Least action principle

Hamilton's formulation of Lagrangian mechanics

Lagrange's equations of motion can alternatively derived as the stationary paths of the following path integral using the Euler-Lagrange equations:

S [L] = \int_{t_{1}}^{t_{2}} L (\vec{q}, \dot{\vec{q}}, t) d t = \int_{t_{1}}^{t_{2}} (T - V) d t

This path integral is commonly called the action, and this principle of defining the motion of the system is termed Hamilton's principle or (somewhat incorrectly) the least action principle. This is conceptually elegant because it frames the mechanics of all holonomic systems with forces derivable from potentials in terms of only coordinate-independent scalar quantities (kinetic and potential energies).

Furthermore, if there are $m$ holonomic constraints on the system,

f_{a} (\vec{q}, t) = 0, a = 1, 2, \dots m

we can either (i) pick an independent coordinate system of $q$ 's as above (more useful in general), or (ii) use Lagrange multipliers to explicitly include (and calculate) the constraints via the modified action integral:

S [L] = \int_{t_{1}}^{t_{2}} (L (\vec{q}, \dot{\vec{q}}, t) + \sum_{a = 1}^{m} λ_{a} f_{a} (\vec{q}, t)) d t

We solve this system by setting the $λ_{a}$ 's for some $m$ coordinates and using those values for the remaining $m - n$ coordinates in the $n$ equations of motion given by:

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{j}}) - \frac{\partial L}{\partial {\dot{q}}_{j}} = - \sum_{a = 1}^{m} λ_{a} \frac{\partial f_{a}}{\partial q_{j}} = Q_{j}

Here the $λ_{a}$ 's are treated as additional state variables and their variation yields the original constraint equations, so we get $n + m$ equations for $n + m$ variables. The $Q_{j}$ 's as defined here then correspond to the generalized constraint forces (upto a sign change).

The Hamiltonian

Following Beltrami's identity from Variational calculus, if $L$ doesn't explicitly depend on time, i.e., $\frac{\partial L}{\partial t} = 0$ , then

H =

Links

Sources

Classical Mechanics by Herbert Goldstein, Charles Poole and John Safko
Classical Mechanics (John R. Taylor)