# Geometric mechanics

Back to my research page.

Foundations for mechanics
Affine differential geometry in mechanics
Constrained mechanical systems

## Foundations for mechanics

1. A comprehensive formulation of mechanics from ``first principles'' using differential geometric methods.
Reference: The physical foundations of geometric mechanics.
2. An apology for geometric mechanics written for a robotics audience.
Reference: Is it worth learning differential geometric methods for modelling and control of mechanical systems?
3. A Riemannian geometric formulation of reduction and stability of relative equilibria.
Reference: Reduction, linearization, and stability of relative equilibria for mechanical systems on Riemannian manifolds.
4. A genuinely Galilean setting for rigid body mechanics which seems strangely absent from the literature.
Reference: Rigid body mechanics in Galilean spacetimes.
5. A general setting for Lagrangian mechanics using the ``Lagrange two-force.'' This is a quite general presentation of Lagrangian mechanics.
Reference: Towards F=ma in general setting for Lagrangian mechanics.
6. An investigation of Lagrangian foliations of the zero energy levels of central force problems.
Reference: Lagrangian submanifolds and an application to the reduced Schrödinger equation in central force problems,

## Affine differential geometry in mechanics

1. Affine connections preserving the kinetic energy of a Riemannian metric are characterised.
Reference: Energy-preserving affine connections.
2. A thorough investigation of how affine connections can be used to describe the unforced equations of motion for a system with constraints linear in velocity. Note that the use of an affine connection to describe constrained mechanical systems dates back as far as the work of Synge (Geodesics in nonholonomic geometry Math. Ann. 99 738-751, 1928). Other authors have since picked up this thread, and my efforts have been motivated by work of Peter Crouch and Tony Bloch.
Reference: Affine connections and distributions.

## Constrained mechanical systems

1. A variational principle for nonlinear constraints, and a Noether theorem for these systems.
Reference: Variational principles for nonlinearly constrained systems in one independent variable.
2. A generalisation of the Gibbs-Appell method from systems of particles and rigid bodies to general Lagrangians, along with the relationship of these to Gauss's Principle of Least Constraint.
Reference: The geometry of the Gibbs-Appell equations and Gauss's Principle of Least Constraint.
3. An investigation of the ``snakeboard'' example.
Reference: Nonholonomic mechanics and locomotion: the snakeboard example.
4. A theoretical and experimental investigation of variational methods, wherein the value of the vakonomic equations are put in doubt, as concerns their description of solutions for mechanical systems.
Reference: Variational principles for constrained systems: theory and experiment.

Andrew D. Lewis (andrew at mast.queensu.ca)