Constrained Geodesic
This paper discusses the geometry of a manifold specified as a set of algebraic equality constraints. The manifold is embedded in a Riemannian space in which the existence of a metric is assumed. The main result in the paper is the construction of the constrained geodesic



Here, uⁱ denote the coordinates of a point along the constrained geodesic and s is arclength. The second derivatives of the coordinate functions along the constrained geodesic [the left hand side of its defining equation] represent the principal normal to the constrained geodesic. The Christoffel symbols are for the embedding space and vanish for an Euclidean space. The linear system specifies the projective decomposition of the normal to the constrained geodesic in terms of the normals to the individual constraint surfaces. The right hand side of this linear system is defined below for both types of constraints.

Two forms of the constraints are considered: holonomic constraints where the constraints depend on space alone and the hybrid case of holonomic and non-holonomic constraints. For both cases, the constrained geodesic is constructed using standard methods from the calculus of variations. The Taylor series expansion of the constrained geodesic is analyzed for both types of constraints.

Holonomic Case
In the case of holonomic constraints we write the constraints as



The Gram matrix of the normals to the constraint surfaces is



and the auxiliary data is

Non-Holonomic Case
In the case of mixed holonomic and non-holonomic constraints we have



In this case, the Gram matrix of the system of constraints is



and the auxiliary data is given as



These systems of ordinary differential equations describe the geodesic on the manifold specified by the holonomic constraints. In the usual case where the embedding space is Euclidean, the Christoffel symbols of the embedding space vanish.

The inverse Gram matrix is also required and it is defined as usual

Geometry of the Manifold
The association of the Christoffel symbols for the constrained geodesic with the manifold of the constraints permits analysis of some of its geometric properties using standard methods from differential geometry. The modified Christoffel symbols for the constrained geodesic may be considered the Christoffel symbols for the manifold



The Riemann curvature tensor is expressible in terms of the Christoffel symbols and their derivatives and is computed as



The Gaussian curvature invariant is found as usual by twice contracting the Riemann tensor



In the usual case where the embedding space is Euclidean the appropriate simplifications are indicated. It is also possible to consider the minimization problem for the principal directions and the corresponding principal curvatures of the manifold in analogy with the classical theory for a surface in three-dimensional space. The structure of the minimization problem for the principal curvatures is such that the case of a single constraint (hypersurface) presents a considerable simplification. The main subject of classical differential geometry which is the analysis of a surface in three dimensional space is the simplest case of such a hypersurface.

Classical Differential Geometry Revisited
The present theory also affords a new look at the classical treatment of a surface in three-dimensional Euclidean space by regarding the surface as a single constraint. This permits the derivation of results without reference to coordinates in the surface. The advantage of this approach is that this construction is in terms of the coordinates of the embedding space and therefore does not require the construction or assumption of a coordinate system in the surface. The geodesics in a surface can now be computed in terms of the coordinates of the embedding space rather than in terms of the coordinates in the surface. The disadvantage relative to using coordinates in the surface is that in all but trivial cases a numerical method must be used to integrate the geodesic differential equations. The numerical error in a method for integrating the system of ordinary differential equations for the constrained geodesic in the embedding space inherently introduces small violations of the constraints.

Applications
The constrained geodesic is the natural basis for the development of methods for the solution of constrained optimization problems with algebraic equality constraints. I have done some work on this problem by considering the variation of an arbitrary scalar function along the constrained geodesic. However, I have not been able to completely solve this problem and present some preliminary results. The modified second derivative matrix governing the variation of a function F along the constrained geodesic is



This is an important result for the constrained optimization problem. In the usual case where the embedding space is Euclidean this reduces to



The constrained geodesic can also be used for relatively simple problems such as computing the system of ordinary differential equations for the curve of intersection of two surfaces in three-space. The constrained geodesic for the case of mixed holonomic and non-holonomic constraints may provide new insight in the solution of the equations of motion of classical mechanics.

Points on the Sphere Example
The optimization of a function associated with an assembly of points on the sphere is used as an example problem for the case of holonomic constraints. This problem is attractive because the holonomic constraints are of two distinct types: unit norm constraints for individual points and conditions to prevent meaningless rotations of the assembly as a whole. The sphere problem is also an excellent test problem for methods for constrained optimization and the resulting optimum configurations are interesting polyhedra. The Taylor series for the constrained geodesic for this system of constraints is given to fourth order as



It is also sometimes useful to track the tangent vector



Here, uⁱ and pⁱ denote the initial point vector and initial direction of the constrained geodesic and uⁱ(s) and pⁱ(s) denote the respective values along the constrained geodesic. The auxiliary matrix M is defined in terms of the coefficients as



This solution has the amazing property that the movement of the individual points on the surface of the sphere is such that the configuration as a whole remains on main axes. Not surprisingly, the equations for the determination of the coefficients and their derivatives are rather complicated. Truncating the Taylor series or the use of any alternative numerical method will of course introduce some numerical error. It is interesting to compare the true geodesics for the constrained problem which includes the inertia constraints with the great circle geodesics for the case without the inertia constraints



These useful equations are not found in any textbook I know off by the way. The Points on the Sphere problem is treated in a separate paper.

Download
The paper is available in Adobe PDF format. Since it is quite large it has been formatted in four different sections. The main theory is in the first section. The section on constrained optimization is unfinished and of an exploratory nature.