Acceleration of Siacci

The components of the Accélération in various frames of reference are well-known.

In the case of curved plane, the polar coordinates are often used (R, $\ theta$ ).

Less known, is the system using the definition of the curve like antipodaire: That is to say O the origin and P the projection of O on the tangent in M with the curve (C): the point P then describes the Podaire of O to the curve (C): one calls p distance COp. Reciprocally the perpendicular with the podaire at the point P will wrap the curve (C) to study. A frame of reference little used is the couple (R, p). The traditional example is: that is to say a circle and a point O interior; the antipodaire is an ellipse of hearth O,

The acceleration of Siacci expresses the acceleration of a point moving, M, on (C) according to its distance to the point O (one poses OM: = R) and of the distance from O to the podaire (one poses COp: = p).

This acceleration is very useful in the case of a central Force.

A second part will give the components of this acceleration in the general case: less useful, it makes it possible nevertheless to the neophyte in mechanics to include/understand well the difference between projections and components, since the base used will be not-orthogonal.

Formulate of Siacci: case of a central Force

Newton is undoubtedly one of the first to have reference mark this formula. But the demonstration that it gives some is purely geometrical.

Here, the formula of Leibniz is used: dW = F (R) .dr = mv.dv, because it leads more simply to the result:

That is to say C the constant of the surfaces (= p.v); then:

$\ vec {\ gamma} = - \ vec {U} \ cdot \ frac {C^2dp} {p^3dr}$

The formula is then remarkable, because time does not intervene there explicitly any more (it is masked in C ²); if one knows the expression podaire trajectory R = F (p) or p = G (R), then the law of force F (R) is obtained directly!

Use

the traditional example is that of Isaac Newton (Nov. 1684): for an ellipse, the equation podaire seen of its hearth is: ² /p ² = 2a/r -1 has (cf (note))

By deriving this equation, one obtains - (² /p ³ has) dp= - (2a/r ²) Dr. From where the result requested by Edmund Halley (August 1684): F (R) is in 1/r ²!

Newton (1687) uses it many times, cf Exégèse of Principia (spiral logarithmic curves, Transmutation of the force,…).

Example: Spiral logarithmic curve traversed at angular velocity W constant. Then central acceleration remains proportional to 1/r ³, since p = r.sin $\ alpha$ = r.k (eadem mutata resurgo, dixit Jacques Bernoulli). It is proposal IX.
Example: ellipse of Robert Hooke: the equation podaire is ab/p ² = has ² + B ² + R ²; it results from it that F (R) = - m (C ² /a ² B ²) OM .c' is proposal X.
Exemple: O on the circle of diameter " vertical" OA has (0,2R). the podaire is immediate p = R ²/2R: from where F (R) = - m 8C ² R ² /r ⁵: proposal VII, corollary I.
various Examples: the curves r^k = a^k cos k $\ theta$ have for podaires:

p = r^ (k+1) /a^k, and thus F (R) = - C ² ac (2k) (k+1)/r^ (2k+3): that is to say cases:

k= 1, the circle and F ~ 1 r^5
k=2, the lemniscate: F ~ 1/r^7
k=-2, the hyperbole équilatère (Symmetry of Corinne of Hooke): F ~ + OM .
k= -1 (n+1) =0 thus for the line, F= 0
k=1/2, ardioid and its origin O: F ~ 1/r^4
k= -1/2: the parabola and O with the hearth (case of Briggs): F ~ 1/r^2.

(Note: in an ellipse there are two hearths, with r+r' = 2a and pp' = have ² and p'/r' =p/r).

Demonstration of the formula

Case of central Force:

Then C = cste. The elementary work of the Force is worth: F (R) .dr = m v.dv, but v ² = C ² /p ², therefore F (R) /m = - C ² /p ³. dp/dr

General formula of Siacci

It breaks up acceleration according to OM and V , which seems rather reasonable like basic choice, and yet, obviously the base of Frenet is largely easier to use!

$\ vec {\ gamma} = - \ vec {U} \ cdot \ frac {C^2dp} {p^3dr} + \ vec {V} \ frac {C \ dowry {C}} {C^2}$

To initially notice the homogeneous homogeneity, (1/C) dC/dt contrary to a time. To notice the Choice to write with C ² rather than with C, because one cannot distinguish between the plan seen from top or seen from lower part. In general, one takes C positive, BUT HERE, C (T) is variable! Moreover the change T in - T, does not change acceleration.

Remarkable also is the fact that the first term (radial) does not change between the formula central Force and the general formula. Admittedly, the second term contains dC/dt in factor.

Known lemma used: (R Dr.) = p. R with R: =rayon of curve. (cf notes)

Demonstration : one will be satisfied to identify the formula with that of Frenet.

The second term (1/C) dC/dt = (1/v) dv/dt + a term has = (1/p) dp/dt.

It thus remains to be shown that V . With + the radial term is normal and equal to v ² /R NR .

To write OM = COp + Pt = - p NR + ( r.T ) T = - p NR + ( r.V ) V /v ².

The projection of Frenet sur' is not written: + (C ² /rp ³) dp/dr .p= v ² .dp/ (r.dr) =v ² /R

The projection of Frenet on T is written: (v/p) dp/dt - (C ² /rp ³). (dp/dr). ( r.V) /v;

However ( r.V ) = (R Dr.) /dt, from where (v/p) dp/dt - (v ² /rp) dp/dt. (R Dr.) /v which is cancelled.

Note: in the second term one can make disappear time: T (1/p ²) dC/ds.

Note: here a demonstration: p = - r.N thus dp = - Dr. NR + r.dN = 0+ r.T d $\ alpha$ = ( r.T ) ds/R = R Dr. R. CQFD.

See too

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