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July 31

Is there any simple necessary sufficient condition that conserves kinetic energy, without mentioning kinetic energy?

Note that the absence of external net force is a simple necessary sufficient condition that conserves momentum, without mentioning momentum (just as the absence of external torque is a simple necessary sufficient condition that conserves angular momentum, without mentioning angular momentum).

However, the absence of external net force is not a necessary condition (that conserves kinetic energy), because a given body's kinetic energy can be conserved even when external net force are exerted on that body, e.g. when it's in a circular orbit (in which case the space must have more than one dimension), or when the body elastically collides with a wall sharing a reference frame with an observer who measures the body's kinetic energy (in which case the space is allowed to have a single dimension).

The absence of external net force is not a sufficient condition (that conserves kinetic energy) either, as can be shown when two bodies inelastically collide with each other: The two body system's kinetic energy is not conserved [when the whole system is not seen at rest], although no external net force is exerted on that two body system.

HOTmag (talk) 07:11, 31 July 2024 (UTC)

In a system of two equal bodies circling each other around a common centre of mass, the net momentum of the system is constant, yet there are forces at play.  --Lambiam 09:18, 31 July 2024 (UTC)

The forces you are talking about, are internal ones, each of which is exerted on only a part of the two body system, on which no external force is exerted. Any way, for the sake of clarity, I've just added the word "external" to my first post. HOTmag (talk) 09:37, 31 July 2024 (UTC)

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Resolved

I've just thought about it, assuming that restmass does not change:

The conservation of velocity is a simple necessary sufficient condition that conserves momentum, without mentioning momentum.

The conservation of speed (i.e. of the absolute value of velocity) is a simple necessary sufficient condition that conserves kinetic energy, without mentioning kinetic energy. HOTmag (talk) 10:49, 31 July 2024 (UTC)

You write ‘The absence of external forces is not a sufficient condition ...’ This is debatable. I say that the kinetic energy of the constituents of the system doesn't necessarily contribute to the kinetic energy of the entire system. The energy associated with the movement of the inelastically colliding bodies relative to their common centre of mass is counted as internal energy of the system, not as kinetic energy. Take for example a bottle of warm gas in a circular orbit around Saturn. After a while, the gas cools down. The kinetic energy of the gas molecules decreases, but the kinetic energy of the bottle of gas remains the same; the thermal energy decreases. PiusImpavidus (talk) 18:08, 31 July 2024 (UTC)

Had the absence of external forces been a sufficient condition that conserves kinetic energy, then under that condition - the kinetic energy would've been conserved in all cases - hence in all reference frames and not only when the whole system is seen at rest. Hence, the absence of external forces can't be a sufficient condition that conserves kinetic energy, because when two bodies inelastically collide with each other while the kinetic energy is measured relative to any point that doesn't see the whole system at rest - then the system's kinetic energy does change after the collision - even though no external force is exerted on the system (because the system's momentum does not change). Any way, for the sake of clarity, I've just added this clarification to my paragraph you've quoted. HOTmag (talk) 20:54, 31 July 2024 (UTC)

We have two bodies, inelastically colliding. The kinetic energy of each of the bodies isn't conserved and external forces act on each of them. Namely, the force exerted by the other body. On the other hand, the kinetic energy of the system of two inelastically colliding bodies is conserved, as no external force acts on the system.

Consider the combined kinetic energy of two bodies with masses $m_{1}$ and $m_{2}$ and velocities ${\vec {v}}_{1}$ and ${\vec {v}}_{2}$:

$K={\frac {1}{2}}m_{1}{\vec {v}}_{1}^{2}+{\frac {1}{2}}m_{2}{\vec {v}}_{2}^{2}$

The velocity of the centre of mass is

${\vec {v}}_{cm}={\frac {m_{1}{\vec {v}}_{1}+m_{2}{\vec {v}}_{2}}{m_{1}+m_{2}}}$

Subtracting ${\vec {v}}_{cm}$ from ${\vec {v}}_{1}$ and ${\vec {v}}_{2}$ and rearranging some terms gives the combined kinetic energy of the two bodies in the centre-of-mass frame as

$K_{cm}={\frac {m_{1}m_{2}}{2\left(m_{1}+m_{2}\right)}}\left({\vec {v}}_{1}^{2}+{\vec {v}}_{2}^{2}-2{\vec {v}}_{1}\cdot {\vec {v}}_{2}\right)$

Now suppose we put the two masses into a black box and calculate the kinetic energy of the system in the box:

$K_{s}={\frac {m_{1}+m_{2}}{2}}{\vec {v}}_{cm}^{2}={\frac {1}{2\left(m_{1}+m_{2}\right)}}\left(m_{1}^{2}{\vec {v}}_{1}^{2}+m_{2}^{2}{\vec {v}}_{2}^{2}+2m_{1}m_{2}{\vec {v}}_{1}\cdot {\vec {v}}_{2}\right)$

Now add $K_{s}$ and $K_{cm}$ together and notice how the dot products cancel. Simple calculation shows that

$K=K_{s}+K_{cm}$

That was the maths, now the physics.

The two bodies each have their own kinetic energy. However, when we package the two bodies into a system, the system as a whole has a kinetic energy less than the kinetic energies of the constituent parts. The rest of the kinetic energy of the constituent parts isn't kinetic energy of the system, but internal energy of the system. And this internal energy is equal to the kinetic energy of the constituent parts in the centre-of-mass frame.

So, in your system of two inelastically colliding bodies, the kinetic energy of the bodies is indeed not conserved, but there's an external force acting on each of them. The kinetic energy of the system of two bodies (on which no external force acts) is conserved; it's the internal energy that decreases. PiusImpavidus (talk) 12:14, 1 August 2024 (UTC)

I agree to all of your maths, as well as to all of your claims about physics, except two sentences: "the kinetic energy of the system of two inelastically colliding bodies is conserved, as no external force acts on the system....The kinetic energy of the system of two bodies (on which no external force acts) is conserved".

The correct fact that "no external force acts on the system", is not sufficient for justifying your claim that the system's kinetic energy is conserved. As I've already pointed out in my previous response, the system's kinetic energy is only conserved when the whole system is seen at rest. For more details, see our article inelastic collision: "inelastic collisions do not conserve kinetic energy" [i.e. not in all reference frames].

Btw, for simplicity, let's assume that ${\vec {v}}={\vec {v}}_{1}=-{\vec {v}}_{2}$ and that $m=m_{1}=m_{2}$ (hence $K=K_{cm},$ and $K_{s}=0).$ HOTmag (talk) 14:23, 1 August 2024 (UTC)

The kinetic energy of the system $K_{s}$ only depends on the total mass of the system and $v_{cm}$. In the absence of external forces, neither of those change, so $K_{s}$ is conserved, in every reference frame. What the bodies do to each other is irrelevant. $K_{cm}$ and $K$, the difference between which is constant, are not conserved in an inelastic collision and the loss of energy is the same in every reference frame, including the centre-of-mass frame.

The nice thing I tried to show you is that the sum of the kinetic energies of the bodies can be broken into two parts, being the kinetic energy of the system of two bodies and the internal energy of the system, which can simply be added together. When you calculate the kinetic energy of a rock, you don't include the kinetic energies of all its vibrating atoms, right? Because that's the thermal energy of the rock. It's exactly the same here. PiusImpavidus (talk) 08:07, 2 August 2024 (UTC)

Thank you ever so much for your clarifications. So, I'm striking out the wrong thing in my first post (See above). HOTmag (talk) 11:12, 2 August 2024 (UTC)

Consider again a system of two equal bodies orbiting around a common centre of mass. Choose the coordinate system such that the common centre of mass of the two-body system is at rest. If one body's momentum equals $\mathbf {p}$ at some instant of time, that of the other at the same instant of time equals $-\mathbf {p} ,$ so the momentum of the system is $\mathbf {0} .$ Now apply external forces rotation-symmetrically to the objects so as to reverse their motion, making them circle again around their common centre of mass – which has not budged – but in the opposite sense. The symmetry guarantees that the momentum of the system remains $\mathbf {0}$ at all times, so the absence of external forces is not a necessary condition for conserving momentum.  --Lambiam 20:06, 31 July 2024 (UTC)

Your case does not involve an external force but rather involves an external torque. Any way, for the sake of clarity, I've just added this clarification to the first paragraph of my first post. HOTmag (talk) 20:54, 31 July 2024 (UTC)

...but equal and opposite applied forces like compression conserves momentum too. In other words, absence of external forces is a sufficient condition but it is not necessary. Modocc (talk) 21:35, 31 July 2024 (UTC)

By "equal", I guess you mean "having the same absolute value".

Anyway, when the external forces are equal [in their absolute value] and opposite, then the sum of those external forces is zero. Hence, saying that the external forces are equal [in their absolute value] and opposite, is like saying that there are no external forces. Hence, if you want to prove that the absence of external forces is not a necessary condition that conserves momentum, you will have to give an example in which the sum of the external forces is not zero. HOTmag (talk) 06:47, 1 August 2024 (UTC)

The fact that external forces can sum to zero does not imply they are absent... For example, without material internal resisting forces submarines and neutron stars implode. Modocc (talk) 08:18, 1 August 2024 (UTC)

All depends on what we mean by "absence of external forces". By "absence of external forces" I intend to include also all cases of external forces summing to zero. HOTmag (talk) 08:41, 1 August 2024 (UTC)

If you mean "absence of external net force" then why not write that instead? Also if F=0 then dP/dt=0 per Newton's second law, see Force. Modocc (talk) 12:09, 1 August 2024 (UTC)

Re. your first sentence: Ok, I'm adopting your current suggestion (See above in my first post).

Re. your second sentence: Of course, but what was wrong in my saying (ibid.) that "the absence of external [net] force is a simple necessary sufficient condition that conserves momentum, without mentioning momentum"? HOTmag (talk) 14:23, 1 August 2024 (UTC)

Nothing wrong, so I deleted the second part. Then in hindsight I deleted the first part because in many contexts it's understood, but not always. Then you restored the entire comment. Sigh. Modocc (talk) 15:02, 1 August 2024 (UTC)

Probably I had started responding to your response before you deleted it? Anyway, surprisingly, I didn't get any warning of "edit conflict" when I responded to your deleted response. HOTmag (talk) 15:56, 1 August 2024 (UTC)

Also, it is best practice to strike the original phrase(s) when modifying them so Lambiam's and my comments retain context. Modocc (talk) 15:54, 1 August 2024 (UTC)

Agree. Next time... HOTmag (talk) 15:57, 1 August 2024 (UTC)

${\vec {F}}\cdot {\vec {v}}=0$ PiusImpavidus (talk) 12:50, 1 August 2024 (UTC)

In my first post I gave two counter-examples:

1. On the one hand, when the body elastically collides with a wall sharing a reference frame with an observer who measures the body's kinetic energy, then the body's kinetic energy is conserved even though this case does not satisfy your condition. Hence it's not a necessary condition.

2. On the other hand, when two bodies inelastically collide with each other, the two body system's kinetic energy is not conserved [when the whole system is not seen at rest], even though this case does satisfy your condition. Hence it's not a sufficient condition. HOTmag (talk) 14:23, 1 August 2024 (UTC)

Both counter-examples fail:

1. While bouncing, ${\vec {v}}$ changes such that it is on average either 0 or parallel to the wall, whilst ${\vec {F}}$, the normal force, is perpendicular to the wall. Therefore, their dot product is zero, so the condition is satisfied.
2. I explained that one above. You have to make a distinction between the sum of the kinetic energies of the two bodies and the kinetic energy of the system of the two bodies. The former decreases in every reference frame, but the latter is conserved in every reference frame, so this doesn't prove that the condition is insufficient.

PiusImpavidus (talk) 08:39, 2 August 2024 (UTC)

So by ${\vec {v}}$ you meant the average velocity. I couldn't understand it before you made it clear. Instead of you, I would write: ${\vec {F}}\cdot ({\vec {v}}_{1}+{\vec {v}}_{2})=0.$

Anyways, your necessary sufficient condition could also be expressed verbally: "a zero external net force - or a zero average velocity - in every coordinate", right? HOTmag (talk) 11:12, 2 August 2024 (UTC)

By ${\vec {v}}$ I mean the instantaneous velocy, so the ${\vec {F}}\cdot {\vec {v}}$ is the instantaneous rate of change of the kinetic energy. If that's 0, the kinetic energy is always conserved.

The thing is, an instantaneous elastic collision is a strange thing. The force is infinite and acts during an infinitesimal time interval, the velocity is a step function changing exactly when the force in infinite. If we multiply the two, the result is zero except at the moment of collision, where it's undefined. So the maths don't work. And of course, no real collision is instantaneous. At the midpoint of any real collision, the kinetic energy (except the part associated with the motion parallel to the wall) has been converted to elastic energy, after the collision it has turned back into kinetic energy. So during a non-instantaneous elastic collision (BTW, no real collision can be fully elastic), kinetic energy isn't conserved at all times, but it is restored afterwards.

If you don't want to look into the details of the collision, you can say that it must be symmetrical, so that $\int {\vec {F}}\cdot {\vec {v}}\,dt=0$. If this integral isn't zero, the collision can't have been elastic. PiusImpavidus (talk) 08:15, 4 August 2024 (UTC)

1. Ok, so during the so-called "an elastic collision between a wall and a body", while the wall is exerting a non-zero external force on the body, the body's kinetic energy is not conserved, because it's converted to elastic energy. Agreed. But why does your condition have to mention the value V of the velocity, rather than only saying that "the kinetic energy is conserved if and only if no external force is exerted on the body along the axis of the body's motion", without having to mention the value V of the velocity? I guess that's because you want to include also the cases of so-called "elastic collisions" in which - the average external force is not zero (because the final momentum does not equal the initial momentum) - but the final kinetic energy does equal the initial kinetic energy, so the condition ${\vec {F}}=0$ (where F denotes the average external force exerted on the body along the axis of the body's motion) wouldn't be a sufficient condition, and that's why you had to strengthen it as: ${\vec {F}}\cdot {\vec {v}}=0$ where both F and V are average values, am I right?

2. I asked: "Is there any simple necessary sufficient condition that conserves kinetic energy, without mentioning kinetic energy?"

By asking that, I meant the following:

Is there any simple necessary sufficient condition, satisfying the following:

If a given system, carried (at the beginning of the process) an initial mass $m_{1}$ and an initial velocity ${\vec {v}}_{1}$ and an initial kinetic energy $E_{K1},$ and is carrying (right now) a final mass $m_{2}$ and a final velocity ${\vec {v}}_{2}$ and a final kinetic energy $E_{K2},$ then $E_{K1}=E_{K2}$ if and only if the condition holds.

Then I added, that the condition was not allowed to mention the kinetic energy.

Please note:

a). The kinetic energy can be defined as $\int {\vec {F}}\cdot d{\vec {s}}$ (when it's defined as zero if the momentum is zero), when ${\vec {F}}$ denotes the force and $d{\vec {s}}$ denotes the infinitesimal displacement, so I don't allow the condition to mention (at once) both ${\vec {F}}$ and $d{\vec {s}}$ (sorry for not making it clear before). Nor do I allow the condition to mention any set of properties (e.g. ${\vec {F}},{\vec {v}},dt$) that determines (at once) both ${\vec {F}}$ and $d{\vec {s}}$.

b). The kinetic energy is also determined once we are given any pair of the following three properties: mass-velocity-momentum, so I don't allow the condition to mention any such a pair either (sorry for not making it clear before). Nor do I allow the condition to mention any set of properties that determines any such a pair.

c). Your condition ${\vec {F}}=0$ is not necessary. Check: $m_{1}={\vec {v}}_{1}=m_{2}=1,{\vec {v}}_{2}=-1.$

d). Your condition ${\vec {F}}=0$ is not sufficient. Check: $m_{1}={\vec {v}}_{2}=1,m_{2}={\vec {v}}_{1}=2.$

HOTmag (talk) 19:28, 5 August 2024 (UTC)

Is there any simple necessary sufficient condition, that conserves a given system's total energy, without mentioning energy/mass?

Just as absence of external force is a simple necessary sufficient condition that conserves momentum, without mentioning momentum. HOTmag (talk) 10:57, 31 July 2024 (UTC)

The net work done by the system is equal to the net heat received. PiusImpavidus (talk) 17:45, 31 July 2024 (UTC)

It's not a necessary condition: Consider a harmonic oscillator, in which the total energy is conserved, even though the (changing) net work done by the system is not equal to the (zero) net heat received. HOTmag (talk) 21:09, 31 July 2024 (UTC)

What work? An isolated harmonic oscillator does no work on its surroundings. The parts forming the isolated harmonic oscillator do work on each other and their energies oscillate, but the whole system does no work and has constant energy.

Like in the above discussion, you don't keep proper track of what's in your system and what's not. PiusImpavidus (talk) 12:25, 1 August 2024 (UTC)

Sorry for not clarifying myself. Anyway, I meant the following:

When a gravity pendulum is influenced by gravitation, the pendulum's total energy is conserved, even though the (changing) net work done by the gravitation on the pendulum is not equal to the (zero) net heat received by the pendulum's movement. HOTmag (talk) 14:23, 1 August 2024 (UTC)

There are two ways to look at this:

1. When the bob of the pendulum climbs, its kinetic energy is converted into potential energy, but still present in the bob. The total energy of the bob doesn't change and no work is done. When the bob descends, the potential energy is converted back into kinetic energy, but again no work is done. This is mathematically the easy way, but not entirely correct.
2. When the bob of the pendulum climbs, it performs work on the gravitational field. The bob loses energy, the field gains energy. When the bob descends, the field performs work on the bob. The field looses energy, the bob gains it. The energy of the bob isn't conserved, but an isolated bob without a gravitational field is no harmonic oscillator. The system of bob+string+gravitational field is a harmonic oscillator and energy in that is conserved. Unfortunately, this approach is mathematically very hard.

In physics, we store energy in fields all the time. PiusImpavidus (talk) 08:22, 2 August 2024 (UTC)

no work is done. What? Don't you agree that the work is equivalent to the change in kinetic energy? HOTmag (talk) 11:18, 2 August 2024 (UTC)

In physics, Work is the product of a force and a lasting, permanent displacement. Philvoids (talk) 18:43, 2 August 2024 (UTC) Underlining added for clarity.

If the force is denoted by ${\vec {F}}$ and the displacement is denoted by ${\vec {s}},$ then the work is defined as $W=\int {\vec {F}}\cdot d{\vec {s}},$ and it's equivalent to the change in the kinetic energy, isn't it? HOTmag (talk) 21:02, 3 August 2024 (UTC)

As I stated, it's not entirely correct, but it's mathematically the easy way. PiusImpavidus (talk) 08:17, 4 August 2024 (UTC)

The more correct alternative is stating that gravity does do work on the bob (approach 2), but then the bob can't have potential energy. PiusImpavidus (talk) 08:24, 4 August 2024 (UTC)

1. I'd asked: "Is there any simple necessary sufficient condition, that conserves a given system's total energy, without mentioning energy/mass?"

2. You'd answered: "The net work done by the system is equal to the net heat received".

I assumed your answer was entirely correct, so I presented a counter example, with a pendulum's bob being the system I'd asked about:

3. When the bob reaches its lowest point, the bob carries a positive kinetic energy. Agree?

4. When the bob reaches its highest point, the bob carries no kinetic energy. Agree?

5. The heat doesn't change. Agree?

6. Hence, the (changing) net work done by the gravitation on the bob is not equal to the (zero) net heat received by the bob's movement. Agree?

7. The bob's total kinetic energy is conserved. Agree?

8. There's a contradiction between: the combination of 1,2, and the combination of 6,7. Agree?

For every statement of the above eight, please indicate if you agree to it. HOTmag (talk) 19:13, 5 August 2024 (UTC)

To be pedantic, I'll assume the point of view where no gravitational potential energy is stored in the bob; it will be stored in the gravitational field instead.

1. Yes, that's what you asked.
2. Yes, that's what I answered.
3. Indeed, at the lowest point the bob has positive kinetic energy.
4. Indeed, at the highest point the bob has no kinetic energy. The energy has been stored in the potential energy of the gravitational field.
5. Heat doesn't change; it flows. But in this case, the heat flow is zero, so I agree.
6. Indeed, during each quarter swing (left–centre–right–centre–left), either positive or negative work is done by the gravitational field on the bob, but no heat ever flows.
7. No, the kinetic energy of the bob isn't conserved. It goes up and down all the time. The total energy of the oscillator (kinetic energy of the bob plus potential energy of the gravitational field) is conserved.
8. No, there's no contradiction. The answer at 2 was about the total energy of the oscillator, the statement at 7 is about the kinetic energy of the bob.

You forgot that the gravitational field is part of the oscillator too. It's as important as the bob. PiusImpavidus (talk) 08:45, 6 August 2024 (UTC)

Oh!!! sorry for the big mistake I made in question #7, when I drafted it, adding the redundant word "kinetic". I've just struck it out. Please answer again #7 and #8, and please explain to me what I still didn't understand: when the bob gets to its highest point, doesn't the bob carry a potential energy? If it doesn't, in your opinion, then could you back it with any authoritative source? From Wikipedia if possible. Thank you for your patience! HOTmag (talk) 12:26, 6 August 2024 (UTC)

There are two ways to view this. The lazy way says that the gravitational potential energy is stored in the bob, which is converted to kinetic energy when moving down. Many beginner's books tell this. But if gravity performs work on the bob, this point of view violates conservation of energy. One solution is to ignore work done by gravity, which is easy and what most physicists do most of the time. The other, and proper, way to look at this, explained above under 2, is that the potential energy is stored in the gravitational field, not in any object. Unfortunately, the maths get really hard if you try it that way. As you wondered about the work done by gravity, that's the point of view I'll use here – but I'll skip the maths. So:

7. The bob's energy isn't conserved. The energy moves back and forth between the kinetic energy of the bob and the potential energy of the gravitational field. The total energy of the oscillator, the sum of these, is conserved.

8. No contradiction. Field and bob do work on each other, but are both part of the system. The system performs no work on the outside world. The energy of the system is conserved. The energy of the bob isn't.

It's just a matter of proper accounting. PiusImpavidus (talk) 20:01, 6 August 2024 (UTC)

I'm pretty surprised to read now, that the attitude I've always been taught about, is only what "Many beginner's books tell".

Do you think Wikipedia may add some info about the new description you support?

Additionally and more important: I still don't understand what's wrong in the attitude you describe as the one presented in the beginner's books. You say that "if gravity performs work on the bob, this point of view violates conservation of energy". So first, please notice that also the new description you support violates conservation of energy, because when the bob gets to the highest point the bob has lost the whole kinetic energy the bob carried when it was at the lowest point. Second, according to the attitude you describe as the one presented in the beginner's books, gravity is not committed to the conservation of kinetic energy, but rather is only committed to the sum of the kinetic energy and potential energy, and this sum is always conserved under gravity, even when the potential energy is attributed to the bob rather than to the gravitational field. HOTmag (talk) 07:40, 7 August 2024 (UTC)

Many more advanced books don't mention the subject at all.

Suppose an elevated object caries gravitational potential energy. When we let it fall down, it looses this potential energy and gains kinetic energy. So the source of the kinetic energy was already in this object; no energy is transferred to the object when it falls. However, gravity provides a force pulling the object down en provides work to accelerate the object. So the energy is transferred to the object in the form of work by gravity and can't have been in the object beforehand. This is a contradiction. There are just two ways to solve it: either gravity does no work or the potential energy was stored somewhere else than in the object. As you asked about the work done by gravity, I was forced to take the latter approach.

First point: when the bob reaches its highest point, it has lost all its kinetic energy. Gravity has done negative work on the bob and the energy has been stored in the gravitational field. So energy has been conserved.

Second point: The sum of potential energy and kinetic energy is conserved, provided no work is performed on the system. But work is performed on the bob, by gravity. So if our system is just the bob, the potential energy can't be stored in the system, so it's not in the bob.

I never really thought about it for a long time after reading my first books on physics a long time ago, but it's actually pretty obvious. PiusImpavidus (talk) 09:41, 8 August 2024 (UTC)

Suppose an elevated object caries gravitational potential energy. When we let it fall down, it looses this potential energy and gains kinetic energy. So the source of the kinetic energy was already in this object.

What? According to the attitude you described as the one presented in the beginner's books, the source of the kinetic energy was not in the object, because every kinetic energy added to any object is given to the object by a force, being the gravity in our case. This is supposed to be obvious in classical mechanics, which really considers gravity to be a "force" .

no energy is transferred to the object when it falls.

I agree, if by "no energy" you mean zero total energy, because in classical mechanics gravity is a conservative force, which actually conserves the sum of potential energy and kinetic energy. However, gravity does not conserve the kinetic energy, and a kinetic energy is transferred to the object when it falls.

First point. When the bob reaches its highest point, it has lost all its kinetic energy. Gravity has done negative work on the bob and the energy has been stored in the gravitational field. So energy has been conserved.

Correct. I apologize for my first point. I confused kinetic energy - which is actually not conserved, with total energy (being the sum of potential energy and kinetic energy in our case) - which is actually conserved.

Second point: The sum of potential energy and kinetic energy is conserved, provided no work is performed on the system.

What? According to the attitude you described as the one presented in the beginner's books, the sum of potential energy and kinetic energy is always conserved under any conservative force, while gravity is really considered to be a conservative force - in classical mechanics, regardless of the work done by that conservative force. According to this attitude, "Work" is equivalent to the change in kinetic energy if there is no change in the potential energy, and is also equivalent to the change in potentia energy if there is no change in the kinetic energy. HOTmag (talk) 11:51, 8 August 2024 (UTC)

First two paragraphs: yes, there's an inconsistency. That's what I was trying to explain. I'll consider it settled now.

Last paragraph: yes, gravity is a conservative force, so the sum of potential and kinetic energy is conserved. I was only acknowledging the possibility that other forces may be present. PiusImpavidus (talk) 16:05, 9 August 2024 (UTC)

I don't see any inconsistency. The attitude you described as the one presented in the beginner's books, does not agree to your claim that "the source of the kinetic energy was already in the object". Where is the inconsistency?

Nor does this attitude agree to your addition "provided no work is performed on the system", so again, where is the inconsistency? HOTmag (talk) 16:29, 9 August 2024 (UTC)

Physicists carefully define Energy a measurable property that is transferred in interactions and Mass the intrinsic quantity of matter in a body because these are fundamental to both our study and understanding of many Systems i.e. functional groupings of elements that act according to a set of rules. Posing yet another challenge to define a physical system "without mentioning those words" is just inviting responders into a handicapped word play that may provide the engaging debate that the OP is seeking to expose their own ideas but that is a misuse of the reference desk that should not continue. Underlining added for clarity. Philvoids (talk) 10:03, 2 August 2024 (UTC)

I didn't ask about defining something (without using specific words). I only asked about a necessary sufficient condition for the conservation of (total) energy (without mentioning energy/mass), which is a definitely different thing!

Just as the absence of external net force is a necessary sufficient condition for the conservation of momentum, without mentioning momentum. This is a definitely legitimate question. HOTmag (talk) 11:27, 2 August 2024 (UTC)