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Derivation of equation of motion from Lagrangian and Hamilton formulation of classical mechanics.<ref name="frenkel">Frenkel and Smit, Understanding Molecular Simulations From algorithm to Applications.</ref>

<span>The Hamiltonian and Lagrangian formulations of classical mechanics yield
identical results. This is not surprising as the Hamiltonian formulation was
derived from the Lagrangian equations (see section A.2). Yet, the forms of
the Lagrangian and Hamiltonian equations of motion are quite different: the
Hamiltonian equations of motion arefirst-order differential equations for the
momenta and coordinates of all particles in the system. The Lagrangian
equations of motion are second-order equations of motion for the coordinates
only. The choice of formalism is dictated by considerations of convenience.
For instance, in order to derive the equations of motion of a system with
constraints, the Lagrangian formalism is more convenient (see 15.1). On the
other hand, the Hamiltonian expressions are to be used when establishing
the connection with statistical mechanics (see Chapter 2).</span>

== Lagrangian  ==

Lagrangian mechanics combines '''conservation of momentum with conservation of energy.''' It was introduced by Lagrange in 1788.<ref>http://en.wikipedia.org/wiki/Lagrangian_mechanics</ref> 

The Lagrangian formulation of classical mechanics is based on a variational principle. The trajectory of a classical system between time interval {tb, te}, an initial position xb and a final position xe, is the one for which the action, S, is a minimum: 

<math>S=\int_{tb}^{te}dt\left [ K - U \right ]</math> 

Define <math>\mathfrak{L}\equiv K(\dot{q})-U(q)</math> 

define a path 

 <math>q(t)=\bar{q}(t)+\eta(t)</math>
 <math>\dot{q}=\dot{\bar{q}}+\dot{\eta}(t)</math>
 <span class="texhtml">η(''t'')</span> is zero at b and e (boundaries)

q is the general coordinate. <math>\dot{q}</math> is the time derivative. 

[[Image:Lagrangian.png|580px]] 

The derivation of Eq. A.1.3 utilizes the partial integration 

<math>\int u\, dv=uv-\int v\, du.\!</math>

<math>\int_{tb}^{te}\frac{d\mathfrak{L}}{d\dot{q}}d\eta  =</math>


'''U(x) is independent of <math>\dot{x}</math>'''

== Hamiltonian  ==

Define <math>H(q,p)\equiv p\dot{q}-\mathfrak{L}(q,\dot{q},t)</math> which means H = K + U 

[[Image:Hamilton1.png]] 

[[Image:Hamilton2.png]] 

[[Image:Hamilton3.png]] 

== References  ==