Physics using combined matrices? [closed]

Matrices, and the concept of applying one to another or multiplying them together is powerfull.

Binary Matrices can be multiplied to achieve encryption, decryption or pathfinding in the same way as matrix that are referred to as transformation matrix can be multiplied to combine many translation, rotation and other orientation operations all at once in any order, taking just the time required to perform one transformation matrix application, a constant time.

Possibly physics could be done with this principle, or even other types of calculations? For example multipling many physics transformation matrices together to create a new matrix that applies all the effects together in one operation?

Could example matrices be given, for example for a point in a 3D enviroment, where the enviroment contains three stationary elements, two that are points that generate an attractive force and the other being a solid object that can be collided with?

particle_vector' = particle_vector × (point1_attraction_matrix ×
point2_attraction_matrix × collide_with_solid_object_matrix)

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I don't quite see what the question is? As for multiplying matrices and all of them happening at once, it is natural. The matrix of composition transformation is the product of matrix of tranformations that were composed. –  user21436 Feb 21 '12 at 17:45
What does "Can Physics be done with this principle...?" even mean? –  user21436 Feb 21 '12 at 17:46
Gravity, other forces, momentum, collisions, etc... Analogous to how space can be transformed with translations and rotations. –  alan2here Feb 21 '12 at 17:50
Many principles in physics are non-linear or irreversible. So I would say no. –  Bernhard Feb 21 '12 at 18:33
Physicists had no use for matrices until Heisenberg used them to do quantum mechanics; physicists have been using matrices ever since. Despite Samuel Reid's informative answer, I must vote to close as "not a real question". –  Gerry Myerson Feb 21 '12 at 22:41

closed as not a real question by Kannappan Sampath, joriki, Gerry Myerson, t.b., Asaf KaragilaFeb 22 '12 at 10:45

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

You may be interested in the idea of a gauge group or in particular a similarity group, which for applications in physics, in short, could consist of some subgroups which all apply matrix operations.

More concretely, the similarity group $\textbf{$S_{d}$}$ is a non-abelian group with $\psi$ generators belonging to the translation subgroup $T_{d}$, rotation subgroup $SO(d)$ plus the constant scaling element $k$, where $d$ is the dimension. The general formula for the number of generators $\psi$ in the similarity group is given, where $d$ is the dimension as: $$\psi = \underbrace{d}_\text{translation} + \underbrace{\frac{d(d-1)}{2}}_\text{rotation} + \underbrace{1}_\text{scale}$$ Thus, the similarity group has 3 elements: the subgroups $T_{d}$ and $SO(d)$ plus the element $k$. To illustrate, the rotation subgroup $SO(3)$ has $\frac{3(3-1)}{2} = 3$ elements in it, and each of these elements consist of a rotation matrix defining one of the group parameters $\theta_{xy}, \theta_{yz}, \theta_{zx}$, while the generators of the subgroup $SO(3)$ define the capacity to rotate in the $xy, yz$ and $zx$ directions.

From this set-up, we can apply this group action to a system of particles and control the way it rotates, translates, scales relative to one another, etc. Or we can introduce some external "principle" or rule that the particles must follow and we use Calculus of Variations on a functional, or maybe simply equating many partial derivatives equal to zero.

Your idea is correct as we would explicitly represent, say the coordinates of some system of particles and have coefficients to these matrices to account for things like Coulombs constant or the universal gravitational constant (you could set this up). Multipy through by rotation matrices in each direction, for $d=3$, $$R_{x}(\theta)=\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \\ \end{array} \right]$$

$$R_{y}(\theta)=\left[ \begin{array}{ccc} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \\ \end{array} \right]$$

$$R_{z}(\theta)=\left[ \begin{array}{ccc} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \\ \end{array} \right]$$

Scaling a system of particles relative to each other would look like, $$k \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right]$$

While a translation row vector would look like adding,

$$\left[ \begin{array}{c} x\\ y\\ z\\ \end{array} \right]$$

In this way, if you determine the specific $\theta$, $k$ and $x,y,z$ translation values, you can simply add and multiply all these matrices together and then have your group "act" on your system of particles.

Hopefully you continue to look up some of this! If you are interested in some high-level applications, look up Julian Barbour's "Shape Dynamics", otherwise take a look at the wikipedia/mathworld articles for group action, calculus of variations, generators, etc.

For example, here is an actual code from Mathematica that I had to solve while finding the minimal potential energy of a changing system of particles using these ideas,

Throw this into Mathematica! \begin{code} {Solve[{!( *SubscriptBox[([PartialD]), ([Theta])]( *UnderoverscriptBox[([Sum]), (i = 1), (3)] *SuperscriptBox[(( *SubscriptBox[(W), (i)] - (((( *SubscriptBox[(x), (i)] k\ Cos[[Theta]])) + (( *SubscriptBox[(y), (i)] k\ Sin[[Theta]])) + c)))), (2)])) + (Subscript[Z, i] - ((-Subscript[x, i] k Sin[[Theta]]) + (Subscript[y, i] k Cos[[Theta]]) + d))^2 = 0, !( *SubscriptBox[([PartialD]), (k)]( *UnderoverscriptBox[([Sum]), (i = 1), (3)] *SuperscriptBox[(( *SubscriptBox[(W), (i)] - (((( *SubscriptBox[(x), (i)] k\ Cos[[Theta]])) + (( *SubscriptBox[(y), (i)] k\ Sin[[Theta]])) + c)))), (2)])) + (Subscript[Z, i] - ((-Subscript[x, i] k Sin[[Theta]]) + (Subscript[y, i] k Cos[[Theta]]) + d))^2 = 0, !( *SubscriptBox[([PartialD]), (c)]( *UnderoverscriptBox[([Sum]), (i = 1), (3)] *SuperscriptBox[(( *SubscriptBox[(W), (i)] - (((( *SubscriptBox[(x), (i)] k\ Cos[[Theta]])) + (( *SubscriptBox[(y), (i)] k\ Sin[[Theta]])) + c)))), (2)])) + (Subscript[Z, i] - ((-Subscript[x, i] k Sin[[Theta]]) + (Subscript[y, i] k Cos[[Theta]]) + d))^2 = 0, !( *SubscriptBox[([PartialD]), (d)]( *UnderoverscriptBox[([Sum]), (i = 1), (3)] *SuperscriptBox[(( *SubscriptBox[(W), (i)] - (((( *SubscriptBox[(x), (i)] k\ Cos[[Theta]])) + (( *SubscriptBox[(y), (i)] k\ Sin[[Theta]])) + c)))), (2)])) + (Subscript[Z, i] - ((-Subscript[x, i] k Sin[[Theta]]) + (Subscript[y, i] k Cos[[Theta]]) + d))^2 = 0}, {[Theta], k, c, d}] Solve[{!( *SubscriptBox[([PartialD]), ([Theta])](( *SuperscriptBox[((1 - ((((k\ Cos[[Theta]]))\ + \ c)))), (2)]\ + \ *SuperscriptBox[((1\ - \ (((((-1)\ k\ Sin[[Theta]])) + d)))), (2)]\ + \ *SuperscriptBox[((3 - ((((2\ k\ Cos[[Theta]])) + ((3\ k\ Sin[\ [Theta]])) + c)))), (2)] + *SuperscriptBox[((4 - (((((-2)\ k\ Sin[[Theta]]))\ + \ ((3\ \ k\ Cos[[Theta]]))\ + \ d)))), (2)]\ + *SuperscriptBox[((4 - ((((4\ k\ Cos[[Theta]]))\ + \ ((k\ Sin[\ [Theta]]))\ + \ c)))), (2)]\ + \ *SuperscriptBox[((2 - (((((-4)\ k\ Sin[[Theta]]))\ + \ ((k\ \ Cos[[Theta]])) + d)))), (2)]))) == 0, !( *SubscriptBox[([PartialD]), (k)](( *SuperscriptBox[((1 - ((((k\ Cos[[Theta]]))\ + \ c)))), (2)]\ + \ *SuperscriptBox[((1\ - \ (((((-1)\ k\ Sin[[Theta]])) + d)))), (2)]\ + \ *SuperscriptBox[((3 - ((((2\ k\ Cos[[Theta]])) + ((3\ k\ Sin[\ [Theta]])) + c)))), (2)] + *SuperscriptBox[((4 - (((((-2)\ k\ Sin[[Theta]]))\ + \ ((3\ \ k\ Cos[[Theta]]))\ + \ d)))), (2)]\ + *SuperscriptBox[((4 - ((((4\ k\ Cos[[Theta]]))\ + \ ((k\ Sin[\ [Theta]]))\ + \ c)))), (2)]\ + \ *SuperscriptBox[((2 - (((((-4)\ k\ Sin[[Theta]]))\ + \ ((k\ \ Cos[[Theta]])) + d)))), (2)]))) == 0, !( *SubscriptBox[([PartialD]), (c)](( *SuperscriptBox[((1 - ((((k\ Cos[[Theta]]))\ + \ c)))), (2)]\ + \ *SuperscriptBox[((1\ - \ (((((-1)\ k\ Sin[[Theta]])) + d)))), (2)]\ + \ *SuperscriptBox[((3 - ((((2\ k\ Cos[[Theta]])) + ((3\ k\ Sin[\ [Theta]])) + c)))), (2)] + *SuperscriptBox[((4 - (((((-2)\ k\ Sin[[Theta]]))\ + \ ((3\ \ k\ Cos[[Theta]]))\ + \ d)))), (2)]\ + *SuperscriptBox[((4 - ((((4\ k\ Cos[[Theta]]))\ + \ ((k\ Sin[\ [Theta]]))\ + \ c)))), (2)]\ + \ *SuperscriptBox[((2 - (((((-4)\ k\ Sin[[Theta]]))\ + \ ((k\ \ Cos[[Theta]])) + d)))), (2)]))) == 0, !( *SubscriptBox[([PartialD]), (d)](( *SuperscriptBox[((1 - ((((k\ Cos[[Theta]]))\ + \ c)))), (2)]\ + \ *SuperscriptBox[((1\ - \ (((((-1)\ k\ Sin[[Theta]])) + d)))), (2)]\ + \ *SuperscriptBox[((3 - ((((2\ k\ Cos[[Theta]])) + ((3\ k\ Sin[\ [Theta]])) + c)))), (2)] + *SuperscriptBox[((4 - (((((-2)\ k\ Sin[[Theta]]))\ + \ ((3\ \ k\ Cos[[Theta]]))\ + \ d)))), (2)]\ + *SuperscriptBox[((4 - ((((4\ k\ Cos[[Theta]]))\ + \ ((k\ Sin[\ [Theta]]))\ + \ c)))), (2)]\ + \ *SuperscriptBox[((2 - (((((-4)\ k\ Sin[[Theta]]))\ + \ ((k\ \ Cos[[Theta]])) + d)))), (2)]))) == 0}, {[Theta], k, c, d}] \end{code}

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