Related papers: Lie Group Spectral Variational Integrators
We show that symplectic Runge-Kutta methods provide effective symplectic integrators for Hamiltonian systems with index one constraints. These include the Hamiltonian description of variational problems subject to position and velocity…
The discrete gradient approach is generalized to yield integral preserving methods for differential equations in Lie groups.
We study the integrability of Poisson and Dirac structures that arise from quotient constructions. From our results we deduce several classical results as well as new applications. We also give explicit constructions of Lie groupoids…
This paper formulates variational integrators for finite element discretizations of deformable bodies with heat conduction in the form of finite speed thermal waves. The cornerstone of the construction consists in taking advantage of the…
We view disentanglement learning as discovering an underlying structure that equivariantly reflects the factorized variations shown in data. Traditionally, such a structure is fixed to be a vector space with data variations represented by…
We present a novel Type II variational principle on the cotangent bundle of a Lie group which enforces Type II boundary conditions, i.e., fixed initial position and final momentum. In general, such Type II variational principles are only…
In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian defined on QxQ, where Q is the configuration manifold, and a…
This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation…
We suggest a numerical integration procedure for solving the equations of motion of certain classical spin systems which preserves the underlying symplectic structure of the phase space. Such symplectic integrators have been successfully…
Elegant integration schemes of second and fourth order for simulations of rigid body systems are presented which treat translational and rotational motion on the same footing. This is made possible by a recent implementation of the exact…
The article considers smooth optimization of functions on Lie groups. By generalizing NAG variational principle in vector space (Wibisono et al., 2016) to Lie groups, continuous Lie-NAG dynamics which are guaranteed to converge to local…
We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine…
We consider the continuous and discrete-time Hamilton's variational principle on phase space, and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete and continuous Hamiltonian mechanics. The…
This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that…
Locally exact integrators preserve linearization of the original system at every point. We construct energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional canonical Hamiltonian systems by modifying…
Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study…
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie…
We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous…
In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of…
As is known that various dynamical systems including all Hamiltonian systems preserve volume in phase space. This qualitative geometrical property of the analytical solution should be respected in the sense of Geometric Integration. This…