Simple Pendulum

The simple pendulum is the model of heavy Pendule simplest: one considers a specific mass at the end of a rigid connection without mass length $l$ being able to turn in a vertical plan. The material point in G, of mass $m$, moves then on an arc of circle of ray OG: the effect of the weight constantly tending to bring back the pendulum towards its position of balance, that Ci oscillates as soon as it was isolated vertical then left with the only action of gravity.
Un real pendulum comparable to a simple pendulum consists of a mass of low dimension at the end of a wire. (See illustration below).

Equations of the movement

Setting in equation

One locates the position of the simple pendulum by the angle which it forms with the downward vertical. A positive orientation is chosen: the position of the mass is thus located by the angular elongation algebraic $\backslash \; theta\; \backslash ,$.
On note $\backslash \; overrightarrow\; \{G\}$ acceleration due to gravity (under our latitudes, $g\; \backslash \; simeq\; 9,81\; \backslash \; m.s^\; \{-\; 2\}$).

Analysis of the forces :

• the weight $\backslash \; overrightarrow\; P\; =\; m\; \backslash \; overrightarrow\; \{G\}$
• the tension $\backslash \; overrightarrow\; T$ of the stem, always perpendicular to the circular motion of G.

In this model the other forces are forgotten, in particular the forces of friction; however a pendulum arrète to oscillate itself under the action of frictions: the Perpetual motion does not exist on this scale of energy.

mechanical Energy of the pendulum :

• the sum of the kinetic energy of the pendulum and its potential energy of gravity, measured starting from the point low is worth (the speed of the mass being worth $v\; =\; L\; \backslash \; frac\; \{D\; \backslash \; theta\}\; \{D\; T\}$):

$E\; =\; E\_c+E\_p=\; \backslash \; frac\; \{1\}\; \{2\}\; m\; l^2\; \backslash \; dowry\; \{\backslash \; theta\}\; ^2+mgl\; (1\; \backslash \; cos\; \backslash \; theta)$ with $\backslash \; dowry\; \{\backslash \; theta\}\; =\; \backslash \; frac\; \{D\; \backslash \; theta\}\; \{D\; T\}$

• Since the tension of the stem is at any moment perpendicular to the circular motion of G, this force exerts a work no one. Moreover as the weight is a conservative Force and than any other force is neglected, the mechanical energy system is preserved. To say that this quantity is preserved during the movement, it is to say that its value is constant during time, or that its variation is null at any moment. This can be translated mathematically by writing that the derivative compared to time is null. One obtains then:

$\backslash \; ddot\; \{\backslash \; theta\}\; +\; \backslash \; omega\_0^2\; \backslash \; sin\; \backslash \; theta\; =\; 0$ with $\backslash \; omega\_0^2\; =\; \backslash \; frac\; \{G\}\; \{L\}$ and $\backslash \; ddot\; \{\backslash \; theta\}\; =\; \backslash \; frac\; \{D\; \{\backslash \; dowry\; \{\backslash \; theta\}\}\}\; \{D\; T\}$

• This equation can also be deduced from the Basic principle of Dynamics, by projecting the two forces $\backslash \; vec\; T$ and $\backslash \; vec\; P$ on the tangent with the movement.

Well of potential :

If one traces according to θ the graph of the potential energy $mgl\; (1\; \backslash \; cos\; \backslash \; theta)\; \backslash ,$, one obtains the following figure. One traced in gray the maximum potential energy level 2 mgl .

• If the mechanical energy E of the pendulum is at a level E ₁ lower than 2 mgl , the pendulum is confined in a Puits of potential. There exists a maximum elongation $\backslash \; theta\_0$ pendulum for which speed is cancelled, and the pendulum oscillates periodically. One has then:

$\backslash \; frac\; \{m\; l^2\; \backslash \; dowry\; \{\backslash \; theta\}\; ^2\}\; \{2\}\; -\; mgl\; \backslash \; cos\; \backslash \; theta\; =\; -\; mgl\; \backslash \; cos\; \backslash \; theta\_0$ who is simplified in:

$\backslash \; dowry\; \{\backslash \; theta^2\}\; +\; 2\; \backslash \; omega\_0^2\; (\backslash \; cos\; \backslash \; theta\_0\; -\; \backslash \; cos\; \backslash \; theta)\; =\; 0$

• If energy E of the pendulum is at a level E ₂ higher than 2 mgl , then the pendulum crosses the barriers of potential, its angular velocity cannot be cancelled and the pendulum turns around the point O.

Resolution

The resolution of the equations of the movement of the simple pendulum is not easy. The cycloidal Pendule of Huygens represents a movement in a Puits of potential easier to solve. The simple Pendule discrete proposes an approach step by step resolution.

1 for small oscillations , one can confuse $\backslash \; sin\; (\backslash \; theta)$ with $\backslash \; theta$. The equation then is obtained:

$\backslash \; ddot\; \{\backslash \; theta\}\; +\; \backslash \; omega\_0^2\; \backslash \; theta\; =\; 0$ with, let us recall it, $\backslash \; omega\_0^2\; =\; \backslash \; frac\; \{G\}\; \{L\}$

whose solution is:

$\backslash \; theta\; (T)\; =\; \backslash \; theta\_0\; \backslash \; cos\; (\backslash \; omega\_0\; T)\; \backslash ,$; from period $T\_0\; =\; \backslash \; frac\; \{2\; \backslash \; pi\}\; \{\backslash \; omega\_0\}\; =\; 2\; \backslash \; pi\; \backslash \; sqrt\; \backslash \; frac\; \{L\}\; \{G\}$.

2 for greater amplitudes , one can use for the period:

• the formula of Bordered : $T\; (\backslash \; theta\_0)\; =\; T\_0\; (1\; +\; \backslash \; frac\; \{\backslash \; theta\_0^2\}\; \{16\})$
• the exact formula : $T\; (\backslash \; theta\_0)\; =\; T\_0\; \{2K\; (\backslash \; sin\; \backslash \; frac\; \{\backslash \; theta\_0\}\; \{2\})\; \backslash \; over\; \backslash \; pi\}$, which uses a elliptic Fonction of Jacobi.

In addition, the periodic oscillation becomes definitely anharmonic , as the harmonic content shows it.

3 for a mechanical energy higher than 2 mgl , the pendule' tournoie' in a periodic way. At high speed V , this period T tends towards $2\; \backslash \; pi\; L\; \backslash \; over\; V$.

Tension of the stem

A physical quantity depends on the mass of the pendulum: the tension of the stem (for its measurement, one can stick on the bar a calibrated strain gauge).

Projection on the normal ($\backslash \; vec\; N$) of the Basic principle of Dynamics makes it possible to obtain the relation:

$m\; a\_\; \backslash \; vec\; NR\; =\; T\; +\; P\_\; \backslash \; vec\; NR$

However the expression of the radial acceleration in polar coordinates with a distance to the constant origin (constant ray) is $l\; \{\backslash \; dowry\; \backslash \; theta^2\}$ from where $T\; =\; Mg\; \backslash \; cos\; \backslash \; theta\; +\; ml\; \{\backslash \; dowry\; \backslash \; theta^2\}$ where

and we saw that

$ml\; \{\backslash \; dowry\; \backslash \; theta^2\}\; =\; 2mg\; (\backslash \; cos\; \backslash \; theta\; -\; \backslash \; cos\; \backslash \; theta\_0)$, from where

$T\; =\; Mg\; (3\; \backslash \; cos\; \backslash \; theta\; -\; 2\; \backslash \; cos\; \backslash \; theta\_0)\; \backslash ,$

T varies between $Mg\; \backslash \; cos\; \backslash \; theta\_0\; \backslash ,$ and $mg\; (3-2\; \backslash \; cos\; \backslash \; theta\_0)\; \backslash ,$. For example, for 90°, T varies between 0 and 3 Mg . If one replaces the stem by a wire, it is necessary to envisage a wire resistant to 3kg for a mass of 1 kg, if not the wire breaks and the mass leaves then in parabolic trajectory. The experiment is easy to show and rather spectacular but it is necessary to find the wire which is not stretched too much before breaking. An easy description of the increase in the tension T is to use an elastic wire. But it is not any more about the same problem and it is not any more elementary (cf Botafumeiro).

To buckle the loop

T is cancelled for certain initial conditions of launching different from that proposed above, and even becomes negative, the stem supporting the mass then. It is traditional to show that, launched point low with an energy 2 mgl , the mass will arrive at the end of an infinite time at the top of the circle (and the case is integrable easily). One suspects that if the stem is replaced by a wire (unilateral connection), the trajectory will not be: gone up at the top, then falls with the vertical; there will be unhooking when T is null, i.e. for $\backslash \; theta$ such as $\backslash \; cos\; (\backslash \; theta)\; =\; -\; \{2\; \backslash \; over\; 3\}$, which corresponds to an angle of amplitude 132° and a height H = L + 2/3 L . The experiment is easy to make with a pendulum whose mass is a perforated part, slipping initially on a rigid half-circle, then finding " in the air" attached to its wire for the second part of the movement (or obviously with the strain gauge!).

Whereas for a stem, it is enough that energy $E$ exceeds $2mgl$ so that the pendulum starts to turn (loop the loop), in the case of a wire one needs an initial kinetic energy higher than $\{5\; \backslash \; over\; 2\}\; mgl$ so that the wire remains tended.

Great amplitudes and not linearity

Non-linearity gradually is introduced:

• initially by considering the second term of the development of the sine.
• then by treating the general case, which requires the use of the elliptic functions of Jacobi K, Sn, Cn, DNN.

Formulate Bordered

One thus considers the differential equation approximate, known as of Duffing, obtained by replacing $\backslash \; sin\; \backslash \; theta$ by $\backslash \; theta\; -\; \{\backslash \; theta^3\; \backslash \; over\; 6\}$:

$\backslash \; ddot\; \{\backslash \; theta\}\; +\; \backslash \; frac\; \{G\}\; \{L\}\; \backslash \; theta\; -\; \backslash \; frac\; \{G\}\; \{6l\}\; \backslash \; theta^3\; =\; 0$

It is shown whereas the period depends on the amplitude. The formula of Bordered gives:

$T\; =\; 2\; \backslash \; pi\; \backslash \; sqrt\; \backslash \; frac\; \{L\}\; \{G\}\; (1+\; \{\backslash \; theta\_0^2\; \backslash \; over\; 16\})$

The neglected term which follows is $\backslash \; frac\; \{11\; \backslash \; theta\_0^4\}\; \{3072\}\; +\; O\; (\backslash \; theta\_0^6)$. This formula is enough until π/2, to 3% of precision (1 + 10/64 = 1.156 instead of 1.18). There are several demonstrations:

• the method of the disturbances of Lindstedt-Poincaré consists in modifying the solution $\backslash \; theta\; =\; \backslash \; theta\_0\; \backslash \; sin\; (\backslash \; omega\_0t)\; \backslash ,$ by adding to him a small disturbance $\backslash \; theta\_1\; \backslash ,$ while also modifying the pulsation of the movement in $\backslash \; omega\_0+\; \backslash \; omega\_1\; \backslash ,$. One seeks the value to give to $\backslash \; omega\_1\; \backslash ,$ so that the differential equation, simplified while being limited to the first order disturbances, gives a solution $\backslash \; theta\_1\; \backslash ,$ limited. This value of $\backslash \; omega\_1\; \backslash ,$ is $-\; \backslash \; omega\_0\; \{\backslash \; theta\_0^2\; \backslash \; over\; 16\}$ so that the pulsation selected is $\backslash \; Omega\; =\; \backslash \; omega\_0\; (1\; -\; \{\backslash \; theta\_0^2\; \backslash \; over\; 16\})$. T being proportional contrary to ω, the formula of Bordered results from this.

• If the quasi-sinusoidal oscillation is supposed, the average stiffness being weaker, one physically expects a reduction in the pulsation. By using the formula of Euler $\backslash \; sin^3x\; =\; \{3\; \backslash \; over\; 4\}\; \backslash \; sin\; X\; -\; \backslash \; over\; 4\}\; \backslash \; sin\; (3x)\; \_\; \backslash \; mathrm\; \{omitted\}$, and while seeking $\backslash \; theta=\; \backslash \; theta\_0\; \backslash \; sin\; (\backslash \; Omega\; T)\; \backslash ,$, it comes $-\; \backslash \; omega^2\; +\; \backslash \; omega\_0^2\; -\; \backslash \; omega\_0^2\; \backslash \; frac\; \{\backslash \; theta\_0^2\}\; \{6\}\; \backslash \; frac\; \{3\}\; \{4\}\; =0$, from where $\backslash \; omega^2\; =\; \backslash \; omega\_0^2\; (1\; -\; \{\backslash \; theta\_0^2\; \backslash \; over\; 8\})$ and $\backslash \; Omega\; =\; \backslash \; omega\_0\; (1\; -\; \{\backslash \; theta\_0^2\; \backslash \; over\; 16\})$.

• One can prefer the following demonstration known as of the virial: ~

from where by the formula of Wallis: $\backslash \; theta\_0^2\; \backslash \; omega^2\; \backslash \; times\; \backslash \; frac\; \{1\}\; \{2\}\; =\; \backslash \; omega\_0^2\; (\backslash \; theta\_0^2\; \backslash \; times\; \backslash \; frac\; \{1\}\; \{2\}\; -\; \backslash \; frac\; \{\backslash \; theta\_0^4\}\; \{6\}\; \backslash \; times\; \backslash \; frac\; \{1\; \backslash \; times\; 3\}\; \{2\; \backslash \; times\; 4\})$, that is to say $\backslash \; omega^2\; =\; \backslash \; omega\_0^2\; (1\; -\; \backslash \; frac\; \{\backslash \; theta\_0^2\}\; \{8\})$.

• the equation of the movement of the pendulum is completely integrable thanks to the elliptic functions, which is the subject of the paragraph which follows. It is then enough to carry out a development with the desired order of the exact solution.

Fully non-linear case

The fully non-linear case is considered. Let us write the mechanical conservation of energy

$E\; =\; \backslash \; frac\; \{1\}\; \{2\}\; m\; l^2\; \backslash \; dowry\; \backslash \; theta^2\; +\; mgl\; (1\; \backslash \; cos\; \backslash \; theta)$

in the form: $l^2\; \backslash \; dowry\; \{\backslash \; theta^2\}\; +\; 2gh\; =\; 2gH$, with $h\; =\; L\; (1\; \backslash \; cos\; \backslash \; theta)\; =\; 2l\; \backslash \; sin^2\; \backslash \; frac\; \{\backslash \; theta\}\; \{2\}$

Let us pose $H\; =\; 2lk^2$. There exist three cases:

• K < 1, the pendulum oscillates : H varies between 0 and $H\; =\; L\; (1\; \backslash \; cos\; \backslash \; theta\_0)$. One a:

$\backslash \; dowry\; \backslash \; theta^2\; =\; 2\; \{G\; \backslash \; over\; L\}\; (\backslash \; cos\; \backslash \; theta\; -\; \backslash \; cos\; \backslash \; theta\_0)$ Between $0$ and $\backslash \; theta\_0$, there is $\backslash \; dowry\; \backslash \; theta\; =\; \backslash \; sqrt\; \{2\; \{G\; \backslash \; over\; L\}\; (\backslash \; cos\; \backslash \; theta\; -\; \backslash \; cos\; \backslash \; theta\_0)\}$. A small elementary angle $d\; \backslash \; theta$ is traversed during an elementary time interval $dt\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{2g\}\}\; \{\backslash \; frac\; \{D\; \backslash \; theta\}\; \{\backslash \; sqrt\; \{\backslash \; cos\; \backslash \; theta\; -\; \backslash \; cos\; \backslash \; theta\_0\}\}\}$. The total period of the oscillations is thus $T\; =\; 4\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{2g\}\}\; \backslash \; int\_0^\; \{\backslash \; theta\_0\}\; \{\backslash \; frac\; \{D\; \backslash \; theta\}\; \{\backslash \; sqrt\; \{\backslash \; cos\; \backslash \; theta\; -\; \backslash \; cos\; \backslash \; theta\_0\}\}\}$, and it is shown that: $T=\; 4\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{G\}\}\; K\; (K)\; =\; T\_0\; \{2K\; (K)\; \backslash \; over\; \backslash \; pi\}$ with $k\; =\; \backslash \; sin\; \backslash \; frac\; \{\backslash \; theta\_0\}\; \{2\}$ $\backslash \; dowry\; \{\backslash \; theta\}\; =\; 2\; K\; \backslash \; omega\_0\; \backslash ,\; \backslash \; mathrm\; \{Cn\}\; (\backslash \; omega\_0t,\; K)$ $h\; =\; H\; \backslash ,\; \backslash \; mathrm\; \{Sn\}\; ^2\; (\backslash \; omega\_0t,\; K)$ where K, Sn and Cn are elliptic functions of Jacobi, K being tabulée below. The function K also admits the following development: $K\; (K)\; =\; \{\backslash \; pi\; \backslash \; over\; 2\}\; (1\; +\; \{k^2\; \backslash \; over\; 4\}\; +\; \{9k^4\; \backslash \; over\; 64\}\; +\; \backslash \; cdots\; +\; \backslash \; frac\; \{16^n\}\; k^\; \{2n\}\; +\; \backslash \; cdots)$, where $\{2n\; \backslash \; choose\; N\}$ is a binomial Coefficient. By replacing K by $\backslash \; sin\; \{\backslash \; theta\_0\; \backslash \; over\; 2\}$ and while being limited to the first two terms, one finds the formula of Bordered.

• K = 1, borderline case correspondent with $\backslash \; theta\_0=\; \backslash \; pi$. One a:

Infinite time to go up to the vertical $\backslash \; theta\; =\; 4\; \backslash \; arctan\; (e^\; \{\backslash \; omega\_0t\})\; -\; \backslash \; pi$ $\backslash \; dowry\; \{\backslash \; theta\}\; =\; \backslash \; frac\; \{2\; \backslash \; omega\_0\}\; \{\backslash \; mathrm\; \{CH\}\; (\backslash \; omega\_0t)\}$ $h\; =\; 2l\; \backslash ,\; \backslash \; mathrm\; \{HT\}\; ^2\; (\backslash \; omega\_0t)$ where CH and HT are respectively the cosine and the hyperbolic tangent.

• K > 1, the pendulum whirls : $v^2$ varies between $2g\; (H-2l)$ and $2gH$. The period to carry out a turn is $T\_0\; \{1\; \backslash \; over\; \backslash \; pi\; K\}\; K\; (\{1\; \backslash \; over\; K\})$. If H is very large, taking into account the fact that K (0) is worth π/2, one will be able to check that the period tends towards $\{2\; \backslash \; pi\; L\}\; \backslash \; over\; V$.

It is sometimes judicious to take for period time put to make two turns. Indeed, for K slightly lower than 1, the pendulum carries out a trajectory length close to 4π. With this convention, there is then $T\; =\; 2T\_0\; \{1\; \backslash \; over\; \backslash \; pi\; K\}\; K\; (\{1\; \backslash \; over\; K\})$ (Voir Chenciner (Pendulum with Gazette).

One also has: $\backslash \; dowry\; \{\backslash \; theta\}\; =\; 2k\; \backslash \; omega\_0\; \backslash ,\; \backslash \; mathrm\; \{DNN\}\; (K\; \backslash \; omega\_0t,\; 1/k)$ $h\; =\; 2l\; \backslash ,\; \backslash \; mathrm\; \{Sn\}\; ^2\; (K\; \backslash \; omega\_0t,\; 1/k)$ where Sn and DNN are elliptic functions of Jacobi.

Plan of phase

One calls orbit of phase the representation parameterized in time of the couple ($\backslash \; theta\; (T)$, $\backslash \; dowry\; \{\backslash \; theta\}\; (T)$), or of monotonous functions of those. In the graph below, $\backslash \; theta$ is in X-coordinate and $\backslash \; dowry\; \backslash \; theta$ in ordinate. One distinguishes:

• the area known as of oscillation (in black), known as of eye of Horus or of almond eye. Each orbit is traversed in the opposite direction with the trigonometrical direction and turns around the points of steady balance S, corresponds to the values 0,2π, 4π, etc of $\backslash \; theta\_0$.
• two areas of revolution (in red), either positive (in top), or negative (in bottom), corresponding if the pendulum turns around the point O.
• the separating one, in blue, corresponding to the borderline case where $\backslash \; theta\_0$ is worth π.
• points of steady balance S already evoked.
• points of unstable balance I corresponding to the values π, 3π, etc of $\backslash \; theta\_0$. It takes an infinite time to traverse an orbit which goes from an item I to another.

to also see for an animation.

It appears clearly henceforth which if one establishes an unspecified mechanism which can withdraw or add a small energy to the pendulum in the vicinity of the elongation π, one will have a phenomenon difficult to envisage even if it is deterministic: example, to place a very small pendulum fixed on the mass m: there is thus a double pendulum; the non-linear oscillations of this pendulum, ballasted of such a tiny pendulum, leave pantois when they are recorded: Poincaré was, with Liapunov, one of the first to consider this kind of problem; then Birkhoff; then the Russian school pulled by the high figure of Kolmogorov, and then that of Bogoliubov and Krylov, then Arnold,… until the moment when an article of 1971 of Lane and Takens came to suggest that the situation was normal as soon as the space of the phases was with three dimensions or more uses sometimes expression 1.5 degree of freedom.

Detailed study in the vicinity of the separating one

One is interested in the spectrum the speed just above and energy level of the separating one. On this separating, the spectrum is described as soliton mode.

Recall: the separating one and the soliton mode

In the case of separating, the first order equation is written:

$\backslash \; dowry\; \{\backslash \; theta^2\}\; =\; 2\; \backslash \; omega\_0^2\; (1+\; \backslash \; cos\; \backslash \; theta)\; =\; 4\; \backslash \; omega\_0^2\; \backslash \; sin^2\; \{\backslash \; theta\; \backslash \; over\; 2\}$ with $\backslash \; theta\; (0)\; =0$ and $\backslash \; dowry\; \{\backslash \; theta\}\; (0)\; =2\; \backslash \; omega\_0$

The solution " soliton" is characterized by the following equations: $\backslash \; theta\; =\; 4\; \backslash \; arctan\; (e^\; \{\backslash \; omega\_0t\})\; -\; \backslash \; pi$ $h\; =\; 2l\; \backslash ,\; \backslash \; mathrm\; \{HT\}\; ^2\; (\backslash \; omega\_0t)$ $\backslash \; dowry\; \{\backslash \; theta\}\; =\; \backslash \; frac\; \{2\; \backslash \; omega\_0\}\; \{\backslash \; mathrm\; \{CH\}\; (\backslash \; omega\_0t)\}$ $v^2\; =\; \{4gl\; \backslash \; over\; \backslash \; mathrm\; \{CH\}\; ^2\; (\backslash \; omega\_0t)\}$

Long oscillations: 1 - k² << 1

If the energy of the pendulum is very slightly lower than 2gH, the difference with the soliton mode is negligible. The value speed is imperceptibly the same one and the movement is thus quasi-identical, EXCEPT for the moment when it will be cancelled. the period is finished east is worth: $T\_0\; \{-\; \backslash \; ln\; (1-k^2)\; +\; \backslash \; ln\; (16))\backslash \; over\; \backslash \; pi\}$, value obtained by using the approximate value $K\; (K)\; =\; \backslash \; ln\; (2\; \backslash \; sqrt\; (2))\; -\; \{1\; \backslash \; over\; 2\}\; \backslash \; ln\; (1-k)$ in the vicinity of K = 1.

Long whirlings: k² - 1 << 1

In the same way, if energy is very slightly higher than 2gH, the movement is quasi-identical (soliton mode), Except that speed is never cancelled and that the elongation becomes monotonous function in quasi-staircase of steps height 2π in form the sigmoid ones (of the " kinks" in English), long of a very large but finished period : $T\_0\; \{-\; \backslash \; ln\; (k^2-1)\; +\; \backslash \; ln16\; \backslash \; over\; 2\; \backslash \; pi\}$. To notice the appearance of one 2 with the denominator, which is an artefact due to the fact that in a case, one calculates the period on an outward journey and return (either 4π approximately), whereas a whirling is carried out on 2π. It is one of the reasons to examine the " pendulum with gazette" carefully.

Anharmonicity

It is thus found that $v^2\; \backslash ,$ or $h\; \backslash ,$ are thus well the same functions of period $4\; \backslash \; ln2\; -\; \backslash \; ln|1-k^2|\backslash over\; \backslash omega\_0$. Below, pace of $v^2\; \backslash ,$ in the vicinity of $k\; =\; 1\; \backslash ,$. The pace of the graph is the same one, whether $k\; \backslash ,$ is slightly higher than 1 or slightly inferior.

One characterizes the rate of anharmonicity by the extent of the spectrum (discrete since the function is periodic). With the limit:

• H = 2l + nothing, |v| is $2l\; \backslash \; omega\_0\; \backslash \; times\; \backslash \; mathrm\; \{PeigneDeDirac\}\; (t/T)$

However, the spectrum of a comb of Dirac is a comb of Dirac (theorem of Poisson)

The simple pendulum is the most elementary example which shows:

• with low amplitude: linearization and thus the monochromatism
• with critical amplitude: all the harmonics are present with same amplitude .

In experiments, one launches a Pendule of Mach in whirling: weak frictions will make forward from one mode to another. The projection of the ball on the bearing axis $h\; \backslash ,$, it, will not express a transition: there is continuity of the phenomenon.

Thorough study of the spectrum

The development in Fourier series of the functions of Jacobi Sn, Cn and DNN are known. One from of deduced a development in Fourier series the angular velocity.

Case K > 1 : that is to say NR = T/To, with T the period to carry out two turns, corresponding to a rotation of 4π. NR is worth $\{2K\; (1/k)\; \backslash \; over\; K\; \backslash \; pi\}$. We will take as fundamental pulsation of the movement $\backslash \; Omega\; =\; \{\backslash \; omega\_0\; \backslash \; over\; NR\}$. One a:

$\backslash \; dowry\; \{\backslash \; theta\}\; =\; \{2\; \backslash \; omega\_0\; \backslash \; over\; NR\}\; (1\; +\; a\_2\; \backslash \; cos\; 2\; \backslash \; Omega\; T\; +\; a\_4\; \backslash \; cos\; 4\; \backslash \; Omega\; T\; +\; \backslash \; cdots)$ with $a\_\; \{2n\}\; =\; 4\; \backslash \; frac\; \{q^n\}\; \{1+q^\; \{2n\}\}$, and $Q\; =\; \backslash \; exp\; (-\; \backslash \; pi\; \{K\; (\backslash \; sqrt\; \{1\; -\; 1/k^2\})\; \backslash \; over\; K\; (1/k)\})$. If K is very large, the movement is a rotation movement around O at very high speed. Q is very small, and the movement is carried out almost according to the law $\backslash \; dowry\; \{\backslash \; theta\}\; =\; \{2\; \backslash \; omega\_0\; \backslash \; over\; NR\}\; =\; \{4\; \backslash \; pi\; \backslash \; over\; T\}$. When K decreases, Q increases, so that the has _{2 N} take importance. When K is very slightly higher than 1, NR is very large, $q\; \backslash \; simeq\; e^\; \{\backslash \; frac\; \{-\; \backslash \; pi\}\; \{NR\}\}$ and is very close to 1. The spectrum is very wide, since, for N ~ NR, has _N is worth approximately 0,17 more.

Below, frequency spectrums, by decreasing values of K , since a great value up to a value slightly higher than 1. In X-coordinate, one carried the even indices 2 N and in ordinates the values of has _{2 N} (one took has ₀ = 2):

Case K < 1 : the value of NR is this time $\{2K\; (K)\; \backslash \; over\; \backslash \; pi\}$. The pulsation of the movement is always $\backslash \; Omega\; =\; \{\backslash \; omega\_0\; \backslash \; over\; NR\}$. One a:

$\backslash \; dowry\; \{\backslash \; theta\}\; =\; \{2\; \backslash \; omega\_0\; \backslash \; over\; NR\}\; (a\_1\; \backslash \; cos\; \backslash \; Omega\; T\; +\; a\_3\; \backslash \; cos\; 3\; \backslash \; Omega\; T\; +\; \backslash \; cdots)$ with $a\_\; \{2n+1\}\; =\; 4\; \backslash \; frac\; \{q^\; \{n+1/2\}\}\; \{1+q^\; \{2n+1\}\}$, and $Q\; =\; \backslash \; exp\; (-\; \backslash \; pi\; \{K\; (\backslash \; sqrt\; \{1\; -\; k^2\})\; \backslash \; over\; K\; (K)\})$. The situation where K is very slightly lower than 1 is comparable with that where K is very slightly higher than 1. When K decreases, Q decrease, and when K is close to 0, the dominating pulsation is that which corresponds to ω.

Below, frequency spectrums, by decreasing values of K , since a value slightly lower than 1 up to a very small value. In X-coordinate, one carried the odd indices 2 N +1 and in ordinates the values of has _{2 N +1}:

Here also the chart of $\backslash \; dowry\; \backslash \; theta$ and the representation of the sums partial of Fourier corresponding, on the one hand for K lower than 1, on the other hand for K higher than 1:

Deadened simple pendulum

• elementary level : in small oscillations, the problem was already studied; it is simple if the mode is of Stokes, or if damping is of solid friction type.

• elevated level : if one takes into account the resistance of the air which, at the speeds concerned, is not in mode of Stokes (in - Kv), but in mode of strong Reynolds number (in - kv^2.sgnv), how to trace the separating ones? how to find how much turns makes the pendulum before oscillating.

And then how to study what was by_passé by Galileo, as indicated seriously today previously?

• Number of revolutions :

it is that this problem is analytically soluble:

If ,

the pendulum will turn N turns before oscillating.

This indication is enough to trace a draft of portrait of rather correct phase.

• the air :

The fact is that the pressure of the air plays a part: a few seconds per day for a clock! And there exists a minimum of the period according to the pressure!

That really any more does not have of importance today, because the pendulums are systematically readjusted on transmitter GPS, and later perhaps on the transmitter of the system Galileo.

History of sciences

Analysis of Evangelista Torricelli

In the case of small oscillations, Torricelli is certainly one of the first to obtain a measurement of the coefficient $2\; \backslash \; pi$ on the basis of considerations on the slowed down fall. (Cf Freefall).

One can to consider the movement of the pendulum of amplitude 3$\backslash \; theta\_0\; \backslash ,$, to approximate it by a fall on a tilted level of 2.$\backslash \; theta\_0\; \backslash ,$, length $BC=\; 2l\; .sin\; \backslash \; theta\_0$, followed by a horizontal trajectory of C in is, length BC/2.

There will be thus the quarter of the trajectory. The period T in this basin BCAC' B' is:

• $T\; =\; 4\; (2\; +\; 1/2)\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{G\}\}\; \backslash \; sqrt\; \{\backslash \; frac\; \{sin\; \backslash \; theta\_0\}\; \{sin\; 2\; \backslash \; theta\_0\}\}\; \backslash ,$

maybe by approximation, $T\; =\; 2.\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{G\}\}\; \backslash \; frac\; \{5\}\; \{\backslash \; sqrt2\}\; \backslash ,$soit an approximation of $\backslash \; pi$:

$\backslash \; pi\; \backslash \; approx\; \backslash \; frac\; \{5\}\; \{\backslash \; sqrt2\}\; =\; 3,53\; \backslash ,$

Another approximation gives $2+\; \backslash \; sqrt2\; =\; 3,414\; \backslash ,$

But better still, Torricelli notices rightly that

$\backslash \; frac\; \{1\}\; \{2\}\; m\; v^2\; +\; Mg\; H\; =\; cste\; \backslash ,$, with $H\; \backslash \; approx\; \backslash \; frac\; \{s^2\}\; \{2l\}\; \backslash ,$, is

$v^2\; +\; (g/l)\; s^2\; =\; cste\; \backslash ,$

It is enough for him to check that the function sine satisfies the equation and it has the result. As a good pupil of Cavalieri, is it able to make this reasoning before 1647? The mysterious cassette having disappeared with its death, one will undoubtedly know never anything ultimate work of Torricelli (1608-1647): raise of Castelli and Galileo, it lived one time when one did not joke with the Enquiry in Italy: abjuration of Galileo, on June 22nd, 1633.

In any case, its disciple (via Mersenne), Huygens, finds the value of $2\; \backslash \; pi$ before 1659, and shows that the curve such as $h\; =\; s^2/2l$ exactly is the Cycloïde. Let us recall that Dettonville publishes its Treaty of the Caster in January 1659].

Note: these terms are anachronistic: G does not exist yet, because there will be units only late in the century but one compares with the time of freefall the H=l height: this famous report/ratio: 1.11 (~ $\backslash \; pi$/(2sqrt2)), which intrigued Mersenne.

Isochronicity

(according to Koyré, studies galiléennes)

It could appear surprising then, that Galileo and his pupils did not see this phenomenon, whereas 4K becomes INFINITE when the pendular amplitudes tend towards 180°.Or, Galileo affirmed that the oscillations of the pendulum were isochronous (see heavy Pendule). It is thus there about an experimental blindness, which is worth the sorrow to be put forward.

1. With the discharge of Galileo, one can notice that it operated probably with wire (unilateral connection), therefore launching without initial speed (falls " free ralentie") was carried out with an amplitude lower than 90°: one will be able to test oneself, gràce with the simulation presented in heavy Pendule: , to recall without cheating (i.e. without looking at the tabulées values) the values of 4K. It is true that to 18% close 4K is constant under these conditions: Galileo thus could be let deceive .

is true also that it could while launching the pendulum by bottom, to try to go until 138°. Did it do it? Would Torricelli have tried the experiment? In any case, the reported experiment, of the nail to the vertical of the point of suspension (see [[principle of Torricelli]) indicates that they had the elements to make it and that probably they did it. But to measure 1/4 of period just was it reasonable; and how this movement " violent" (they are the terms of the time) could have been attached to a slowed down fall? Therefore, it is more advisable to draw aside this argument ].

2. With the load of Galileo, Koyré points out that it is not very probable: if one has several identical pendulums, one notes not-isochronism immediately: dephasing is very visible at the end of 10 oscillations, but he claims to have observed the oscillations on greater numbers. BUT, it had a thesis to defend : isochronism. More probably, as a good lawyer, it cheating (one knows, in addition, that Galileo has " triché" same manner on other occasions: deviation towards the East; tides; answers to Kepler; …) :

• the fact fascinating to defend is isochronism.
• the other made attractive is the not-dependence in mass.

3/.Compte-held of the resistance of the air and the real problem of the pseudo-period of the damped oscillations,

Taking into account the fact that this same problem of the resistance of the air had to be isolated with the freefall,

Taking into account the fact that with 90°, a pendulum with cork ball and a pendulum with steel ball do not behave same manner (it is immediately visible, as in the freefall, notwithstanding the push of Archimedes) it was necessary to defend the second thesis (just, it!),

it is probable that " this cheating was honnête" , within the meaning of the 17th century: it was carried to the account of the resistance of the air.

Galileo should not have let himself misuse; it had to decide, as a good lawyer, to plead what it wrote.

4/.Comme all the opinions in epistemology, they are opinions ; and one can only calculate: the quoted text of Galileo in the " dialogo" is thus to take with precaution (cf heavy Pendule), as well as the conclusion which of it is drawn. A proof is the letter of Mersenne to the Huygens young person: after having known as great wonder of Torricelli, the question is put: what is it of this factor K (K) /K (0) (known as in modern notations)? reference Web the preceding one (of the {{S|XXI|E}}) announces in letters violets that there does not exist explicit formula for K (K), which surprises obviously, since it IS the complete elliptic integral of first species of Legendre, and that it is perfectly tabulée just like the table of the sines! ''. One imagines, in 1645, the Huygens young person in catch with this problem arising just after the death of the Master (1642), (respect obliges), by its Torricelli pupil. Apparently this factor 1.18 posed problem (ref. to him: Chenciner, do you know the simple pendulum?)

See too

• heavy Pendulum: the analysis is detailed and more excavated than this article which starts on a level b.a.basic
• made up heavy Pendule
• Pendule doubles
• Puits of potential: it is advised to treat the cycloidal pendulum before the simple pendulum.
• simple Pendulum discrete: seemingly of elementary level
• Pendulum of Kater: examine the pendular metrology of measurement of G
• Pendule elliptic: is reduced easily to the simple pendulum
• pendular Oscillation of tide: it is about the libration of the satellites
• simple Pendule variable length: one obtains there the general equation of small oscillations.
• Pendulum of Bessel: here, L (T) = l0 + v0.t
• adiabatic Clock: a prequantic curiosity.
• simple Pendulum with parametric resonance: a contrario studies the equations type of Mathieu.
• Swing: traditional of a case L ($\backslash \; theta$) resonant
• Botafumeiro analyzes: Censer of Saint-Jacob de Compostelle: parametric resonance
• Pendulum reversed: on the condition of controlling, via the theorem of Kapitza
• Pendulum maintained by anchor: base ballistic clock industry
• Pendule: seemingly elementary
• Pendulum of Mach: easy modification of G in k.g
• spherical Pendulum: difficult, but integrable via the functions of Jacobi presented above.
• magnetic spherical Pendulum: add with the precedent the force of a monopoly located out of O.
• Pendule conical and Pendule of Huygens: curiosity of history.
• Pendulum of Newton: System made up of several pendulums

• Oscillating harmonic

• oscillating harmonic and simple pendulum in Hamiltonian mechanics:

• Integrating symplectic

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