rr’ sin(rr’), which will be carried perpendicularly to the plane of the two straight lines r and r’ and will be situated on the same side, with respect to the senses that go from r to r’, that the axis of z is with respect to the senses that go from y to x.

The above expressions in p, q, r, dx, dy, dz, will be the doubles of the components following the axes of a force directed perpendicularly to the plane of the axis of rotation and of the velocity are, 1° those which are opposed to the forces capable of producing on each point the motion it would have if it were linked to the moving planes, 2° the doubles of the composite centrifugal forces.

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