- Theoretical Fluid Mechanics — vortex dynamics, specifically the dynamics of singular coherent vortical structures such as point vortices, vortex filaments and vortex patches.
- Dynamical Systems – geometric mechanics, the study of Lagrangian and Hamiltonian systems and of the role that geometry plays in the behavior of nonlinear systems.
- Nonlinear Control Theory – optimal control theory and geometric control theory and their applications to finite-dimensional systems
- Modeling, simulation and understanding of the complex nonlinear phenomena in the dynamics and control of coupled fluid-solid systems. The focus is on inviscid, incompressible flows–endowed with vorticity–dynamically interacting with rigid and deformable solids. Hamiltonian models of such problems are being currently investigated. Applications are to biomimetic locomotion and biological swimming problems (such as fish swimming and bird flight).
- Hamiltonian and dissipative models for interacting vortex structures in incompressible flows. Recent modeling work incorporates dissipative effects, which respect Navier-Stokes symmetry properties, into Hamiltonian models of singular coherent vortical structures such as point vortices and vortex filaments.
- Optimal control problems involving ideal fluids or/and rigid bodies: applications, mainly of Pontryagin’s Maximum Principle along with ideas in geometric control, to different problems.
- B.N. Shashikanth, Dissipative N-point-vortex models in the plane, Journal of Nonlinear Science, 20(1), 81-103, 2010.
- B. N. Shashikanth, A. Sheshmani, S. D. Kelly and M. J. Wei, Hamiltonian Structure and Dynamics of a Neutrally Buoyant Rigid Sphere Interacting with Thin Vortex Rings, Journal of Mathematical Fluid Mechanics, DOI 10.1007/s00021-008-0291-0, 2008.
- B. N. Shashikanth, A. Sheshmani, S. D. Kelly and J. E. Marsden, Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape, Theoretical and Computational Fluid Dynamics, vol. 22, pp.37-64, 2008.