Theorem of Siacci - SpeedyLook encyclopedia

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The theorem of Siacci is a Théorème of Cinématique which applies to the plane curved parameterized by the distance to the origin and the distance from the origin to the Podaire.

It can serve usefully central Force in the case of. Isaac Newton used much the formulas of Siacci in this simple case.

Statement

The components of acceleration $\ vec {has}$ are according to radial direction OM and the tangent:

\ vec {U} \ cdot \ frac {C^2dp} {p^3 Dr.} + \ vec {T} \ cdot \ frac {1} {2p^2} \ frac {D (C^2)}{ds}

where R indicates the radius of curvature, S indicates the curvilinear X-coordinate and C = L/m is the kinetic moment per unit of mass, nonconstant a priori (but only C ² intervenes, as usual (invariance D-if chemistry!).

The first component is called formula of Newton. Like dp/dr = r/R, it notes also -1/2.C ² .d (1/p ²) /dr. Alors the Old ones noticed that the data of the trajectory in podaire p = F (R) solved the direct problem of the central forces: the trajectory (T) gives the central force.

Example: Newtonian central field

In the case of a central field, the theorem of Siacci gives the central component and C = cste. One often poses W= C/p.

Then in the elliptic case, with for center of force the hearth, it comes W ² /2 = 1/r -1/2a (obvious formula if one thinks A contrario of the theorem of the kinetic energy of Leibniz). Then this formula returned celebrates Newton, since it gave him, in the of Motu of 1684, the first demonstration of the direct problem of the Lois of Kepler: the force is in -1/r ², by direct application of the theorem of Siacci.

In the case of Hooke, has ² B ². W ² = has ² + B ² - R ², Newton obtained the force $\ vec F = - K \ cdot \ vec OM$

in the case of the circle, with the center of force on the circle: F = - k/r^5 (because p ~ R ²)

in the case of the spiral logarithmic curve, p = K R, therefore F ~ - k/r^3

in the case of the curves $r^n = b^n \ cdot \ cos N \ theta$ , $p = \ frac {r^ {n+1}} {b^n}$ ,

then: F = k/r^ (2n+3)

that is to say n=-2 (hyperbole: Hooke in Symmetry of Corinne): F = +k R

n=-1: line and null force

n= -1/2: parabola and hearth: F = 1/r ² (case of null energy)

n= 1/2: it is the reverse of the parabola, therefore it is the cardioîde: F = k/r^4

n= 1, already considering, it is the circle.

n= 2 it is the lemniscate: F= k/r^7

Almost all these cases are treated in Principia! It is clear that Newton, without saying it explicitly, had perfectly included/understood this theorem on the podaires.

Demonstration

That is to say (T) the trajectory of point running P, of close Q, FORMER point of dt (all the easy way is there!), of center of curve C, and OH the parallel with CP which cuts the " tangente" PQ out of H; then CP: = R, radius of curvature; and OH: = p, H describing the podaire (T) catch compared to the origin O. And QR the parallel with CP which cuts COp in O'. The not-constant of the surfaces is C = statement infinitesimal triangle PQO' is similar to PHO:

thus dp/r = dr/R !

Lemma: Kinematic recall of of Frenet:

if $\ vec {U}: = \ frac {\ vec {OM}} {OM}$ , then $\ dowry {\ vec {U}} = \ frac {(\ vec {R} \ wedge \ vec {v}) \ wedge \ vec {R}} {r^3}$ . end of lemma.

Acceleration according to Frenet is on the tangent dv/dt = 1/2. D (v ²) /ds; on normal v ² /R.

To break up the normal vector according to the radial one and the tangent: $\ vec {R} = - p \ vec {N} + \ frac {Dr.} {ds} \ vec {T}$

From where the radial component: (v ² /R). (r/p) = C ² .r/(p ³. R) (formula of Newton)

And the tangential component:

1/2.d (v ²) /ds + (v ² /R) (r/p) (dr/ds) = (1/2p ²). + v ² .2p.dp /ds = 1/2. D (C ²) /p ² ds.

CQFD

the preceding formula of Frenet (which was used) is often used also to show the existence of the Invariant of Runge Lenz: indeed, it immediately gives the circular Hodograph of the Keplerian Mouvement by integration, R \ v = Lo/m being constant.

Recall: for any movement on conical of parameter Po, |\ v has|. Po = C ³ /r^3 (cf Theorem of Newton-Hamilton). That gives the following demonstration: like |\ v has|= aC/r if acceleration is central, then has ~ 1/r ².

Newton and podaires

The Keplerian Mouvement is shown by Newton in 1684 in the deMotu. Then in corollary 1 of Proposals 11-13 of section III of book 1. October 11th, 1709, written Newton with Roger Dimensions, for the republication (1713) of Pricipia, some precise details: That is to say S sun, P the planet, V its speed. Then there exists only one conical having S like centers of central force in 1/r ².

This was necessary to counter the quibbles of Bernouilli, concerning the direct problem and the opposite problem. (Obviously, the case of the Spirale logarithmic curve of Newton posed problem! ).

Of course, if one multiplies acceleration by speed, one must find the theorem of Leibniz:

indeed 2 a.v = -2C ² /p^3 .dp/dt + 1/p ² .dC ² /dt = D (C ² /p2) /dt

It is included/understood why the couple W: = C/p and R went well: it expressed dand the central case the theorem of Leibniz, i.e. the conservation of energy.

See too

External bonds and documents

Newton and the podaires , Pourciau in arch. off Hist. sciences, 1996 (contains 22 references on this subject).
The sheer joy off celestial mechanics , Grossman, 1996, ED. Birkhauser, ISBN 3-7643-3832-6, p. 28-30.

Laplace, celestial mechanics, 1798-1825, ED Graduate.

Hérivel, background to Newton' S Principia, 1965; ED OUP.
Cushing, American Newspaper off physics, 50 (1982) 617-628.

Category: Mechanics Sciacci

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