### Hamiltonian

**(Hamiltonian function)**
(mathematical transform useful in mechanics)

**Hamiltonian** (or **Hamiltonian function**) is the name of a function
used in **Hamiltonian mechanics**,
consisting of the total energy of a closed system
(one in which energy is not at the time being exchanged with
the outside world):
its kinetic energy plus its potential energy.
Hamiltonian mechanics can solve classical mechanical problems and
was useful as a step toward statistical mechanics
and quantum mechanics.
The Hamiltonian function, ℋ*(q,p,t)*, of space coordinates
*q*, momentum coordinates *p* and time *t* satisfies:

*dp* ∂ℋ
—— = - ——
*dt* ∂*q*
*dq* ∂ℋ
—— = + ——
*dt* ∂*p*

(These are derived from Newton's laws.)
In a simple system, i.e., single particle traveling on a single dimension:

ℋ = *T* + *V*
where
*p*²
*T* = ——
2*m*
*V* = *V*(*q*)

*T* - kinetic energy.
*m* - mass of particle.
*V* - potential energy, a function of space-coordinates q.

(*mathematics,mechanics,function*)
**Further reading:**

https://en.wikipedia.org/wiki/Hamiltonian_mechanics

**Referenced by pages:**

Lie transform

perturbation theory

Index