Submersible vehicles may be regarded as complex systems because of their complex interaction with the surrounding fluid. This paper presents a mathematical model of a submersible vehicle formulated in a coordinate-free manner through the language of Lie groups and Lie algebras. The d'Alembert virtual-work principle was applied in conjunction with the minimal-action principle for a rigid body in order to incorporate into the mathematical model external influences such as fluid-current-induced deflection and control inputs. Such a method from mathematical physics can also take into consideration how a vehicle interacts with the fluid it is immersed in under the form of added (or virtual) mass. The resulting equations of motion were given over the Lie group of three-dimensional rotations as (non-pure) Euler-Poincaré relations. A numerical simulation technique based on Lie-group integrators was also briefly recalled and deployed to simulate the behavior of such mathematical model of an existing, academic-design-type submersible vehicle.

This paper presents a survey of a geometrically exact beam theory formulated within the framework of Lie groups, aimed at providing a mathematically consistent description of slender structures undergoing large displacements and rotations. The beam configuration is modeled as a curve in the space $\mathbb{S}\mathbb{O}(3)\times {\mathbb{R}}^3$, enabling a coordinate-free expression of the governing equations. A variational formulation serves as the basis for deriving the equations of motion, which emerge as nonstandard Euler-Lagrange equations on the configuration space. Strain measures arising from the group structure define the internal forces and moments, which couple to the dynamics via balance laws. The formulation automatically incorporates conservation of energy, linear momentum, and angular momentum, and reveals the underlying geometric structure through the appearance of Lie brackets in the angular momentum equation. This framework emphasizes the connection between geometry and mechanics, offering advantages in both physical fidelity and computational stability.

Riemannian cubics are critical points of the total squared norm of the accelerations of the curves on Riemannian manifolds with given end-points and end-velocities. Riemannian cubic splines are $C^2$ track sums of Riemannian cubics. Determining Riemannian cubics with given boundary data is equivalent to solving fourth-order ordinary differential equations with boundary conditions, while finding Riemannian cubic splines is tantamount to solving boundary and interior value problems, which are both hard in practice. The present paper proposes Riemannian gradient-based methods to tackle the former problem by taking advantage of the variational principle and the shooting method. The core idea is to add a number of junctions and associated velocities between given boundary points and to adjust such junctions and associated velocities according to a variational principle. Based on the obtained Riemannian cubics, Riemannian cubic splines are determined by constructing piecewise Riemannian cubics of class $C^2$ at interior points. The effectiveness of the proposed method is assessed by numerical experiments on a unit sphere.