Pendulum of Bessel

When the length of a simple Pendule varies in a way closely connected: L (T) = lo + vt, one says that it is about a Pendule of Bessel, because the solution (for small oscillations) is expressed using the functions of Bessel. If v are weak, one finds the adiabatic invariant E (T) T (T) (see adiabatic Pendule).

Equation of the variable pendulum length

The simple Pendule variable length has as an equation for this temporal law of L (T) where its derivative second is null:

$l\; (T)\; \backslash \; ddot\; \{X\}\; (T)\; +\; G\; (T)\; X\; (T)\; =\; 0$

In the usual case G is constant. Let us return to the angular function $\backslash \; theta=\; \backslash \; frac\; xl$:

$L\; (T)\; \backslash \; ddot\; \{\backslash \; theta\}\; (T)\; +2v\; \backslash \; dowry\; \{\backslash \; theta\}\; (T)\; +\; G\; \backslash \; theta\; (T)\; =0$

Let us pose like new variable without units $u\; =\; G\; \backslash \; frac\; TV$:

$L\; (U)\; \backslash \; frac\; G\; \{v^2\}\; \backslash \; ddot\; \{\backslash \; theta\}\; (U)\; +2\; \backslash \; dowry\; \{\backslash \; theta\}\; (U)\; +\; \backslash \; theta\; (U)$

Then to simplify, let us pose like length $L=\; \backslash \; frac\; \{v^2\}\; \{G\}$:

$\backslash \; frac\; \{L\; (U)\}\{L\}\; \backslash \; ddot\; \{\backslash \; theta\}\; (U)\; +\; 2\; \backslash \; dowry\; \{\backslash \; theta\}\; (U)\; +\; \backslash \; theta\; (U)\; =\; 0$

One recognizes the equation circuit RLC with variable coil linearly, it is a traditional problem (see obtaining intense magnetic fields). One can still transform this equation.

Equation of Bessel

One makes a new change of variable without units $\backslash \; alpha=\; 2\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\; (U)\}\{L\}\}$:

$2\; \backslash \; ddot\; \{\backslash \; theta\}\; (\backslash \; alpha)\; +2\; \backslash \; frac\; \{\backslash \; dowry\; \{\backslash \; theta\}\; (\backslash \; alpha)\}\{\backslash \; alpha\}\; +\; \backslash \; theta\; (\backslash \; alpha)\; =0$

One changes function $there\; (\backslash \; alpha)\; =\; \backslash \; alpha\; \backslash \; theta\; (\backslash \; alpha)$, and one ends up finding an equation of Bessel with n=1:

$\backslash \; alpha^2\; \backslash \; ddot\; \{there\}\; (\backslash \; alpha)\; +\; \backslash \; alpha\; \backslash \; dowry\; \{there\}\; +\; (\backslash \; alpha^2-1)\; Y\; =\; 0$

The solutions are the functions of Bessel, functions traditional of the mathematical physics (cf Campbell, for example):

$there\; (\backslash \; alpha)\; =A\; \backslash \; quad\; J\_1\; (\backslash \; alpha)\; +B\; \backslash \; quad\; Y\_1\; (\backslash \; alpha)$

From where: $\backslash \; theta\; (\backslash \; alpha)\; =A\; \backslash \; quad\; \backslash \; frac\; \{J\_1\; (\backslash \; alpha)\}\{\backslash \; alpha\}\; +B\; \backslash \; quad\; \backslash \; frac\; \{Y\_1\; (\backslash \; alpha)\}\{\backslash \; alpha\}$

Resolution with the initial conditions

$\backslash \; theta\_0=\; has\; \backslash \; quad\; \backslash \; frac\; \{J\_1\; (\backslash \; alpha\_0)\}\{\backslash \; alpha\_0\}\; +B\; \backslash \; quad\; \backslash \; frac\; \{Y\_1\; (\backslash \; alpha\_0)\}\{\backslash alpha\_0\}$

$\backslash \; dowry\; \{\backslash \; theta\_0\}\; =\; -\; has\; \backslash \; quad\; \backslash \; frac\; \{J\_2\; (\backslash \; alpha\_0)\}\{\backslash \; alpha\_0\}\; -\; B\; \backslash \; quad\; \backslash \; frac\; \{Y\_2\; (\backslash \; alpha\_0)\}\{\backslash alpha\_0\}$

One solves this linear system of two equations to two unknown factors and which gives finally:

$has\; =\; \backslash \; alpha\_0\; \backslash \; frac\; \{\backslash \; theta\_0\; Y\_2\; (\backslash \; alpha\_0)\; +\; \backslash \; dowry\; \{\backslash \; theta\_0\}\; Y\_1\; (\backslash \; alpha\_0)\}\{J\_1\; (\backslash \; alpha\_0)\; Y\_2\; (\backslash \; alpha\_0)\; -\; J\_2\; (\backslash \; alpha\_0)\; Y\_1\; (\backslash \; alpha\_0)\}$ $B\; =\; \backslash \; alpha\_0\; \backslash \; frac\; \{\backslash \; theta\_0\; J\_2\; (\backslash \; alpha\_0)\; +\; \backslash \; dowry\; \{\backslash \; theta\_0\}\; J\_1\; (\backslash \; alpha\_0)\}\{J\_1\; (\backslash \; alpha\_0)\; Y\_2\; (\backslash \; alpha\_0)\; -\; J\_2\; (\backslash \; alpha\_0)\; Y\_1\; (\backslash \; alpha\_0)\}$

One can then trace $\backslash \; theta\; (T)$ using a software.

One can even check graphically, that if v are weak, E (T) T (T) is a constant (adiabatic Pendule). It should nevertheless be remembered that one always placed in the case of the small oscillations. Of course, any standard numerical method Runge-Kutta gives the same results without obtaining the general formula.

See too

• variable Pendulum length
• adiabatic Clock

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