Spatial translation invariance implies conservation of linear momentum, while rotational invariance implies conservation of angular momentum. Time-translation invariance implies conservation of energy when the force derives from a time-independent potential.

For a particle obeying $m\mathbf{a}=\mathbf{F}$, dotting with $\mathbf{v}$ gives $dK/dt=\mathbf{F}\cdot\mathbf{v}$. If $\mathbf{F}=-\nabla V$, then $K+V$ is conserved: $$E=\frac12 mv^2+V(\mathbf{r}).$$