Mechanics is the branch of physics that studies the motion of particles or large bodies. Currently, mechanics can be divided into two main branches: classical and quantum mechanics. The two branches take different mathematical approaches reflecting the different philosophies, though some of the tools overlap. Further, each type of mechanics can be done using more than one mathematical route to get to the same answer. In mechanics, the "answer" comes in the solution of the equations of motion, which describe the motion of the particle with respect to time.
Classical mechanics uses traditional calculus in either of its formulations. It assumes that the behavior of particles is deterministic. The method of solution usually taught in elementary physics is the solution of the force equation. Using Newton's F = dp/dt, where F is the vector sum of the forces on the particle and p is the momentum vector, a second-order differential equation is obtained, which is then solved to determine the position over time. The Lagrangian method for solving mechanics problems deals with mechanics in an energy formulation, making the quantities calculated scalars. The Hamiltonian method even further reduces complications by presenting a first-order scalar differential equation. The classical formulation of mechanics is valid when the object in question is sufficiently large, or as an average result of quantum mechanical situations.
Quantum mechanics, on the other hand, is the science of the very small levels of matter and their behavior. Unlike the macroscopic situations we observe with the naked eye, quantum mechanical situations behave probabilistically. That is, even if all initial conditions are known, only probabilities may be stated about the particle's further behavior. However, quantum mechanics provides several alternatives for calculating this probable behavior. They are mathematically equivalent but often involve quite different procedures in arriving at the results. The three main options in quantum mechanics are Schrödinger's integral formulation, Heisenberg's matrix formulation, and Feynman's sum-over histories. While the consequences of this theory are often counterintuitive, they have been proven to work on the physical level, and approximations demonstrate that quantum behavior will "average out" to classical behavior.
While much of mechanics is already solved, it is still necessary and useful for students of physics to see how things are moving and what they can expect of basic force-influenced motion in experiments on other topics. There are also elements of mathematics that tie in with current "hot" topics in mechanics, notably chaos theory. In the real world, chaotic equations often express the situation most accurately, complicating the physics and making it more interesting to physicists.
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