Classical Non-Relativistic Particle

Let us consider a non-relativistic particle of mass m, moving non-relativistical in a flat (D-1)-dimensional space ${\mathbb{R}}^{D-1}$ with orthonormal coordinates $\vec{x}=\{x^1,...,x^{D-1}\}$. The action principle is

$S=\frac{1}{2}m\int dt|\dot{\vec{x}}{|}^2$.

The canonical momenta are $\vec{p}=\frac{\partial \mathcal{L}}{\partial \dot{\vec{x}}}=m\dot{\vec{x}}$ and the energy is $E=\dot{\vec{x}}.\vec{p}-\mathcal{L}=\frac{|\vec{p}{|}^2}{2m}$.

The change in action is $\delta S=-m\int dt\delta \vec{x}.\ddot{\vec{x}}=-\int dt\delta \vec{x}.\dot{\vec{p}}$. Thus, the equations of motion are $\dot{\vec{p}}=0$.