Geometric-mechanical origin of global planetary angular momentum dynamics Ramses van der Toorn ⁎ Faculty of Electrical Engineering, Math & Computer Science, Delft University of Technology, Mekelweg 4, 2628CD Delft, The Netherlands abstract article info Article history: Received 24 August 2011 Received in revised form 31 July 2012 Accepted 9 August 2012 Available online 21 August 2012 Keywords: Planetary Fluid Dynamics Angular Momentum Dynamics Variational Principle Symmetry In this paper we shall derive integral angular momentum equations for shallow water dynamics on a rotating planet and in doing so we shall show that these equations are a consequence of the partial spherical symme- try of an appropriate Lagrangian density for this dynamics. © 2012 Elsevier B.V. All rights reserved. 1. Introduction In this paper we shall derive integral angular momentum equa- tions for shallow water dynamics on a rotating planet. In particular, we shall show that these equations are a consequence of the partial spherical symmetry of an appropriate Lagrangian density for plane- tary shallow water dynamics. Our derivation is inspired by the methodology behind Noether's the- orem. Noether's theorem presents conservation laws as consequences of symmetries of a dynamical system. We will show that by applying the methodology behind Noether's theorem to the specific case of plan- etary shallow water dynamics, one can derive a set of criteria, that must be simultaneously met by a transformation for that transformation to be a symmetry. We shall subsequently show that in the specific case at hand, as a next step, the notion of symmetry can actually be relaxed, simply by exploring transformations that meet not all, but only some of the criteria for symmetry. Obviously, with such transformations, no conservation laws will be associated. As we shall explicitly show in the present paper however, by the geometrical, spherical symmetry of a planet, a set of three equations for the balance of integral angular mo- mentum of a shallow fluid layer on a rotating planet is implied in this manner. The spherical symmetry of a planet, which as we shall show is essentially three dimensional, does not correspond to a three dimen- sional symmetry of planetary fluid dynamics. This is mainly because the rotation of the planet defines an axis, and hence a direction in space. Therefore, a single conservation law only can be expected to be associated with rotations, namely one associated with rotations about the axis of the planet. As we shall show in the present paper however, the full three dimensional spherical symmetry, associated with only the spherical shape of the planet, still implies a set of three equations for integral angular momentum. Noether's theorem builds on the variational formulation of mechan- ics, which was first introduced by Lagrange (1736–1813). This formal- ism has the so called Lagrangian function L as its central quantity. According to the Lagrangian formulation of mechanics (Goldstein, 1980; Lanczos, 1970), motions that are realized in nature always corre- spond to extremes of a so-called action integral, which in turn has the Lagrangian L as its argument. The corresponding (differential) equa- tions of motion are the Euler–Lagrange equations associated with the same Lagrangian function L. Because the Lagrangian function is a key in- gredient to Noether's theorem, and hence to our analysis, in Section 2 we shall introduce a Lagrangian density function L, appropriate for plan- etary shallow water dynamics. Given the material discussed so far, a first issue to be clarified is the issue of spherical symmetry of a rotating planet. In an explicit analysis of the approximate spherical symmetry of rotating planets, Van der Toorn and Zimmerman (2008) demonstrated that due to a delicate balance between (a) Newtonian gravitational forces, as induced by the mass distribution of the planet, and (b) centrifugal forces as observed in a co- ordinate system that is co-rotating with the planet, at least in the con- text of planetary fluid dynamics (PFD), the mean geometry of the planet may indeed be approximated by that of a simple sphere. This means that, in terms of e.g. momentum balance equations, with respect to a co-rotating coordinate system the only consequence for PFD in- duced by the rotation of the planet is the occurrence of Coriolis terms. In co-rotating coordinate systems, centrifugal effects are balanced by gravitational effects induced by the oblateness of the planet, while other dynamical effects induced by the oblateness can be shown to be negligible compared to e.g. Coriolis effects (van der Toorn, 1997). A second issue to be clarified at this stage concerns the motivation behind the key subject of this paper. We shall address this issue in Journal of Sea Research 74 (2012) 45–51 ⁎ Tel.: +31 152787281. E-mail address: R.vanderToorn@TUDelft.nl. 1385-1101/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.seares.2012.08.006 Contents lists available at SciVerse ScienceDirect Journal of Sea Research journal homepage: www.elsevier.com/locate/seares