A New Concept in Mechanics Based on the Notions of Space, Time, and Energy

A new concept in mechanics is proposed based on the notions of space, time, and energy. The energy is represented by the sum of thirteen terms expressed as products of invariants of the Lagrangian equations of motion and scalar factors corresponding to the physical properties of materials. Differential equations of motion and equilibrium are derived from the frame indifference condition for energy, in which the Eulerian variables are the sought functions and the Lagrangian variables are the arguments. Assumptions are made under which these equations can be transformed into the Poisson and Laplace equations. The law of conservation of energy is used to obtain dependences between the Lagrangian stresses and strains in the reversible deformation region; they are compared with the expressions used in the theory of elasticity. It is discussed whether the point of reference for the mean stresses may be chosen with regard to the volume energy density of particles in their initial state, as well as whether elastic deformation can be described using a single constant. As an example, the energy model and the equations of motion in the Lagrangian form are applied to describe the transition from reversible to irreversible deformation. The developed model differs from the classical one by using two independent infinitesimal operators for the time and space. It is shown that Newton’s law of inertia can be regarded as a variant for the determination of the generalized forces characterizing the kinetic energy change in a body with increments in the distance between the origin of the observer coordinate system and the center of mass of the body. The application of the Lagrangian variables and the superposition principle for describing any spatial motions (including for absolutely rigid bodies), which can be used in the dynamic analysis of linkages and other mechanisms, is validated. The multiple choice of generalized forces for absolutely rigid bodies, including when passive forces arise, is considered. The method of dynamic analysis of mechanisms developed on the basis of the energy model allows the law of conservation of energy to be satisfied for any part of the studied system in an arbitrary time interval.