Spherical pendulum

A spherical pendulum also Kugelpendel or spatial pendulum is a pendulum whose suspension allows deflections in different directions. In contrast to the (flat) circular pendulum , in which the movement of the pendulum mass is limited to a vertical circle, in the (spatial) spherical pendulum the pendulum mass moves on a spherical surface .

A special case of the spherical pendulum is the conical pendulum , also conical pendulum , circular pendulum , rotary pendulum or Zentrifugalpendel , the pendulum mass moves in the circular path on a horizontal, and the yarn thus describes a conical surface.

In the theoretical treatment of the spherical pendulum, the suspension is often assumed to be massless and the body of the pendulum as point-shaped, and the influence of friction is neglected. In addition to the conservation of energy , the conservation of angular momentum is also important for the spherical pendulum . In the projection onto a horizontal plane, the pendulum thread therefore sweeps over the same areas at the same time (see area theorem ).

One application of the spherical pendulum is the Foucault pendulum , with the help of which the earth's rotation can be clearly demonstrated without reference to observations in the sky .

Lagrange treatment

General case

Since the pendulum mass of the spherical pendulum moves on a spherical surface, its movement can best be described in spherical coordinates :

${\displaystyle {\vec {r}}(t)=(r,\theta ,\phi )}$

The point of suspension is the origin and the z-axis points to the stable lower rest position. Then ${\displaystyle r=0}$

• ${\displaystyle r=R}$ the length of the pendulum, which cannot change due to the rigid connection between the suspension point and the pendulum body
• the polar angle is the deflection from the lower equilibrium position ${\displaystyle \theta }$
• the azimuth angle is the rotation around the vertical axis. ${\displaystyle \phi }$ ${\displaystyle z}$

Since the length is kept constant, the two angles are the only free variables, i.e. the generalized coordinates for this system. It is now the Lagrange function ${\displaystyle r}$

${\displaystyle L=T-V}$

to form, where the kinetic energy and the potential energy denote depending on the two generalized coordinates and their time derivatives. ${\displaystyle T}$ ${\displaystyle V}$

The potential energy of the pendulum in relation to the suspension point is

${\displaystyle V(\theta ,\phi )=-mgR\cos \theta }$

and has its minimum at . The kinetic energy is ${\displaystyle \theta =0}$

${\displaystyle {\begin{aligned}T(\theta ,\phi ,{\dot {\theta }},{\dot {\phi }})&={\frac {1}{2}}mv^{2}\\&={\frac {1}{2}}mR^{2}({\dot {\theta }}^{2}+{\dot {\phi }}^{2}\sin ^{2}\theta )\end{aligned}}}$.

The equations of motion then result from the Lagrangian equations of the 2nd type:

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}{\frac {\partial L}{\partial {\dot {\theta }}}}={\frac {\partial {L}}{\partial \theta }}}$

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}{\frac {\partial L}{\partial {\dot {\phi }}}}={\frac {\partial {L}}{\partial \phi }}\,.}$

The Lagrange equations give (after shortening ): ${\displaystyle mR^{2}}$

${\displaystyle {\ddot {\theta }}={\dot {\phi }}^{2}\sin \theta \cos \theta -{\frac {g}{R}}\sin \theta }$.

${\displaystyle {\frac {\text{d}}{{\text{d}}t}}({\dot {\phi }}\;\sin ^{2}\theta )=0}$.

The second Lagrange equation immediately lists

${\displaystyle {\ddot {\phi }}=-2{\dot {\phi }}\;{\dot {\theta }}{\frac {\cos \theta }{\sin \theta }}}$.

These equations form a system of two coupled differential equations of the 2nd order, of which the second can, however, be integrated immediately, as can be seen from the Lagrange equation above it, from which it emerged.

According to this second Lagrange equation, the associated conjugate momentum is constant. It is the component of angular momentum ${\displaystyle \phi }$ ${\displaystyle {\tfrac {\partial L}{\partial {\dot {\phi }}}}}$ ${\displaystyle z}$

${\displaystyle {\frac {\partial L}{\partial {\dot {\phi }}}}=mR^{2}{\dot {\phi }}\sin ^{2}\theta =L_{z}={\text{const.}}}$

( does not appear in and is therefore a cyclic variable . This is an example of Noether's theorem .) ${\displaystyle \phi }$ ${\displaystyle L}$

We can thus eliminate from the differential equation for : ${\displaystyle {\dot {\phi }}}$ ${\displaystyle \theta }$

${\displaystyle {\ddot {\theta }}={\frac {L_{z}^{2}}{m^{2}R^{4}}}{\frac {\cos \theta }{\sin ^{3}\theta }}-{\frac {g}{R}}\sin \theta }$.

In general, this equation of motion for can not be solved elementarily, and complex motions can result. An easily solvable case is the conical pendulum (see below). ${\displaystyle \theta }$

Statements on general properties of the movement can be obtained if the constancy of the total energy is also taken into account, which results from the fact that there is no explicit dependence on time. It follows : ${\displaystyle E=T+V}$

• The movement is restricted to one area , i.e. it takes place between two circles of latitude. ${\displaystyle \theta _{min}\leq \theta \leq \theta _{max}}$
• The up and down movement between the circles of latitude is periodic (but not harmonic).
• The azimuthal angular velocity is either constant zero (if ) or has the fixed sign of . The direction of rotation of the pendulum movement around the z-axis can therefore not be reversed. ${\displaystyle {\dot {\phi }}}$ ${\displaystyle L_{z}=0}$ ${\displaystyle L_{z}\neq 0}$
• At, the spherical pendulum swings exactly periodically through the rest position like a flat mathematical pendulum . ${\displaystyle L_{z}=0}$
• At is and . The pendulum maintains a minimum distance from both the lowest and the highest point of the sphere. The periodic up and down movement is superimposed on an azimuthal rotation so that the successive points with (as well as the points with ) are offset by one . ${\displaystyle L_{z}\neq 0}$ ${\displaystyle 0<\theta _{min}<90^{\circ }}$ ${\displaystyle \theta _{max}<180^{\circ }}$ ${\displaystyle \theta =\theta _{min}}$ ${\displaystyle \theta =\theta _{max}}$ ${\displaystyle \Delta \phi }$
• Then the movement as a whole is only periodic if the offset is a rational fraction of the full rotation through 360 °. ${\displaystyle \Delta \phi }$

By taking into account the constancy of energy, the equation of motion for can be converted into a first-order differential equation, which, however, cannot be solved elementarily: ${\displaystyle \theta }$

${\displaystyle {\dot {\theta }}=\pm {\sqrt {{\frac {2E}{mR^{2}}}-{\frac {L_{z}^{2}}{m^{2}R^{4}\sin ^{2}\theta }}+2{\frac {g}{R}}\cos \theta }}}$,

Conical pendulum

The conical pendulum is resolved with

${\displaystyle \theta ={\text{const.}}}$

described. Then and consequently according to the above equation of motion ${\displaystyle {\ddot {\theta }}=0}$

${\displaystyle {\dot {\phi }}^{2}\cos \theta ={\frac {g}{R}}}$.

Accordingly, the pendulum describes with the constant angular velocity

${\displaystyle \omega ={\dot {\phi }}=\pm {\sqrt {\frac {g}{R\cos \theta }}}}$

a cone envelope, where it must be, the constant deflection angle is therefore restricted to the area . ${\displaystyle \cos \theta >0}$ ${\displaystyle 0\leq \theta <90^{\circ }}$

Treatment in Newtonian mechanics

General case

Trajectory of a spherical pendulum

Click to animate

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The pendulum (with a rod instead of a thread between the mass and the suspension point) is pushed from a horizontal position with a speed directed upwards and backwards.

The trajectory of the pendulum mass results from Newton's mechanics as a solution of the vectorial differential equation for the acceleration ${\displaystyle {\vec {r}}(t)}$

${\displaystyle {\ddot {\vec {r}}}={\vec {g}}+{\vec {a}}_{Z}}$.

The first summand on the right is the acceleration due to gravity , which is the applied force here . The second term comes from the constraining force exerted by the rod . It has to hold the body on the spherical shell with the radius , i.e. - at every position and speed of the body - cancel the radial component of gravity and exert the centripetal force necessary for the orbital curvature with the radius of curvature . The constraining force therefore acts in the radial direction and is given by: ${\displaystyle m{\vec {g}}}$ ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle {\vec {a}}_{Z}=\left[-({\vec {g}}\cdot {\vec {e}}_{r})-{\frac {{\dot {\vec {r}}}^{2}}{R}}\right]{\vec {e}}_{r}}$.

${\displaystyle {\vec {e}}_{r}={\vec {r}}/R}$ denotes the radial unit vector directed away from the suspension point.

Together with the impressed force, one can write:

${\displaystyle {\ddot {\vec {r}}}=\left[{\vec {g}}-({\vec {g}}\cdot {\vec {e}}_{r}){\vec {e}}_{r}\right]-{\frac {{\dot {\vec {r}}}^{2}}{R}}{\vec {e}}_{r}}$,

Here it can be seen that the entire acceleration is caused by the tangential component of gravity (term in square brackets) and the radial centripetal force. ${\displaystyle {\ddot {\vec {r}}}}$

Expressing this equation in spherical coordinates results in the differential equations for the angles and , which - as noted above - cannot be solved in a closed manner. Cartesian coordinates are more favorable for a numerical solution, because the angle at the position of rest is not defined in spherical coordinates . The animation opposite, which shows a complicated sequence of movements, was created in this way with a SciLab script. ${\displaystyle \theta }$ ${\displaystyle \phi }$ ${\displaystyle \phi }$

Harmonic and anharmonic approximation for small deflections

Qualitative description

With small swings, the movements of the spherical pendulum are simple: If the swings are infinitesimally small, it oscillates like an isotropic two-dimensional harmonic oscillator with the same frequency as the flat mathematical pendulum in harmonic approximation. That is, the trajectories are space-fixed ellipses, including the limit cases of linear oscillation and the circle. For this see the special section in the article Harmonic Oscillator . In the case of small but finite excursions, anharmonic effects occur, which lead to a reduction in the orbital frequency and a precession of the orbital ellipse (in the direction of rotation of the orbit). Both are due to the fact that the frequency of the planar mathematical pendulum is only independent of the size of the swing in the infinitesimal range , but decreases with increasing swing. The lowest approximation applies (see in mathematical pendulum ) ${\displaystyle \omega _{0}={\sqrt {g/R}}}$ ${\displaystyle \theta _{\text{max}}}$

${\displaystyle \omega (\theta _{\text{max}})=\omega _{0}\left(1-{\frac {1}{16}}\sin ^{2}(\theta _{\text{max}})\right)}$

An elliptical oscillation can be viewed as a superposition of two linear oscillations of the same frequency with differently large excursions, which are offset by a quarter period and at right angles to each other along the major and minor semi-axes of the ellipse. This possibility is given with infinitesimally small deflections, so that the linear oscillations remain synchronous and form a spatially fixed ellipse. With real deflections, however, with the spherical pendulum the oscillation along the small semiaxis is somewhat faster than the oscillation along the major semiaxis, so that it is already beyond its zero point when the other is only at its maximum deflection, i.e. H. at the apex of the trajectory, arrives. Put together it results that the vertex wanders around on a circle.

calculation

The movements with small deflections are most easily treated in Cartesian coordinates by expanding according to powers. The origin lies at the suspension point and the z-axis is directed downwards. Small deviations from the rest position are defined by and and . There are two coupled differential equations for and , which can be developed in power series. ${\displaystyle |x|/R\ll 1}$ ${\displaystyle |y|/R\ll 1}$ ${\displaystyle |z-R|/R\ll 1}$ ${\displaystyle {\ddot {x}}}$ ${\displaystyle {\ddot {y}}}$

1st approximation - linearization

If only the terms of the lowest power are taken into account for infinitesimal excursions, two decoupled differential equations are obtained for a pair of harmonic oscillators with the same frequency

${\displaystyle {\begin{array}{ll}{\ddot {x}}&=-\omega _{0}^{2}\,x\\{\ddot {y}}&=-\omega _{0}^{2}\,y\end{array}}}$

For the solution and solution see harmonic oscillator . The same differential equations are obtained for small deflections from the physically based approximation that the movement only takes place in the plane and the force driving back towards the rest position is given by the tangential component of gravity, whereby this (for deflection in the x-direction) is given by ${\displaystyle z=R}$ ${\displaystyle x=y=0}$

${\displaystyle -m\,g\,{\text{sin}}(\theta )\approx -m\,g\,\theta \approx -{\frac {m\,g}{R}}x}$

is approximated (correspondingly for y-direction). The trajectories are spatially fixed ellipses with any orientation of the axes in the plane of oscillation, including the borderline cases of distance and circle.

2nd approximation - cubic members, rotating ellipse

As a closest approximation, cubic terms appear, via which the two differential equations are also coupled. A closed solution is not possible. In accordance with the qualitative discussion above, an approximate solution is based on the approach of a slowly rotating elliptical path. Accordingly, the pendulum body runs through an ellipse with the semi-axes and with the circular frequency ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle \omega (a,b)=\omega _{0}\left(1-{\frac {1}{8}}{\frac {a^{2}+b^{2}}{R^{2}}}\right)}$.

The ellipse rotates in the sense of the revolution so that the apex is around the angle with each revolution

${\displaystyle \Delta \phi (a,b)={\frac {3}{4}}{\frac {\pi ab}{R^{2}}}}$

is moved. This corresponds to a rotation of the path with an angular velocity

${\displaystyle \Omega (a,b)={\frac {3}{8}}\omega _{0}{\frac {ab}{R^{2}}}}$.

This precession motion is, for example, a common perturbation in Foucault's pendulum because it easily reaches the size of the precession due to the rotation of the earth.

Individual evidence

1. ↑ Bergmann-Schaefer Textbook of Experimental Physics , Volume 1: Mechanics, Acoustics, Warmth , Chapter IV, Section 35
2. ↑ A. Budó: Theoretical Mechanics . 4th edition. VEB Deutscher Verlag der Wissenschaften, Berlin 1967, § 23, p. 117-118 .
3. ↑ ^{a b} MG Olsson: Spherical Pendulum revisited , American Journal of Physics 49, 531 (1981); doi: 10.1119 / 1.12666
4. ↑  
   The SciLab script is given here.

   ---

   // STARTING OF THE SCRIPT. This script can be executed as a sce file with the free mathematics software "Scilab"

      // Autor Modalanalytiker 20.08.2018
      //Numerische Lösung der vektoriellen Bewegungsgleichung des sphärischen Pendels
      //Bahnkurve für Wikimedia-Bild "SphaerischesPendelxyz.svg"
      //Durchgängig SI-Einheiten!
        d2r=%pi/180;//degree to radian
   

     //########## [ EINGABE ##############################################
        g=[0;0;-9.81]//Erdbeschleunigung, kartesische Koordinaten
        L=1//Pendellänge
        TSim=3// Dauer der Simulation
        deltaT=0.01//Zeitschritt der Ausgabe
        //Anfangszustand in sph. Koord., um unrealistische Zustände auszuschließen
          Th0=90*d2r;     //Polarwinkel
          dTh0=-200*d2r;  //Polarwinkelgeschwindigkeit
          Ph0=0*d2r;      //Azimut
          dPh0=90.*d2r;   //Azimutwinkelgeschwindigkeit
     //########## ] EINGABE ENDE #########################################
   

      function dz=xyzPendel(t,z,g)//
          //Ableitung des Zustandsvektors z=[x;y;z;dx;dy;dz],  6 x 1
          //g=[gx; gx; gz]: Schwerebeschleunigung
          dz(1:3)=z(4:6)//Geschwindigkeit 3 x 1
          r=norm(z(1:3))//Abstand der Masse vom Ursprung, Länge der Pendelstange, 1 x 1
          er=z(1:3)/r //Radialer Einheitsvektor der Massenposition, 3 x 1
          dz(4:6)=g-er*(g'*er + z(4:6)'*z(4:6)/r) //Beschleunigung, 3 x 1
      endfunction
   

      t=0:deltaT:TSim;//Zeitspanne der Simulation
    //Umrechnung der AW in kartesische Koordinaten
      //Anfangsposition
      x0=L*sin(Th0)*cos(Ph0);
      y0=L*sin(Th0)*sin(Ph0);
      z0=L*cos(Th0);
      Z0(1:3)=[x0;y0;z0];//Anfangsposition der Masse, Großbuchstabe Z0!
    //Anfangsgeschwindigkeit
      eTh0=[cos(Th0)*cos(Ph0);cos(Th0)*sin(Ph0);-sin(Th0)]//polarer Einheitsvektor
      ePh0=[-sin(Ph0);cos(Ph0);0]//azimutaler Einheitsvektor
      Z0(4:6)=L*dTh0 *eTh0 + L*sin(Th0)*dPh0 *ePh0//Anfangsgeschwindigkeit der Masse
   

    //Integration der Dgl.
    //Solver automatically selects between nonstiff predictor-corrector Adams method and
      //stiff Backward Differentiation Formula (BDF) method
      zk=ode(Z0, t(1), t, list(xyzPendel,g));//###### Dgl.-Lösung ###########
   

    //Graphik
      xdel();
      param3d(zk(1,:),zk(2,:),zk(3,:))//####################################
      title('Sphärisches Pendel mit Aufhängung im Ursprung ""o""')
        ce=gce(); ce.foreground=5; ce.thickness=3; //rote Kurve
      param3d(zk(1,1),zk(2,1),zk(3,1))//ausgefüllter roter Kreis für Anfang
        ce=gce();ce.mark_mode="on"; ce.mark_style=0; ce.mark_size_unit = "point";
        ce.mark_size = 10; ce.mark_foreground = 5;  ce.mark_background = 5;
     param3d(zk(1,:),zk(2,:),-L*ones(zk(1,:)))//Projektion z=konst
        ce=gce(); ce.foreground=13;  ce.line_style=9;ce.thickness=2;
     param3d(zk(1,1),zk(2,1),-L*ones(zk(1,1)))//Projektion z=konst, Anfang
        ce=gce();ce.mark_mode="on"; ce.mark_style=0; ce.mark_size_unit = "point";
        ce.mark_size = 5; ce.mark_foreground = 5;  ce.mark_background = 5;
     param3d(-L*ones(zk(1,:)),zk(2,:),zk(3,:))//Projektion x=konst
        ce=gce(); ce.foreground=5;  ce.line_style=9;ce.thickness=2;
     param3d(-L*ones(zk(1,1)),zk(2,1),zk(3,1))//Projektion, Anfang
        ce=gce();ce.mark_mode="on"; ce.mark_style=0; ce.mark_size_unit = "point";
        ce.mark_size = 5; ce.mark_foreground = 5;  ce.mark_background = 5;
     param3d(zk(1,:),+L*ones(zk(1,:)),zk(3,:))//Projektion y=konst
        ce=gce(); ce.foreground=5;  ce.line_style=9;ce.thickness=2;
     param3d(zk(1,1),+L*ones(zk(1,1)),zk(3,1))//Projektion, Anfang
        ce=gce();ce.mark_mode="on"; ce.mark_style=0; ce.mark_size_unit = "point";
        ce.mark_size = 5; ce.mark_foreground = 5;  ce.mark_background = 5;
     param3d(0,0,0) //schwarzes O-Zeichen für Ursprung
        ce=gce();ce.mark_mode="on"; ce.mark_style=9; ce.mark_size_unit = "point";
        ce.mark_size = 5; ce.mark_foreground = 1;
     ca=gca();
        ca.rotation_angles=[83.25 280];
        ca.data_bounds = 1.03*[-1,-1,-1.;1,1,1];
        ca.tight_limits = ["on","on","on"];
   

   // SCRIPTORS
5. ↑ Szostak, Roland: A permanently swinging Foucault pendulum for schools, The mathematical and natural science lessons, PLUS LUCIS 2 / 2002-1 / 2003, pp. 11–15, html

Web links

• Physics and Mathematics with Maple, spherical pendulum , accessed December 22, 2014