Hamel’s Equations

Leonhard Euler introduced the use of the body frame to describe the angular velocity components of a rotating rigid body in 1752; such a frame simplifies the equations of motion of the rigid body system and yet is decoupled from the local configuration coordinates describing the orientation and position of the body. The general form of the equations of motion for an arbitrary choice of non-coordinate frame originated in a 1904 work of the German mathematician Georg Hamel and are referred to as Hamel’s equations.

A Moving Frame

A mechanical system with n degrees of freedom can be represented (locally) by n generalized coordinates (q^i)_{i=1}^n that are a local coordinate chart on an n-dimensional differentiable manifold that we call the configuration space. The coordinate-induced frame is a bundle of the set of n vectors in each tangent space (fiber) that project down to the direction of the coordinates. This is easy to picture in 2 dimensions: think of the configuration space as a curved surface covered with grid lines that correspond to a choice of generalized coordinates. Then a tangent space is a flat piece of paper that touches the curved surface at one point, and the coordinate-induced frame at that point is the set of 2 vectors that—if we look at the paper from directly above the point of intersection—line up with the directions of the grid lines below.

Given any selection of generalized coordinates there is a natural choice of a coordinate-induced frame, and when we use this frame as the basis to describe velocity vectors the equations of evolution of a mechanical system are the well-known Euler–Lagrange equations. We often use the coordinate-induced frame without even thinking about it, but there are circumstances in which a different choice of frame can make the equations of motion much simpler to analyze. Such an alternate frame can be represented by a smoothly varying linear transformation of the coordinate-induced frame, and the resulting equations of motion for any particular choice are called the Hamel equations. The Euler–Lagrange equations can be thought of as a special case of the Hamel equations, where the linear transformation that we use to describe our choice of frame is the identity.

My Central Question: Discrete Hamel Equations

As Euler saw in the case of a rigid body, the use of a non-coordinate frame to measure velocity may simplify the equations of motion for certain types of systems. However, the act of decoupling the components of velocity and position introduces a new structure called the Jacobi-Lie bracket into the general form of Hamel’s equations that is difficult to discretize. While numerical integration of Hamel’s equations for rigid body type systems is well understood, it is not clear how to derive symplectic integrators that approximate the equations of motion of any mechanical system given an arbitrary choice of frame because of the Jacobi-Lie bracket term.

The variational approach to the derivation of the Euler–Lagrange equations (Hamilton’s principle) depends upon the commutativity of two differential operators: the time derivative \frac{d}{dt} and the variational derivative \delta. When we use a general (non-coordinate) frame to describe vectors this commutativity disappears, giving rise to the Jacobi-Lie bracket term in the equations of motion. It is this bracket term (in addition to a directional derivative) that distinguishes the Hamel equations from the Euler–Lagrange equations, and it is this bracket term that is the main obstacle to a discrete model of the Hamel equations.

Up to this point it had not been clear how to derive a discrete model of this bracket term, and hence a discrete model of the Hamel equations. However, since the bracket term may be derived from a variational principle in the continuous case we have been able to use variational integration as a means to find this discretization.

It was my idea of discrete path parameterization within the framework of variational integration that has led us towards a consistent discrete version of the bracket term in the Hamel equations. Prior to my work, while the variational derivative \delta would survive when we would transition to the discrete setting, there was no clear discrete analogue of the time-derivative \frac{d}{dt}. I was able to find such an analogue by paying attention to how we actually defined parameterized paths between steps in our discrete curve and then describe it’s noncommutativity with \delta, hence providing us with a variational derivation of the bracket in the discrete setting.

There is quite a bit of current interest in variational integrators related to numerical modeling of relativistic systems, computational control theory, and computer animation. My work on variational integration of Hamel’s equation will provide a means of incorporating the method of moving frames into these models. A clever choice of frame based on symmetries or constraints of a mechanical system can entail significant simplifications in the equations of motion, it is likely that these simplifications will carry over to the symplectic integrators derived from variational principles.