A review on extension of Lagrangian-Hamiltonian mechanics

This paper presents a brief review on Lagrangian-Hamiltonian Mechanics and deals with the several developments and extensions in this area, which have been based upon the principle of D'Alambert or the other. It is not the intention of the authors to attempt to provide a detailed coverage of all the extensions of Lagrangian-Hamiltonian Mechanics, whereas detailed consideration is given to the extension of Noether's theorem for nonconservative systems only. The paper incorporates a candid commentary on various extensions including extension of Noether's theorem through differential variational principle. The paper further deals with an extended Lagrangian formulation for general class of dynamical systems with dissipative, non-potential fields with an aim to obtain invariants of motion for such systems. This new extension is based on a new concept of umbra-time, which leads to a peculiar form of equations termed as 'umbra-Lagrange's equation'. This equation leads to a simple and new fundamental view on Lagrangian Mechanics and is applied to investigate the dynamics of asymmetric and continuous systems. This will provide help to understand physical interpretations of various extensions of Lagrangian-Hamiltonian Mechanics.