In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite.
In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by Newton's laws of motion.
In such systems, time evolution can also refer to the change in observable values.
Consider a system with state space X for which evolution is deterministic and reversible.
In classical mechanics, the propagators are functions that operate on the phase space of a physical system.
[1] A state space with a distinguished propagator is also called a dynamical system.
To say time evolution is homogeneous means that In the case of a homogeneous system, the mappings Gt = Ft,0 form a one-parameter group of transformations of X, that is For non-reversible systems, the propagation operators Ft, s are defined whenever t ≥ s and satisfy the propagation identity In the homogeneous case the propagators are exponentials of the Hamiltonian.
In the Schrödinger picture, the Hamiltonian operator generates the time evolution of quantum states.