Talk:2904: Physics vs. Magic

Isn't the first law of thermodynamics a conservation law? 172.69.134.217 21:27, 8 March 2024 (UTC)

In Lagrangian Mechanics,the Lagrangian is a function of time, position and speed. The action of the system is defined as the integral of the Lagrangian between the initial and final time. Movement equations are derived as those that minimize action. In that sense it can be loosely interpreted that by only setting initial condition and outcome you can get the full picture of all intermediate events. 198.41.230.215 22:46, 8 March 2024 (UTC)

This is why statistics is magical Phlaxyr (talk) 23:33, 8 March 2024 (UTC)

Both thermodynamics and conservation laws make predictions without telling us anything about what exactly is happening in the intermediate steps. In that sense they're no different from the curse in the comic. An example for thermodynamics could be: your coffee cup will get cold if left on your desk (zeroth law). And an example from conservation laws could be: it doesn't matter what method you're going to use to stop a moving car, in all cases the car has lost the same amount of energy (1/2mv^2). 141.101.99.110 00:33, 9 March 2024 (UTC)

I've always been a little bit annoyed by thermodynamics. I mean it has a temperature, it has energy, why can't I have the energy without something colder lying around? "Remove heat energy from this object and charge a battery with it"... It sucks because the inverse is true, I can certainly discharge a battery and make heat energy from chemical... Anyway back on topic, can someone magic me such a device? I promise to share 50% of the big oil hush money. 172.68.210.23 04:04, 9 March 2024 (UTC)

What exactly would be your contribution? Anyone with such device would be already swimming in money from U.S. department of defense. Or, more likely, killed by them. Because it certainly can be used as a bomb. -- Hkmaly (talk) 20:48, 9 March 2024 (UTC)

Anonymity. With magic up their sleeve they probably want to solve a few more world problems too, this allows them to get one out of the way without drawing attention to themselves. 172.68.210.73 01:17, 10 March 2024 (UTC)

The talk had a better explanation of why Thermodynamics, Conservation laws and Lagrangians are 'magic' than the actual explanation. I added a few paragraphs briefly explaining to the explanation, I hope that's helpful, but I left the paragraph about scientific laws being empirical themselves in place despite the fact that I'm pretty dubious about whether that's actually part of the joke. 141.101.99.111 16:46, 9 March 2024 (UTC)

To be frank I just think part of the joke is how naive definitions of science can lead to baffling counterexamples 198.41.230.214 08:03, 10 March 2024 (UTC)

About the stationary action concept

At first sight it looks as if Hamilton's stationary action implies some form of teleology. On closer inspection that turns out not to be the case.

I will use the following case as example of application of calculus or variations in physics: the catenary problem. The properties of the catenary problem that make it lend itself to variational treatment generalize to other areas of physics in which calculus of variations is applied

Take a catenary and divide it into subsections. Here's the thing: each of those subsections is an instance of the catenary problem. The ratio of horizontal and vertical displacement is different for each subsection, of course, but that is not an obstacle.

Solving the catenary problem with calculus of variations consists of the following: you subdivide the total length in infinitesimally small subsections. You then set up an equation that addresses all subsections concurrently.

That equation-for-every-infinitesimal-subsection-concurrently is the Euler-Lagrange equation. You solve the problem by restating the equation as a differential equation.

A differential equation is non-local in the sense that to solve the problem you require that the equation is to be satisfied over the whole domain concurrently.

The derivation of the Euler-Lagrange equation is a generic derivation. That is, the result of that derivation is applicable for any problem that is stated in variational form.

Stating a problem in variational form means that it is stated as an integral. (In the case of the catenary problem that integral is the integral of the potential energy from one point of suspension to the next point of suspension.) The problem statement is then: which curve has the property that for that curve the derivative of the integral of the potential energy is zero.

In the case of the catenary problem:
The integral is integration with respect to the horizontal coordinate. The variation that is applied is perpendicular to that; the variation is applied in the vertical direction. The derivative-is-zero criterion is for the derivative of the integral with respect to that vertical direction.

Key to the derivation of the Euler-Lagrange equation is that it works towards the goal of transforming the integral expression to a differential expression. That is essential: in order to make progress the integration must be replaced with differentiation.

The result of the transformation, the Euler-Lagrange equation, imposes a constraint that is just as demanding as the initial formulation with an integral. The differential equation is to be satisfied concurrently over the whole domain.

There is a derivation of the Euler-Lagrange equation that just skips stating the integral; it goes straight to the differential expression. Geometric derivation of the Euler-Lagrange equation Author: Preetum Nakkiran.

(Preetum Nakkiran uses the catenary problem as motivating example, the result has general validity.)

Further reading: discussion of Hamilton's stationary action in an answer I submitted to physics.stackexchange: Hamilton's stationary action

The relation between Newtonian mechanics and conservation of energy

We have that in order to formulate a theory of mechanics we must at minimum use these three quantities: position, velocity, acceleration. These three are in a cascading relation: velocity is the time derivative of position, acceleration is the time derivative of velocity.

v = ds/dt, a = dv/dt      (1)

In the case of uniform acceleration from a starting velocity of zero we have:
v = at      (2)
s = ½at²      (3)

Take (3), multiply both sides with acceleration a, and substitute according to (2):

as = a½at² = ½a²t² = ½(at)² = ½v²

as = ½v²    (5)

The relation (5) is known as Torricelli's formula.
In case the initial position coordinate and the initial velocity are non-zero:

a(s-s₀) = ½v² - ½(v₀)²    (6)

Interestingly, in the case of a non-uniform acceleration the result of the integration identical to (6)

∫ a ds = ½v² - ½(v₀)²    (7)

(To understand (7): we have that integration is summation of infinitesimal strips. The integration consists of concatenating instances of (6), in the limit of infinitesimal increments. All of the in-between terms drop away against each other, resulting in (7))

The work energy theorem is obtained as follows: start with F=ma, and integrate both sides with respect to the position coordinate.

∫ F ds = ∫ ma ds    (8)

Use (7) to process the right hand side:

∫ F ds = ½mv² - ½m(v₀)²    (9)

(9) is the work-energy theorem

The work-energy theorem is the reason that it is useful to formulate the concepts of potential energy and kinetic energy. If we formulate potential energy and kinetic energy in accordance with the work-energy theorem then we have that the sum of potential energy and kinetic energy is a conserved quantity.

The work-energy theorem consists of two elements: F=ma, and (7).

Here (7) was stated in terms of the familiar quantities of mechanics: position, velocity, acceleration. (7) generalizes to any set of three quantites that features that cascading relation: state, first time derivative, second time derivative.

Example: electric current and electromotive force in an LC circuit
Amount of current is a first derivative (displacement of charge per unit of time)
Change of current strength is a second derivative
For current through an inductor: the rate of change of current strength (second time derivative) is proportional to the electromotive force.
So we see that in the case of an LC circuit the elements necessary to result in a conservation property are present.
Cleonis (talk) 11:50, 10 March 2024 (UTC)