Extension of Lagrangian-Hamiltonian mechanics: Umbra Poisson bracket using bond graphs

In mathematical as well as in classical mechanics, the Poisson brackets are one of the binary operation properties in Hamiltonian Mechanics, which generally govern the Hamiltonian dynamics system. The present paper deals with the development of umbra-Poisson bracket for extended Lagrangian-Hamiltonian Mechanics, where a new time of umbra is applied in extended form and umbra-Lagrangian is obtained through bondgraphs. Some significant insight of the system has been achieved through some useful theorems of Poisson Brackets. It also proves that if Hamiltonian does not depend explicitly on time, the time derivative of umbra-Hamiltonian is zero. In all such analysis, the umbra-Lagrangian is generated through bond-graphs as it provided the interior as well as exterior information of the system.

Further, it is also be proved that if any dynamical system with internal, compliant, dissipative, gyroscope elements and external source with sufficiently smooth real time variations of parameters has a umbra-Hamiltonian, then each of its component umbra-Poisson bracket with itself will be zero for all real time.