Do angular equivalents of linear equations in physics reveal extra information?

Question

First of all, This question is completely based on intuition and some concepts of mathematics. I have been thinking about this now for 5 months and haven't figured it out yet. I am beginner in physics and about to complete Classical Physics Course. But I have always seen two forms of Newton's laws, one that applies to linear motion and other which applies to Rotational motion. There are several comparable variables and equations as well such as Torque for Force, etc. Now we also know from mathematics that one equation can provide a constraint for one variable only ( By constraint I mean solution or limiting the possible values) and once a particular equation has been used to limit the values of a variable then it has revealed all information which it could and can not be further used to reveal extra information. For example,

$$ a+b = 10$$ Now this equation can only reveal single information and can limit the values of either only $a$ or $b$ in terms of each $a$ and $b$, but then once this info is used up this equation then can not reveal any extra information no matter whatever form it is written in.

For example, all the following equations are different forms of above equation but will not reveal any extra information if used with the above given equation

$$ a = 10 - b $$ $$ a/10 + b/10 = 1 $$ $$ ca +cb = 10c$$

Now using this concept, if angular equivalents of linear equation in physics are really just another form of linear equations then how do they sometime reveal extra information eventhough their linear counterpart has already been used up in the solving of problem. To clarify, I have seen several problems where both linear equations are used and then Angular equations are used and Angular ones reveal extra information with their linear counterparts which mathematically tells us that these angular equations can not be any form of linear equations but are completely different equations having their unique identity. Think of comparison between Angular Momentum Conservation Equation and Linear Momentum Conservation Equation For Example.

Answer 1

There is a setup that offers quite a good window onto the relation between linear mechanics and angular mechanics.

That setup is uniform circular motion.

Uniform circular motion can be through of as a single motion, with a uniform rate of change, but at the same time you have the option of decomposing the motion into to perpendicular components.

The components are harmonic oscillation. That is, uniform circular motion can also be thought of as a composition of two harmonic oscillations, perpendicular to each other.

We have for position vectors, velocity vectors, and acceleration vectors alike that they can always be composed or decomposed according to the rules of vector composition in Euclidean space. That composition/decomposition property is enormously powerful.

Whenever dealing with angular mechanics it is a good exercise to work out how the motion can be decomposed in motion components. Independent of whether angular mechanics or linear mechanics is better suited for the case at hand; seeing how the motion can be decomposed will always be helpful for understanding.

All instances of angular mechanics have in common that the minimum of spatial dimensions needed for the motion to happen at all is two spatial dimensions.

As to linear mechanics, here is a way to see that when linear mechanics is applicable, the case can always be narrowed down to one spatial dimension.

Take the case of two billiard balls, both are moving along the cloth, they collide, and proceed on after that collision.

Of course, if you use the frame of the table as reference of the motion of the billiard balls then it was motion in two spatial dimensions. However, choice of inertial coordinate system is arbitrary.

You have the option of using an inertial coordinate system such that the origin of that inertial coordinate system coincides with the common center of mass of the two billiard balls.

Next you fix the orientation of the inertial coordinate system by aligning one of the axes of the inertial coordinate system with the line that connects the two billiard balls. When using that particular coordinate systeme the motion of the two billiard balls is motion along a single spatial dimension.

Summerizing:
In terms of Newtonian mechanics angular mechanics is not distinct. It is not distinct in the sense that all angular motions can be represented as a combination of linear motions.

It is a case of linear mechanics when the motion is such that there is a transformation to an inertial coordinate system such that the motion ends up being motion along a single spatial dimension.

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LATER EDIT

The following diagram expresses Newton's first law, and it also expresses an area law. Newton used that area law for his derivation of Kepler's law of areas from first principles.

First we express that space and time are uniform and correlated by asserting a rule of proportion:

In the absence of a force that changes direction of motion an object will move along a straight line, covering equal intervals of distance in equal intervals of time.

In the diagram the positions A, B, C, D, and E are spaced equally.

Let the point S be the origing of an inertial coordinate system. For every member of the equivalence class of inertial coordinate systems we have the following:

For any such point S the following area law obtains:
In the absence of a force that changes direction of motion an object will sweep out equal areas in equal intervals of time.

As first noticed by Kepler, and derived from first principles by Newton, when there is a force that is continuously directed towards point S then - when using point S as the origing of the inertial coordinate system - the rule of sweeping out equal areas in equal intervals of time still obtains.

This shows unequivocally that in newtonian mechanics linear momentum and angular momentum are fundamentally connected.

Geometrically:
Linear momentum is naturally represented with a one-dimensional geometric entity: a vector
Angular momentum is naturally represented with a two-dimensional geometric entity: a bi-vector (A bi-vector has the property of having an area.)

See also:
My answer to a question on why Angular momentum is represented the way it is

Answer 2

Your basic confusion comes from the fact that in Physics are not only important mathematical form of equations, but measurement units as well. Consider primitive case of linear speed definition $$v = \dot x$$ and angular counterpart: $$ \omega = \dot \theta $$

Mathematically both equations will have same form. However first one measures how fast object travels a linear distance, which is measured in $[m/s]$, while second measures how fast object rotates, which is measured in $[rad/s]$. So they represents different features of physical world processes. Physics is not about plain math, but rather about explanation about how world operates using math equations as a tool. Almost all equations in Physics must be validated by experiments, this is why we need measurement units.

Given this I hope you can now understand why we need linear and angular Newton second law forms: $$ F=ma \\ \tau = I \alpha $$ Or linear and angular momentum conservation : $$ \sum_i m_i v_i = \text {const} \\ \sum_i I_i \omega_i = \text {const} $$ Hint: There might be non-equilibrium systems where linear momentum is conserved, but angular - is not. Or vice-versa. In physics, one needs so much distinct descriptors which fully defines an operational system.

Answer 3

Mathematically speaking the two equations are linearly independent which means each is a different combination of the variables involved. As such you can solve for two variables with two scalar equations. Or solve for 6 variables with two sets of 3-vector equations.

As far as equations of motion goes, the linear equations that relate forces speak for the motion of the center of mass only. And the rotational equations that relate torques speak for the motion about the center of mass.