# Maths

Discussion in 'General Advice' started by Imoyram, Nov 1, 2015.

1. ### spockandawesoft and woolen and writhing with curiosity

I wrapped up an mechanical engineering degree a few years back, so I'm rusty, but I've done actually well up through differential equations and system dynamics, and I've done at-least-I-passed well up through control systems, plus a lot of high level statistics work. Linear algebra was a while ago, but I was good at it, and I'd be willing to take a stab at anything you want to post! OH, and I just read your other post, dynamics was also a while ago, but I LOVED it, and I did well in that class! If you post any questions there, I'm more than happy to try helping! The proudest moment of my grad school was when we had a statics/dynamics question on a class final about loading on the human spine during a lift, and they never actually taught us that so my classmates were lost, but I nailed it because engineering and wrote a snarky mini essay about my units and assumptions :')

Last edited: Nov 2, 2015
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2. ### TheSeer37 Bright Visionary Crushes The Doubtful

I'm good at LinAlg and differential equations too, and complex analysis and a scattering of set theory, stats, and discrete math. What's dynamics? It sounds like physics, in which case WithAnH and I would be excellent help there also, but I thought you were studying pure math.

3. ### spockandawesoft and woolen and writhing with curiosity

It's a lot to do with physics! In terms of my classes, we started with statics, which was an analysis of the forces (and moments) acting on static bodies. So it started with things like you've got a car, how much force is exerted on each tire. And it expands out to things like bridge with trusses, how much force of this joint, things like that. Then the program branched into deformable bodies (I hated that and was never very good at it, that one edges towards materials science engineering) and dynamics, which is analysis of the forces and moments acting on bodies in motion. You get like, mechanical arms with joints, and turn it into a series of equations, based on how the discrete rigid bodies within the system relate to each other. We didn't do anything with non-rigid bodies, thank god. Depending on where your fixed points are and whether something is constrained to move along a certain axis, you simplify things out and let your data echo through the system of equations and figure out the answer to what you've been asked. It's fun stuff!!

4. ### ImoyramWell-Known Member

@EulersBidentity

I'd think it would be for anyone to use, regardless of level.

*shrug*

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5. ### EulersBidentitye^i*[bi] + 1

My dynamics course is kind of a mishmash: some basic Newtonian stuff, study of dynamical systems, Hamiltonian systems, Lagrangian mechanics. Work, energy, paths, central forces. Stuff.

My maths course is even more of a mishmash of pure and applied. Meant to ground us in theory & techniques before we specialise more.

Clarity edit.

Last edited: Nov 2, 2015
6. ### TheSeer37 Bright Visionary Crushes The Doubtful

That is a mishmash, for me those topics were spread out from high school to grad school. But I could be at least some help on all of it.

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7. This is semi-related but I'm finding that I can do the math just fine when I can look at formulas, but I'm not allowed to do that on tests. During tests, I know what I need but I keep drawing a blank. Anyone have any tips on making formulas stick in my head?

8. ### TheSeer37 Bright Visionary Crushes The Doubtful

It depends on the formulas, but yeah, there's ways. The simplest way (though a bit cheap) is to make a formula sheet as though you were going to get to use it as a reference on the test, read it over and over on the way to class, and then throw it out or otherwise make it clearly inaccessible right before the test starts. Then write down the formulas on the top of the test while they're still in short-term memory.

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9. ### EulersBidentitye^i*[bi] + 1

Some of it I studied at A-level: most of my "Mechanics" module for A-level maths was about modelling simple systems using Newton's laws and suvat. But a lot of it is either new this year, new in terms of this application, or now being defined more rigorously.

@Mentarnes I write little tunes for them, and sing them as I revise until they stick in my head. Got the idea from a brilliant YouTube video about the dot product. I've spent four years of exam seasons going "ay dot bee equals magnitude of ay, times by magnitude of bee, cosine theeeeta".

Last edited: Nov 2, 2015
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10. ### TheSeer37 Bright Visionary Crushes The Doubtful

twinkle twinkle little star, power equals I squared R

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11. To the tune of Pop Goes the Weasel

Opposite b plus or minus the squaare root of b squared minus four a c ALL over two aaaa!

I just need a bunch of songs that I can fit integral into

12. ### EulersBidentitye^i*[bi] + 1

You need to...integrate it...into the lyrics...

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13. ### EulersBidentitye^i*[bi] + 1

Can anyone recommend an interesting book or web series or...anything...that's a good intro to multi-particle systems? My eyes are just sliding over my lecture notes without understanding them.

14. ### EulersBidentitye^i*[bi] + 1

Uh. I'm maybe just being a bit Sunday evening about this, but could someone explain to me Lagrangians and action? In small words. I'll copy the section of my lecture notes so you can see what I'm looking at.

There is something else that we can consider:
L = T - V = 1/2(m||2) - V(r).
This is not conserved, it changes in time as the particle moves. But we can consider instead the functional
S[r] = ∫ L[r,]dt [definite integral between t1, t2 - EB]
S is called the action and L is the Lagrangian. The action is a functional in the sense that it is a function of a function: given a function, namely the entire path r(t) of a particle from t1 to t2 then S[r] gives a number. This number depends on the whole path not just any given point on it. Note that the Lagrangian is not strictly speaking a functional. Rather one tends to think of it as a formal function of r and as independent variables, without thinking of the fact that r and are also themselves functions of time.
[...]
We now state the Principle of Least Action: Particles move so as to extremize the action S as a functional of all possible paths between r(t1) and r(t2). That is to say Newton's Laws of motion are equivalent to the statement that
δS = 0,​
where δS is the first order variation of the action obtained by shifting the path r -> r + δr, subject to the condition that the end points of the path are fixed: δr(t1) = δr(t2) = 0.

/lecture notes.
Then there follows a proof of the above. But, er, what?
Things I don't understand:
• Defn of functionals, although this may just be a conceptual thing that I'll need time to get accustomed to
• Everything from "Particles move so as to extremize the action S" onwards.

15. Hey - I didn't have time to respond to this today, but I did see it. I will see if I can put together an explanation tomorrow night if someone else doesn't get to it first.

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16. ### TheSeer37 Bright Visionary Crushes The Doubtful

Oooh, physics! Pick me, pick me!

First, functionals.

You can think of a function as a set of ordered pairs (x, y). For example, the function f(x) = x^2 would be the set {(0,0), (1,1), (-1,1), (2,4), (-2,4), etc etc} or, more formally, {(x,y) | x is real, y=x^2}. Assuming x and y are both real numbers, you could graph all those points on a coordinate plane and get the usual y=x^2 parabola.

We're used to thinking of functions with a domain and range over the real numbers, by default. But the domain and range could be any two sets. The factorial function (as in 3! = 3*2*1) is a function whose domain and range are both limited to the whole numbers. (We say it is a function from W -> W.) The complex magnitude (f(a+bi) = a^2 + b^2) is a function from the complex numbers to the real numbers, C -> R. "How many likes do you have?" is a function from the set of kintsugijin to the set of whole numbers. And so on.

A functional is a function whose input is an entire other function. It's not like a composition of functions such as f(g(x)), where you take one particular value of g and use that as the input of f. Rather, to calculate a functional you need to look at the entire input function to get one output number. For example, "what is the degree of this polynomial" would be a functional - it doesn't use any particular value of the polynomial, it looks at the entire thing.

The Lagrangian is a regular function. All you need to calculate it is the displacement and the velocity of the number at a particular time. Action is a functional because you need to know the entire path of the object r(t) to calculate it.

Now for the physics. (This is the mind-blowy part. I hate calculating Lagrangians but the concept of Lagrangians makes me feel like a god.)

If an object is at position r0 at time t0 and it's going to be at at position r1 at time t1, then out of all the possible paths from (t0, r0) to (t1, r1) it will take the one with the smallest action. For example, suppose a particle on a coordinate plane with no forces acting on it is at (-1,0) at t=0 and will be at (1,0) at t=2. There's a lot of possible paths it could take in between, but let's look at just three.
1. Moving along a semicircle with radius 1, with constant speed.
2. Moving to the right with v=1.5 for 0<t<1, and v=.5 for 1<t<2.
3. Moving to the right with constant v=1.
For path 1, the particle needs to cover pi units of distance in 2 seconds, so its speed is pi/2. Its kinetic energy is pi^2/4 times its mass, about 2.46m. With no forces in play, potential energy is always 0, so L=2.46 times the mass for the whole trip. Integrate that over two seconds of time, and you get the action S=4.92m.

For path 2, kinetic energy (and thus the Lagrangian) is 2.25m for 0<t<1 and .25m for the 1<t<2. The action works out to S=2.5m.

For path 3, kinetic energy is 1m all the time. Action is S=2m. With the smallest value for S, this is the preferred path.

In other words, this math recreates Newton's first law. In the absence of any forces, an object will move in a straight line at constant speed.

Okay, but wait. I knew that in advance, and made sure to include the right answer in my list of possibilities. How do you find the path of least action if you don't already know it? There's infinite possibilities, you can't actually try them all.

So, remember back in Calc 1 when you studied optimization problems? There was some function you wanted to maximize or minimize, so you took its first derivative. The maximum or minimum had to occur in a place where the first derivative was zero. We're using the same idea here. The path with the minimum action is the path where the "derivative" of the action with respect to a tiny change in the path is zero. It's not exactly a derivative, hence all that jabber about "the first order variation of the path etc. etc." But the idea is the same - we find a minimum where the rate of change is zero.

The trippy part is, how does the particle know to take the path that will minimize action? Remember, the action can only be calculated using the entire path. So what makes the particle, at the beginning of that path, take a course that will minimize its action through a potential field that it's not even in yet? I know two answers to this question, take your pick.
1. It's just a mathematical artifact. Lagrange found a handy equation that happens to be equivalent to the laws of motion, and is sometimes easier to calculate, but doesn't model what's actually going on.
2. Cause and effect is not fundamental. The universe actually runs as a big transcendental optimization/continuity problem across space and time, and cause-and-effect is what this looks like from our limited perspective.

17. ### EulersBidentitye^i*[bi] + 1

I like this example! This example makes sense to me!
Oh that's what that was getting at.
Why not? In case the function is non-differentiable?

Thanks so much! This already makes a lot more sense to me.
Thank you @WithAnH, too.

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18. ### TheSeer37 Bright Visionary Crushes The Doubtful

Remember, the formal definition of a derivative involves adding tiny changes to the input of a function. If the input of a function is a number, this makes sense: x becomes (x+h).

However, the input of a functional is another function. Adding a tiny number to a function doesn't really work the same way, and anyway it would make the function move away from its fixed beginning and end points. Instead we add a tiny deviation to the path, without moving its endpoints. So we're doing something derivative-ish, but formally not quite the same thing.

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19. ### EulersBidentitye^i*[bi] + 1

Oh, cool! Okay, that makes sense.

20. ### kastilinget in the fucking crayfish shinji

so this thermodynamics question is giving me a headache & i was wondering if i could get some help if possible. the question i'm struggling with is question 3. I know the pressure at the beginning of the first isothermal process is 5 bar & at the beginning of the second it's 0.8 bar (point 1 & point 3 in the cycle).
i think i may be misunderstanding something fundamental here Last edited: Nov 19, 2015