Nonlocal Linear Second-Order "equations of Motion" and Their Balance Laws and Conservation Laws: a Technique Bypassing Lagrangians.
Nonlocality is a concept which has acquired increasing importance for physical theories, as witnessed by its application to continuum mechanics, Fokker-type interactions, and quantum mechanics. Mathematically, nonlocality usually is expressed by means of integro-differential equations. Frequently, nonlocal theories start from general variational principles, which, using a Lagrangian that contains functionals, lead to nonlocal Euler equations --which are called the "equations of motion" for a particular physical system, although they do not necessarily describe physical motion. It is also known how use of the nonlocal Lagrangian can result in the formulation of quantities, the energy-momentum complex, which are related to nonlocal balance laws, continuity equations or conservation laws. However, it is desirable to relate the equations of motion to these quantities and their associated laws and equations more directly. The main result of this thesis has been to obtain such relations for a subclass of nonlocal equations of motion; relating linear second-order integro-differential equations of motion to nonlocal balance laws, continuity equations or conservation laws without explicitly introducing a Lagrangian. The results can be applied to physical systems whose nonlocal equations of motion may be relativistic and either classical or quantum. They can also be applied to obtain global conservation laws for such quantities as energy, momentum, angular momentum, probability and charge. Applications of the formalism are given for both local and nonlocal equations of motion. They are chosen from elasticity theory, nonrelativistic quantum mechanics, and relativistic field theory. The incorporation of boundary conditions is described, and some limitations of the formalism are discussed.
- Pub Date:
- Physics: General