In this master's thesis we explore the mathematical model of classical Lagrangian mechanics with constraints. The main focus is on the nonholonomic case which is obtained by letting the constraint distribution to be nonintegrable. Motivation for the study arises from various physical examples, such as a rolling rigid body or a snakeboard.
In Chapter 2, we introduce the model and derive the associated equations of motion in several different forms while using the Lagrangian variational principle as a basis for the kinematics. We also show how nonintegrability of the constraint distribution is linked to some external forces via the Frobenius theorem.
Symmetric mechanical systems are discussed in Chapter 3. We define the concept for a Lagrangian system with constraints and show how any free and proper Lie group action induces an intrinsic vertical structure to the tangent bundle of the configuration manifold. The associated bundle is used to define the nonholonomic momentum which is a constrained version of the form that appears in the modern formulation of the classical Noether's theorem.
One applies the classical Noether's theorem to a symmetric system with integrable constraints by restricting observation to an integral submanifold. This procedure, however, is not always possible. In nonholonomic mechanics, a Lie group symmetry implies only an additional equation of motion rather than actual conservation law. In Chapter 4, we introduce a coordinate free technique to split the Lagrangian variational principle in two equations, based on the Lie group invariance. The equations are intrinsic, that is to say, independent of the choice of connections, related parallel transports and covariant differentiation. The vertical projection, associated to the symmetry, may be varied to alter the representation and shift balance between the two equations.
In Chapter 5, the results are applied to the rattleback which is a Lagrangian model for a rigid, convex object that rolls without sliding on a plane. We calculate the nonholonomic momentum and state the equations of motion for a pair of simple connections. One of the equation is also solved with respect to a given solution for the other one.
The thesis is mainly based on the articles 'Nonholonomic Mechanical Systems with Symmetry' (A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden, and R M. Murray, 1996), 'Lagrangian reduction by stages' (H. Cendra, J.E. Marsden, and T.S. Ratiu, 2001), 'Geometric Mechanics, Lagrangian Reduction and Nonholonomic Systems' (H. Cendra, J.E. Marsden, and T.S. Ratiu, 2001) and the book 'Nonholonomic mechanics and control' (A.M. Bloch, 2003).
