There is an explicit resolution of the Poisson reduction of four planar point vortices, in the case that three of the vortex strengths are equal and the total vorticity is zero. The resolution, a Hamiltonian system on a unified symplectic phase space with a symmetry breaking parameter, is obtained by appending redundant states. Though single point vortices do not have the attribute of mass, there are circular assemblages with the collective dynamics of free massive particles, demonstrating a finite dimensional dynamics where mass emerges from a gauge symmetry breaking. The internal vibration of these assemblages is coupled to their collective motion and has the same functional form as the de Broglie wavelength.

The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies geometrically generalize the usual numerical analysis definitions of convergence.

Implicit methods for the numerical solution of initial-value problems may admit multiple solutions at any given time step. Accordingly, their nonlinear solvers may converge to any of these solutions. Below a critical timestep, exactly one of the solutions (the consistent solution) occurs on a solution branch (the principal branch) that can be continuously and monotonically continued back to zero timestep. Standard step-size control can promote convergence to consistent solutions by adjusting the timestep to maintain an error estimate below a given tolerance. However, simulations for symplectic systems or large physical systems are often run with constant timesteps and are thus more susceptible to convergence to inconsistent solutions. Because simulations cannot be reliably continued from inconsistent solutions, the critical timestep is a theoretical upper bound for valid timesteps.

Numerical approximations of differential equations are maps with domains strict subsets of the underlying space of independent variables i.e. partial maps. Such spaces admit geometric topologies defined solely in terms of the topologies of the underlying spaces. It is shown that one such, the Fell topology, is natural for the numerical analysis of differential equations on differentiable manifolds.

The linearization of equilibria of Hamiltonian systems is Hamiltonian; this has well-known and important implications for the spectrum. The analogous statement for nonholonomic systems is provided. It follows, for example, that the linearization of the ground state of a nonholonomic system is always Hamiltonian.

Nonholonomic systems are variational models commonly used for mechanical systems with ideal no-slip constraints. This note provides a differential-geometric derivation of the nonholonomic equations of motion for an arbitrary rigid body rolling on an arbitrary surface, via the semi-symplectic formalism, and in terms of shape operators (a.k.a. Weingarten maps). By a semi-symplectic reduction, the well-known differential equations in the case where the surface is a horizontal plane are shown to be semi-symplectic.

In the formalism of constrained mechanics, such as that which underlies the SHAKE and RATTLE methods of molecular dynamics, we present an algorithm to convert any one-step integration method to a variational integrator of the same order The one-step method is arbitrary, and the conversion can be automated, resulting in a powerful and flexible approach to the generation of novel variational integrators.

The mechanical system of two disks, moving freely in the plane, while in contact and rolling against each other without slipping, may be written as a Lagrangian system with three degrees of freedom and one holonomic rolling constraint. We derive simple geometric criteria for the rotational relative equilibria and their stability. Extending to three dimensions, we derive the kinematics of the analogous system where two spheres replace two disks, and we verify that the rolling disk system occurs as a holonomic subsystem of the rolling sphere system.

Lagrangian systems with constraints are common models of fundamental or idealized physical systems. Holonomic constraints, typified in the example of a freely moving rigid body, give rise to systems with certain special properties, such as symplectic Hamiltonian systems. Nonholonomic constraints arise in such systems as a disk which rolls without slipping. Holonomic and nonholonomic systems have differing mathematical structures, and they have different behaviors [1, 3, 4, 9, 21, 22, 24]. For example, when a vehicle has good contact with a road, then it is behaving as a nonholonomic system. Angular momentum is not conserved; otherwise, the vehicle could not be steered into a turn. Under icy conditions, the vehicle is essentially a holonomic system; then steering cannot change its angular momentum, the vehicle cannot be turned, and whatever spin it has will persist. Conservation of energy, however, is a dominant feature of both systems. A rattleback is a toy top in the shape of long, narrow boat, with a slight, usually imperceptible asymmetry, either in its shape, or in its mass distribution. Many people anticipate that, when spun on a table, the rattleback will behave as other tops do i.e. holonomically. And, when spun in one direction, the rattleback will behave like this. When spun in the opposite direction, rattlebacks will spontaneously reverse direction, exhibiting nonholonomic non-conservation of angular momentum. As it turns out, because of the asymmetry, the table and the rattleback are coupled nonholonomically. Angular momentum is not conserved: some time after spinning in the unstable direction, the rattleback is observed to be spinning, at nearly the same rate, in the opposite direction. The transition between the two spins is dynamically complicated: it occurs through a non-spinning longitudinal wobbling motion. This is the rattleback’s spin reversal. Rattlebacks have been observed, and the basic mathematical model obtained, for over a century [26, 27], and the have been researched off-an-on since then [2, 5, 6, 7, 8, 15, 16, 17, 19, 20, 23, 25, 28]. But the spin reversal is a global dynamical feature, and its understanding is incomplete. Part of the problem is the sheer complexity of the system. The (reduced) rattleback equations of motion, for a body with surfaceM rolling on the plane, are

An error analysis of variational integrators is obtained, by blowing up the discrete variational principles, all of which have a singularity at zero time-step. Divisions by the time step lead to an order that is one less than observed in simulations, a deficit that is repaired with the help of a new past–future symmetry.

Discretizations of variational principles of physical systems are towards discrete models that have a theoretical status equivalent to the continuous models. Practically, such variational discretizations lead to a class of geometric numerical integrators, called variational integrators. To construct variational discretizations of Hamilton's principle of mechanics, we develop geometric discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. In the formalism of constrained mechanics, such as that which underlies the SHAKE and RATTLE methods of molecular dynamics, we develop a method to convert any one-step integrator to a variational integrator of the same order. Existence and uniqueness, and accuracy, of variational integrators, require due consideration of singularities at zero time-step. We show existence and uniqueness for variational integrators by blowing up the variational principle at zero time-step. The straight-forward computation gives an accuracy one less than is observed in simulations, a deficit that is recovered by a past-future symmetry of the blown-up principle.

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vector. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.

In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy–momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between them. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced instability of KAM stable relative equilibria for this system.

The geometry of constrained Lagrangian systems is developed using the Lagrange-d'Alembert principle, extending the variational approach of Marsden, Patrick and Shkoller [26] from holonomic to nonholonomic systems. It emerges that the instrinsic geometry of nonholonomic systems corresponds to the geometry of the distributional Hamiltonian systems of Sniatycki [35], here called semi-Hamiltonian. The principle physical reason that nonholonomic systems exhibit non-symplectic dynamics is exposed, leading to curvature conditions for the presence of holonomic subsystems. The underlying geometry of semi-Hamiltonian systems is semi-symplectic. An abstract exposition of the semi-symplectic category is developed. This is closed under a reduction scheme able to incorporate conserved quantities which are not momenta but are often structurally implied by symmetry. The variational development is continued to include nonlinear constraints irrespective of whether or not they are obtained from Chetaev's rule. Even though these systems are not semi-Hamiltonian, their geometry is still semi-symplectic.

Skew critical problems occur in continuous and discrete nonholonomic Lagrangian systems. They are analogues of constrained optimization problems, where the objective is differentiated in directions given by an apriori distribution, instead of tangent directions to the constraint. We show semiglobal existence and uniqueness for nondegenerate skew critical problems, and show that the solutions of two skew critical problems have the same contact as the problems themselves. Also, we develop some infrastructure that is necessary to compute with contact order geometrically, directly on manifolds.

Relative equilibria of symmetric Lagrangian systems correspond to critical points of a function on configuration space called the amended Lagrangian. This is proved by calculating the action on orbits of one parameter subgroups of the symmetry group.

A variational proof is provided of the existence and uniqueness of evolutions of regular Lagrangian systems.

We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum Methods. Using a topological generalisation of Lyapunov’s result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point. This criterion is applied to generalise the energy-momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a G-stable relative equilibrium satisfies the stronger condition of being A-stable, where A is a specific group-theoretically defined subset of G which contains the momentum isotropy subgroup of the relative equilibrium. The results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid.

Purely rotational relative equilibria of an ellipsoidal underwater vehicle occur at nongeneric momentum where the symplectic reduced spaces change dimension. The stability of these relative equilibria under momentum changing perturbations is not accessible by Lyapunov functions obtained from energy and momentum. A blow-up construction transforms the stability problem to the analysis of symmetry-breaking perturbations of Hamiltonian relative equilibria. As such, the stability follows by KAM theory rather than energy-momentum confinement.

A Lagrangian is derived for the motion small filaments and spheres moving within a Riemannian manifold. The motion of a small sphere generates parallel translation along the curve traced by its center.