In physics, a spherical pendulum is a higher dimensional analogue of the pendulum. It consists of a massm moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity.
Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of , where r is fixed such that .
Routinely, in order to write down the kinetic and potential parts of the Lagrangian in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. Here, following the conventions shown in the diagram,
.
Next, time derivatives of these coordinates are taken, to obtain velocities along the axes
.
Thus,
and
The Lagrangian, with constant parts removed, is[1]
Similarly, the Euler–Lagrange equation involving the azimuth ,
gives
.
The last equation shows that angular momentum around the vertical axis, is conserved. The factor will play a role in the Hamiltonian formulation below.
The second order differential equation determining the evolution of is thus
Hamilton's equations will give time evolution of coordinates and momenta in four first-order differential equations
Momentum is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis.[dubious – discuss]
Trajectory
Trajectory of a spherical pendulum.
Trajectory of the mass on the sphere can be obtained from the expression for the total energy
by noting that the horizontal component of angular momentum is a constant of motion, independent of time.[1] This is true because neither gravity nor the reaction from the sphere act in directions that would affect this component of angular momentum.
^ abcdLandau, Lev Davidovich; Evgenii Mikhailovich Lifshitz (1976). Course of Theoretical Physics: Volume 1 Mechanics. Butterworth-Heinenann. pp. 33–34. ISBN0750628960.
Further reading
Weinstein, Alexander (1942). "The spherical pendulum and complex integration". The American Mathematical Monthly. 49 (8): 521–523. doi:10.1080/00029890.1942.11991275.
Shiriaev, A. S.; Ludvigsen, H.; Egeland, O. (2004). "Swinging up the spherical pendulum via stabilization of its first integrals". Automatica. 40: 73–85. doi:10.1016/j.automatica.2003.07.009.