Maybe it's useful to think this question from the Hamiltonian formalism. The Hamiltonian $\cal{H}$ of a system is, in general, the total energy of the system, expressed with a particular set of variables. For example, for a point particle moving in the presence of a constatn gravitational field, the energy is $$E = \frac{mv^2}{2} + mgx$$ but these variables are not suitable for expressing the Hamiltonian. Instead of the velocity, the apropriate variable is momentum $$\cal{H} = \frac{p^2}{2m} + mgx$$ This choice is not random, but it's based on certain relations between momentum and position. In general, two variables meeting these relations are called canonical coordinates, and can be thought as one representing a "position" $\mathbb{q}$ and the other a "momentum" $\mathbf{p}$ (eg: position $x$ and linear momentum $p$, angle $\theta$ and angular momentum $L$, etc.)
Based on this, the equations of movement of the particle can be found from the <PERSON> equations $$\frac{d\mathbf{p}}{dt} = -\frac{d\cal{H}}{d\mathbf{q}},\;\;\frac{d\mathbf{q}}{dt} = +\frac{d\cal{H}}{d\mathbf{p}}$$
Let's use this in the simplest case possible: a point mass moving with velocity $v$ in free space, without the action of any force. 499
The Hamiltonian in this case is simply $\cal{H}=p^2/2m$, and applying <PERSON> equations $$\frac{dp}{dt} = -\frac{d\cal{H}}{dq} = 0,\;\;\frac{dq}{dt} = +\frac{d\cal{H}}{dp} = \frac{p}{m}$$
Translating this to "normal" variables $q\rightarrow$ position, $p\rightarrow$ momentum, the second equation reads "the variation of position is the velocity", which sounds pretty reasonable, and the first is "the velocity is constant", as we already knew. However, from the first equation you also get the conservation of momentum. This is caused by the Hamiltonian being independent of the position, which can be thought as "no matter where you are in the space, the Hamiltonian remains the same". This "no matter where you are" is called a symmetry in the Hamiltonian, and represent the validity of the equations representing the system, even if you move it anywhere in the universe (where the conditions are still met). All the conservation laws can be represented as symmetries in the Hamiltonian, always involving two related canonical coordinates (position and linear momentum, angle and angular momentum, time and energy, etc.). The formalism of this is the <PERSON> theorem. 499