PX285 - B3 - newtonian mechanics - the principle of least action

the principle of least action

in newtonian mechanics, 'classical' trajectories are those that extremize the action, which is called 'the principle of least action' :

\begin{aligned} A & = \int_{t_{0}}^{t_{1}} L d t \\ where, L & = T - V \end{aligned}

this can be used to rederive newton's laws, and to formulate a new, more general, statement of mechanics

modelling the trajectory of a particle

a particle moving along a trajectory $(x_{0}, t_{0}) \to (x_{1}, t_{1})$
let, $(x_{0}, t_{0}) = (0, 0)$ , and $(x_{1}, t_{1}) = (X, τ)$
the particle can take multiple paths, one with a constant velocity, many with varying velocities
parameterizing the paths using one scalar parameter, $b = c o n s t a n t$ , an equation is formulated, which represents the 'candidate' paths:

x (t) = \frac{t}{τ} X + b (τ - t) t

checking whether the equation starts and finishes at the correct fixed points: $x (0) = 0$ ; $x (τ) = X$
$b$ allows inclusion of acceleration

a falling particle

for a mass, $m$ , falling under gravity from a height, $x$ , the lagrangian is:

L = T - V = \frac{1}{2} m v^{2} - m g x

using the candidate paths equation:

\begin{aligned} x (t) & = \frac{t}{τ} X + b (τ - t) t \\ ⟹ v = \dot{x} (t) & = \frac{X}{τ} + b (τ - 2 t) \end{aligned}

using the principle of least action:

\begin{matrix} A (b) = \int_{t_{0}}^{t_{1}} [\frac{1}{2} m {[\frac{X}{τ} + b (τ - 2 t)]}^{2} - m g [\frac{t}{τ} X + b (τ - t) t]] d t \\ ⋮ \\ A (b) = \frac{m}{2} (\frac{b^{2} τ^{3}}{3} - \frac{b g τ^{3}}{3} - g τ X + \frac{X^{2}}{τ}) \end{matrix}

the action has an extremum (here, a minimum) when:

\begin{matrix} \frac{d A}{d b} = \frac{m τ^{3} (2 b - g)}{6} = 0 \\ ⟹ b = \frac{g}{2} \end{matrix}

therefore, it is found that:

\begin{aligned} x (t) & = \frac{t}{τ} X + \frac{g}{2} (τ - t) t = v_{0} t + \frac{1}{2} g t^{2} \\ ⟹ v & = \dot{x} (t) = \frac{X}{τ} + \frac{g τ}{2} + g t \end{aligned}