Ammm:Md ham

Derivation of equation of motion from Lagrangian and Hamilton formulation of classical mechanics.

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).

Lagrangian

Lagrangian mechanics combines conservation of momentum with conservation of energy. It was introduced by Lagrange in 1788.

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:

$S=\int _{tb}^{te}dt\left[K-U\right]$

Define ${\mathfrak {L}}\equiv K({\dot {q}})-U(q)$

define a path

$q(t)={\bar {q}}(t)+\eta (t)$
${\dot {q}}={\dot {\bar {q}}}+{\dot {\eta }}(t)$
η(t) is zero at b and e (boundaries)

q is the general coordinate. ${\dot {q}}$ is the time derivative.

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

$\int u\,dv=uv-\int v\,du.\!$

$\int _{tb}^{te}{\frac {d{\mathfrak {L}}}{d{\dot {q}}}}d\eta =$

U(x) is independent of ${\dot {x}}$

Hamiltonian

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