the sum of the kinetic and potential energies of all particles in the system is called the
internal energy
title: Internal energy text: the reference state to the given state. It is the energy needed to create the given state of the system from the reference state. From a non-relativistic microscopic point of view, it may be divided into microscopic potential energy, , and microscopic kinetic energy, , components: The microscopic kinetic energy of a system arises as the sum of the motions of all the system's particles with respect to the center-of-mass frame, whether it be the motion of atoms, molecules, atomic nuclei, electrons, or other particles. The microscopic potential energy algebraic summative components are those of the chemical and nuclear particle title: Hamiltonian (quantum mechanics) text: physics. The Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system. For different situations or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation. By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system in the form where is the potential energy operator and is the kinetic energy operator in which "m" is title: Lagrangian mechanics text: all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The "non-relativistic" Lagrangian for a system of particles can be defined by where is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, and "V" is the potential energy of the system. Kinetic energy is the energy of the system's motion, and "v" = v · v is the magnitude squared of velocity, equivalent to title: Energy text: or lose energy in association with matter transfer (both of these process are illustrated by fueling an auto, a system which gains in energy thereby, without addition of either work or heat). Denoting this energy by formula_16, one may write Internal energy is the sum of all microscopic forms of energy of a system. It is the energy needed to create the system. It is related to the potential energy, e.g., molecular structure, crystal structure, and other geometric aspects, as well as the motion of the particles, in form of kinetic energy. Thermodynamics is chiefly concerned with changes in internal title: Hamiltonian mechanics text: coordinates , where each component of the coordinate is indexed to the frame of reference of the system. The time evolution of the system is uniquely defined by Hamilton's equations: </math> where is the Hamiltonian, which often corresponds to the total energy of the system. For a closed system, it is the sum of the kinetic and potential energy in the system. In Newtonian mechanics, the time evolution is obtained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time-evolutions of both position and velocity are computed. In contrast, in