This study proposes to improve the Lagrangian energy equations for instructed complex jet bundles on Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated and applications to complex bundle structures are carried out. The energy equations have been applied to the numerical example in order to test its performance. Moreover, velocity and time dimensions for energy movement equations have been presented as a new concept. This study shows some physical applications of those equations and interpretations are made. Results show that the imaginary coordinates are same as real coordinates. One of the interesting conclusion of this study is that the Lagrangian energy of a movement particle is static when the particle moves in a large velocity.

The aim of this paper is to improve Hamiltonian energy equations for the complex jet bundles using Kähler manifolds. The coordinates on the bundle structure of Kähler manifolds have been given for real and imaginary dimensions. For given bundle structures, all fundamental geometrical properties have been investigated in Hamiltonian energy equations and applications to complex bundle structures. The improved Hamiltonian energy equations have been applied to the presented example in order to test its performance. Moreover, we have presented a new concept of velocity and time dimensions for energy movement equations. Results showed that imaginary Hamiltonian values are opposed to real Hamiltonian values in 2n dimensions. Finally, this study showed a physical application and interpretation of velocity and time dimensions in Hamiltonian energy equations for given examples.