Editing Energy (section)

=== Classical mechanics ===
{{Classical mechanics}}
{{Main|Mechanics|Mechanical work|Thermodynamics}}

In classical mechanics, energy is a conceptually and mathematically useful property, as it is a [[conserved quantity]]. Several formulations of mechanics have been developed using energy as a core concept.

[[Work (physics)|Work]], a function of energy, is force times distance.
: <math> W = \int_C \mathbf{F} \cdot \mathrm{d} \mathbf{s}</math>

This says that the work (<math>W</math>) is equal to the [[line integral]] of the [[force]] '''F''' along a path ''C''; for details see the [[mechanical work]] article. Work and thus energy is [[frame dependent]]. For example, consider a ball being hit by a bat. In the center-of-mass reference frame, the bat does no work on the ball. But, in the reference frame of the person swinging the bat, considerable work is done on the ball.

The total energy of a system is sometimes called the [[Hamilton's equations|Hamiltonian]], after [[William Rowan Hamilton]]. The classical equations of motion can be written in terms of the Hamiltonian, even for highly complex or abstract systems. These classical equations have direct analogs in nonrelativistic quantum mechanics.<ref>[https://web.archive.org/web/20071011135413/http://www.sustech.edu/OCWExternal/Akamai/18/18.013a/textbook/HTML/chapter16/section03.html The Hamiltonian] MIT OpenCourseWare website 18.013A Chapter 16.3 Accessed February 2007</ref>

Another energy-related concept is called the [[Lagrangian mechanics|Lagrangian]], after [[Joseph-Louis Lagrange]]. This formalism is as fundamental as the Hamiltonian, and both can be used to derive the equations of motion or be derived from them. It was invented in the context of [[classical mechanics]], but is generally useful in modern physics. The Lagrangian is defined as the kinetic energy ''minus'' the potential energy. Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (such as systems with friction).

[[Noether's theorem]] (1918) states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalisation of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.