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  1. - Top - End - #1
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    Default Derived v. axiomatic laws of physics

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    All my degrees are in pure math. By a grand coincidence, I never took Diff Eq. I thought to myself... I have a week off, my wife got The Great Courses Plus, there's a Diff Eq course, sure I'll try to learn Diff Eq this week. I taught *very* basic Diff Eq to my Calc students (was a teacher once upon a time), baby stuff like dy/dx=2x, find y, stuff I wouldn't really call Diff Eq. I was always interested in learning it, Heat Equation, Navier-Stokes, etc., just never got around to it.

    So I start the first video. All is well. Then he gets to examples of differential equations, I'm super excited.

    First one he throws up on the screen... Newton's Second Law. I'm familiar, memorized such things, but never thought of it as a differential equation. I'm like wtf.

    So I rewatch the video, do some googling, and man am I struggling.

    In my mind differential equations were... definition-wise, I'd say equations involving derivatives of a function, the solution a (family of) function(s). You know, things like y'+ay=b or something.

    So I'm staring at, on the screen, F(y)=my'' (1-dimensional, constant mass).

    Trying to wrap my head around this... I'll call the 1 dimension the vertical dimension, so y=height, independent variable t, maybe rearrange it a little, m*d^2y/dt^2=...??? I guess F(t), since y=y(t).

    Then I get to wondering... man, what *is* force? Like the definition. Or an axiom. I think of most things in physics in terms of their units. Looks like dp/dt, change in momentum over time. Which I'm familiar with, again Newton's Second Law.

    So I'm thinking... there's nothing to prove in Newton's Second Law. You don't start with some equation, do some derivation/solving, and end up with Newton's Second Law. It's like Newton's Second Law is a *definition* of force (in formula form), a handy expression detailing how mass, position, and time are related; you know, the whole double mass->half acceleration with same force applied. Detailing based on extensive observation. Like maybe in a different universe, had physics happened differently (lulz), we would've gotten F=d(m^.5*y/t)/dt or F=d^2(m*y)/dt^2 or something.

    So I google some laws of physics and find a link like this. Some of these I know, some I don't. Got things like I=V/R, F=G*M*m/r^2, a^3/t^2=G(M+m)/(4p^2) (not that the formulae are included in that link). Which leads me to my question...

    Of the laws of physics that you know (at least the ones with an equation), which are derived v. axiomatic?

    Derived meaning... maybe you start with a differential equation, work on it, out pops a law of physics. Axiomatic meaning... like m*a (dp/dt, rather) defines this thing we call force (not that they're necessarily definitions).
    Last edited by danzibr; 2021-03-24 at 04:40 AM. Reason: Embarrassing

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    Default Re: Derived v. axiomatic laws of physics

    I think there is a lot of stuff that boils down to just establishing constants which convert between systems of units, though once you have a few equations in the mix there are constraints that aren't local to any one equation which make the set of equations as a whole disprovable. Something like E0=mc^2 just establishes a conversion constant, but once you assert that it's the same c that governs the electromagnetic field, it becomes non-trivial.

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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by NichG View Post
    I think there is a lot of stuff that boils down to just establishing constants which convert between systems of units, though once you have a few equations in the mix there are constraints that aren't local to any one equation which make the set of equations as a whole disprovable. Something like E0=mc^2 just establishes a conversion constant, but once you assert that it's the same c that governs the electromagnetic field, it becomes non-trivial.
    E0=mc^2 first of all establishes that mass and energy can be converted or that they are always directly connected. IMO this is far more important than the value of c used there.

    Deciding which equations are axiomatic and which are derived is fairly difficult in physics for a few reasons. For one, a lot of laws were obtained experimentally: does it count as derivation in a strict sense? There is also an important fact that previously axiomatic laws can be currently derived from a broader theory obtained later. When established, Newton's mechanic was indeed axiomatic, but it can be easily obtained for example from quantum mechanics as a specific limit. The whole field of thermodynamics is now a result of statistical mechanics. Electrodynamics can be obtained as a gauge field resulting from phase invariance of wave functions in quantum mechanics.

    Sometimes things are even more mixed up as for example Maxwell's electrodynamics is mostly a collection of known empirical laws with one additional term put in because Maxwell has seen that it would complete the whole system. It's not an axiom establishing some new quantity and it is not something properly derived - it is a well educated guess verified experimentally soon after.
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    Default Re: Derived v. axiomatic laws of physics

    Something else to consider is that most of physics interrelates at some level (and if it doesn't there's stuff in between that does) and this makes defining "Axiomatic" vs "Derived" a lot harder.

    Consider a system of three laws, any two of which can be used to derive the third, but as a set cannot be derived from elsewhere - are any of them Axiomatic or Derived?

    In one sense all three laws are Derived, but the set of them is Axiomatic so really at least 2 of the laws are also Axiomatic except they can be derived...

    I strongly suspect that a huge amount of physics resembles those three laws (except there are way more than three involved) - a lot of the laws can be derived from others (they are restating the same things for a different medium) but, with most of the laws originating through observation, there's no axiomatic starting point.
    It's made worse by the non-scalable nature of a lot of physics - by which I mean how Newtonian physics (which is pretty close to having axioms) isn't actually accurate when you look at the relativistic scale, but neither is Relativity accurate at the quantum level etc.

    The mathematical approach of deriving from first principles is great (I also have most of my degrees in mathematics) but true "first principles" to me is what physics is lacking (and a fair bit of cutting edge physics is trying to work towards). This isn't a criticism of physics and physicists, but it's the problem with applying a pure mathematical approach to physics.
    An analogy - you are trying to find a solid foundation to build on, the quantum physicists are trying to cut off the branch they stand on to see what is beneath it.

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    Default Re: Derived v. axiomatic laws of physics

    As I see it, most axioms are definitions. Take V=IR, for instance - it says that there is a relationship between V (potential difference) and I (charge transferred per unit time), and the ratio between them can be usefully called "resistance". Or the second law of thermodynamics, which asserts the existence of something called "entropy". These laws define (quantify) ideas, such that you can then go and do engineering with them.

    Thinking about it, can you offer any examples of what you would call a "derived" law of physics? I'm thinking maybe the law that you can't accelerate to light speed, which is derived from special relativity - but in truth we don't know if it is a real law, it's an implication of a model, and we know a priori that all models break at some point...
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    Default Re: Derived v. axiomatic laws of physics

    I remember hearing the world "law" is also misleading. The "law" in physic is just a theory that was proven very, very good and that people letter on though that they represent law of nature. But most of them was in XX century proven to be a special cases of some more general "law" and now scientist try to avoid that term, for example quantum mechanic and general relative are still called "theories" even if they are proven more accurate then Newton law's of gravity. The word "law" is still used in names only because of custom.
    Moreover those theories are usually based on observation so axiom here is simple stuff like when you throw something it will fall, and then you try to find explanations. Mathematical derivation are then used to see what this observation can produce and whether this theory broke any other known theories/observations. Or if you have few equations which are based on observation you use math to see whether they are somewhat connected and can be generalize.


    As for question what is force, that's one that I'm also trying to grasp. I'm no physicist, only watched some you-tube shows, including Feyman lectures and Space-time, and my current understanding is that there is no such "thing" as force, this is term used in equations to show somehow changes/interactions and maybe ease up some calculation.
    Ohh and from Feyman interviews on electromagnetism was this great explanation for "what electromagnetism is", which basically concluded that it's impossible to explain what exactly electormagetism with understandable terms because electromagetism is in fact basic element of universe which we need to use to explain other phenomenon in physics.
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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by Khedrac View Post
    Consider a system of three laws, any two of which can be used to derive the third, but as a set cannot be derived from elsewhere - are any of them Axiomatic or Derived?
    They are certainly redundant.

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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by asda fasda View Post
    I remember hearing the world "law" is also misleading. The "law" in physic is just a theory that was proven very, very good and that people letter on though that they represent law of nature. But most of them was in XX century proven to be a special cases of some more general "law" and now scientist try to avoid that term, for example quantum mechanic and general relative are still called "theories" even if they are proven more accurate then Newton law's of gravity. The word "law" is still used in names only because of custom.
    Not quite.

    A law in science is a relationship that can be defined by a mathematical equation.

    A theory is an explanation for why things are the way they are, which makes testable predictions about the world and has withstood rigorous attempts to disprove it.

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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by GloatingSwine View Post
    Not quite.

    A law in science is a relationship that can be defined by a mathematical equation.

    A theory is an explanation for why things are the way they are, which makes testable predictions about the world and has withstood rigorous attempts to disprove it.
    Hmm, to be honest I was sure that all theories in physics are based on mathematical equation at this point.
    Additionally now when I think about it all theories are kind of relations (IF A then B) so they can be defined in some form of mathematical equation : P
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    Default Re: Derived v. axiomatic laws of physics

    From your perspective, I'd expect mechanical physics to be derived from Newton's laws.
    Then electrodynamics from Maxwell's Equations. Although I've heard that the one time somebody tried to teach electrodynamics this way it failed miserably. They really teach it the other way around. Note that the during the Feyman lectures, graduate students quickly replaced the undergrads, so they are really only useful for better understanding of physics once you've already learned the basics (for all I know he derived everything from Maxwell's Equations).

    The catch here is that math is math and physics is a science. "Axioms" rely on observation, and that all laws derived from them equally match observations. The famous F=ma you mention is sometimes taught as F=dp/dt (derivative of momentum over time) as that matches General Relativity. Things don't work if you try to kludge F=ma into GR (assuming "relative mass" is a thing makes you expect things at relativistic velocity to become black holes and then come back to reality to an outside observer. They do nothing of the sort, as there isn't any "relativistic mass" to create gravity).

    When I was learning physics as an engineering undergraduate, one of the other professors absolutely refused to teach F=ma, and plenty of students in later classes would object to that equation. It wasn't much later that I learned how it failed at relativistic speeds.

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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by wumpus View Post
    From your perspective, I'd expect mechanical physics to be derived from Newton's laws.
    Then electrodynamics from Maxwell's Equations. Although I've heard that the one time somebody tried to teach electrodynamics this way it failed miserably. They really teach it the other way around. Note that the during the Feyman lectures, graduate students quickly replaced the undergrads, so they are really only useful for better understanding of physics once you've already learned the basics (for all I know he derived everything from Maxwell's Equations).
    In one of Feyman's public lectures he makes a big deal about there being 3 ways of looking at a moving object (one being Newton's Laws, one being the Langragian, I can't remember what the 3rd was, possibly Conservation), and of course in nice cases they are all mathematically equivalent. And in different not nice cases they each become awkward in different ways.
    In a similar example to the momentum one there's a historic split to whether it's best to say mass is constant and that E=mc^2 only applies at rest or whether to say 'mass' depends on the frame of reference [which the post actually covered, as well as why one works better]
    And again the quantum mechanic interpretations, whichever one is 'real' has massively different philosophical consequences, but little practical difference.

    OTOH Thermodynamics is now totally derived from Statistical Mechanics (although of course the discovery and teaching go the other way). Of course it's possible that may change if we do find something that makes that position untenable (although I can't imagine anything that would)
    Last edited by jayem; 2021-03-24 at 04:53 PM.

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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by danzibr View Post
    Spoiler: Boring background
    Show

    All my degrees are in pure math. By a grand coincidence, I never took Diff Eq. I thought to myself... I have a week off, my wife got The Great Courses Plus, there's a Diff Eq course, sure I'll try to learn Diff Eq this week. I taught *very* basic Diff Eq to my Calc students (was a teacher once upon a time), baby stuff like dy/dx=2x, find y, stuff I wouldn't really call Diff Eq. I was always interested in learning it, Heat Equation, Navier-Stokes, etc., just never got around to it.

    So I start the first video. All is well. Then he gets to examples of differential equations, I'm super excited.

    First one he throws up on the screen... Newton's Second Law. I'm familiar, memorized such things, but never thought of it as a differential equation. I'm like wtf.

    So I rewatch the video, do some googling, and man am I struggling.

    In my mind differential equations were... definition-wise, I'd say equations involving derivatives of a function, the solution a (family of) function(s). You know, things like y'+ay=b or something.

    So I'm staring at, on the screen, F(y)=my'' (1-dimensional, constant mass).

    Trying to wrap my head around this... I'll call the 1 dimension the vertical dimension, so y=height, independent variable t, maybe rearrange it a little, m*d^2y/dt^2=...??? I guess F(t), since y=y(t).

    Then I get to wondering... man, what *is* force? Like the definition. Or an axiom. I think of most things in physics in terms of their units. Looks like dp/dt, change in momentum over time. Which I'm familiar with, again Newton's Second Law.

    So I'm thinking... there's nothing to prove in Newton's Second Law. You don't start with some equation, do some derivation/solving, and end up with Newton's Second Law. It's like Newton's Second Law is a *definition* of force (in formula form), a handy expression detailing how mass, position, and time are related; you know, the whole double mass->half acceleration with same force applied. Detailing based on extensive observation. Like maybe in a different universe, had physics happened differently (lulz), we would've gotten F=d(m^.5*y/t)/dt or F=d^2(m*y)/dt^2 or something.

    So I google some laws of physics and find a link like this. Some of these I know, some I don't. Got things like I=V/R, F=G*M*m/r^2, a^3/t^2=G(M+m)/(4p^2) (not that the formulae are included in that link). Which leads me to my question...

    Of the laws of physics that you know (at least the ones with an equation), which are derived v. axiomatic?

    Derived meaning... maybe you start with a differential equation, work on it, out pops a law of physics. Axiomatic meaning... like m*a (dp/dt, rather) defines this thing we call force (not that they're necessarily definitions).
    Well, there's a part of Newton's laws that is a definition, and a part that is derived. One thing you may not have noticed is that it is also a definition of mass.

    Even before Newton, there was some concept - and measurement - of force. Newton's second law then says that every object has a (mostly unchanging) number which describes how it will react to any force, and that that response is in the form of acceleration proportional to the measured force.

    For the laws you listed: I = V/R is a similar mix which is derived from the claim that response current is proportional to voltage, and defines resistance as the proportion constant.

    F = G*M*m/r^2 is a mix which says that gravitational force is proportional to the mass of the attracting object, the mass of the attracted object, and the inverse square of the distance between them, and defines the gravitational constant as the proportion.

    a^3/t^2=G(M+m)/(4p^2) is entirely derived. This shouldn't be surprising; there's no "new" term (other than possibly the period, which is clearly defined outside of the equation) to be defined.

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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by uncool View Post
    Well, there's a part of Newton's laws that is a definition, and a part that is derived. One thing you may not have noticed is that it is also a definition of mass.

    Even before Newton, there was some concept - and measurement - of force. Newton's second law then says that every object has a (mostly unchanging) number which describes how it will react to any force, and that that response is in the form of acceleration proportional to the measured force.

    For the laws you listed: I = V/R is a similar mix which is derived from the claim that response current is proportional to voltage, and defines resistance as the proportion constant.

    F = G*M*m/r^2 is a mix which says that gravitational force is proportional to the mass of the attracting object, the mass of the attracted object, and the inverse square of the distance between them, and defines the gravitational constant as the proportion.

    a^3/t^2=G(M+m)/(4p^2) is entirely derived. This shouldn't be surprising; there's no "new" term (other than possibly the period, which is clearly defined outside of the equation) to be defined.
    And it's still technically a semi-open question whether the m of F=ma and m of g=GMm/r^2 are the same.
    It has been confirmed to ridiculous precision by every experiment done so far.
    And if we ever find that something funny happens, something is going to have to be tweaked depending on what they actually find.

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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by uncool View Post
    F = G*M*m/r^2 is a mix which says that gravitational force is proportional to the mass of the attracting object, the mass of the attracted object, and the inverse square of the distance between them, and defines the gravitational constant as the proportion.
    This is one of those bits where math and science diverge. Newton's gravitation equation is derived from Kepler's laws of planetary motion. Which were based on the most accurate astronomical observations around (Tycho Brahe's, who Kepler worked under) but not derived from any physical laws.

    If you assume circular orbits it is pretty easy to derive Newton's equation using nothing but algebra (I managed to quickly write it down during an exam). Doing so for elliptical orbits (which is what really happens, circular orbits [Venus] just happen to be ellipses with zero eccentricity) should require pairs of partial differential equations (one for each axes), but I think Newton managed to do it with geometry for his Principia Mathematica (it helps to be a Newton to pull that off).

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    Default Re: Derived v. axiomatic laws of physics

    Quote Originally Posted by danzibr View Post
    Spoiler: Boring background
    Show

    All my degrees are in pure math. By a grand coincidence, I never took Diff Eq. I thought to myself... I have a week off, my wife got The Great Courses Plus, there's a Diff Eq course, sure I'll try to learn Diff Eq this week. I taught *very* basic Diff Eq to my Calc students (was a teacher once upon a time), baby stuff like dy/dx=2x, find y, stuff I wouldn't really call Diff Eq. I was always interested in learning it, Heat Equation, Navier-Stokes, etc., just never got around to it.

    So I start the first video. All is well. Then he gets to examples of differential equations, I'm super excited.

    First one he throws up on the screen... Newton's Second Law. I'm familiar, memorized such things, but never thought of it as a differential equation. I'm like wtf.

    So I rewatch the video, do some googling, and man am I struggling.

    In my mind differential equations were... definition-wise, I'd say equations involving derivatives of a function, the solution a (family of) function(s). You know, things like y'+ay=b or something.

    So I'm staring at, on the screen, F(y)=my'' (1-dimensional, constant mass).

    Trying to wrap my head around this... I'll call the 1 dimension the vertical dimension, so y=height, independent variable t, maybe rearrange it a little, m*d^2y/dt^2=...??? I guess F(t), since y=y(t).

    Then I get to wondering... man, what *is* force? Like the definition. Or an axiom. I think of most things in physics in terms of their units. Looks like dp/dt, change in momentum over time. Which I'm familiar with, again Newton's Second Law.

    So I'm thinking... there's nothing to prove in Newton's Second Law. You don't start with some equation, do some derivation/solving, and end up with Newton's Second Law. It's like Newton's Second Law is a *definition* of force (in formula form), a handy expression detailing how mass, position, and time are related; you know, the whole double mass->half acceleration with same force applied. Detailing based on extensive observation. Like maybe in a different universe, had physics happened differently (lulz), we would've gotten F=d(m^.5*y/t)/dt or F=d^2(m*y)/dt^2 or something.

    So I google some laws of physics and find a link like this. Some of these I know, some I don't. Got things like I=V/R, F=G*M*m/r^2, a^3/t^2=G(M+m)/(4p^2) (not that the formulae are included in that link). Which leads me to my question...

    Of the laws of physics that you know (at least the ones with an equation), which are derived v. axiomatic?

    Derived meaning... maybe you start with a differential equation, work on it, out pops a law of physics. Axiomatic meaning... like m*a (dp/dt, rather) defines this thing we call force (not that they're necessarily definitions).
    I'll note that one can derive Newton's second law using Lagrangian mechanics and the calculus of variations, though all that does is kick the can down the road (where the question shifts to the origin of the Lagrangian rather than Newton's second law.)

    Almost every equation in physics is pulled from thin air, being a post-hoc solution that fits and then predicts experimental data. Off the top of my head, I can only think of one exception: taking Lorentz invariance of spacetime to be axiomatic produces special relativity. (Note that this is a different approach than Einstein's original formulation, where Lorentz invariance arises from axiomatizing that the speed of light and the laws of physics are the same in every non-inertial reference frame.)

    There really isn't anything in physics like say the group axioms, which lead to everything in group theory. Attempts have been made to axiomatize physics (it is the 6th of Hilbert's problems) by trying to find a collection of axioms that give rise to a fundamental theory (quantum field theory and string theory are the most common targets), though I don't believe they have been very successful.
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    Default Re: Derived v. axiomatic laws of physics

    Not exactly a Law but Noether's Theorem is derived.

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    Default Re: Derived v. axiomatic laws of physics

    The axioms of physics don't belong under the label "physics" the belong under the label "metaphysics".
    These axioms are things like:

    There is an mind-external world.
    We can learn by observations.
    Each time a situation reoccurs exactly, it will have exactly the same probability distribution of outcomes.

    For the large part, metaphysics is considered to be "solved" in the sense that scientists don't need to think about which axioms they use, or be explicitly taught them. As for most metaphysics questions still under discussion, physics "brackets" them, which is to say they've checked that the questions don't affect them and therefore can be ignored.

    All of physics "laws" are (in principle) derived from data (which the axioms say is important). Basically all physics laws are of the form "given the universe -> X'.

    There are definitions, which are ideally called out as such when formally doing physics. For example newton defined "energy", "potential energy", "kinetic energy", and bunch more terms. When teaching students established material, this is often glosses over, especially when there is an intuitive version of the term that the definition is just formalizing.

    So I'm thinking... there's nothing to prove in Newton's Second Law
    There's nothing to analytically prove any scientific law. There's just a lot of data and a lack of clear counter-examples. Consider the sunrise problem. Basically we establish really good odds that the thing is true.

    More recently, we think of every other way it could work, and show that it's extremely likely that those ways are wrong.

    I'd say equations involving derivatives of a function, the solution a (family of) function(s). You know, things like y'+ay=b or something.
    Just a wild guess here, but I think things might be easier for you starting from more advanced examples, where it's clear why the machinery of differential equations is used (in fact, why they were invented), rather than the simplified examples were equations are isolated from each other.

    For example, the differential equation for force in a variable mass system is

    F + u(dm/dt) = m (dv/dt)

    Where u is the velocity of the entering or leaving mass relative to the main body of the system. I think this is assuming u is constant with respect to time, like would happen with a rocket.
    Last edited by Quizatzhaderac; 2021-05-03 at 11:56 AM.
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