@123 No, the horizontal force too

@Azmuth :P neglect air resistance or any drag force

k... ok

One object on the floor and another on top of ball we can explain this example by height , gravity , normal etc.. why we need another quantity (work, KE PE) to explain the same phenomenon?

Just to solve the equations of the system?

A state of a system can be fully represented using KE and PE. (except position)

It's extremely practical. The conservation of energy allows for a lot of calculations to be much simpler than trying to solve them with forces, time, and position and it's time derivatives.

@123 The Newtonian equations of motion require only that you know the net force on the object, this doesn't reference potential or kinetic energy at all

@JMac Exactly! Convservation laws help us to avoid complex paths and stuff.

Why we dot product Force and displacement. Not Force and velocity. Any special reason friends.

hyperphysics.phy-astr.gsu.edu/hbase/pow.html

@123 Dot product times displacement helps us to define PE and PE helps us to define KE.

That's it.

@Azmuth ...that's not what "state of a system" means. The full state of a system is enough data to completely specify the initial conditions needed for the equations of motion.

If we dot product Force and velocity we can also derive some more physical quantities.

Boooo

@123 Force and velocity gives you the change in energy over time.

Or Power

@123 You can dot product any two vectors you want, you'll get different quantities with varying degrees of physical meaning/usefulness

as Charlie linked

Thanks @Charlie you remind me three years ago i derived this expression by my own. using power

Once again after derived i thought what is the meaning of this force with velocity.

You could take the dot product of an object's velocity with the vector denoting its relative position to the sun, there's nothing that stops you doing this, it's just not a very useful physical quantity

Sometimes you meet your destiny in the path you chose to avoid it.

@Charlie Hmm... That's GooD. It means dot force with displacement gives us more fundamental quantity than any other that's why we stick to this?

@123 It's pretty intuitive really. If change in energy is Force dot position, and velocity is the change in position over time. The force dot velocity being the same as the change in energy over time fits with that. (when energy changes with position)

"More fundamental" is not necessarily true, "more useful" perhaps

@Azmuth :P

:)

"fundamental" is an ill-defined notion really. There are various ways to build physics from the ground up that consider different things to be "fundamental" and "derived".

In Lagrangian and Hamiltonian mechanics the state of the system is determined uniquely by its coordinates $(q,\dot q)$ and $(q,p)$ respectively

@Charlie +1 for using $q$ instead of $x$

only nerds use cartesian coordinates

yep! :)

Problem is that my friends when i write $\vec{P} = m\vec{v}$ i can actually measure the velocity and mass of object in real time movement. The system in actual showing this behavior. At the same time i write $KE = \frac{1}{2}mv^2 = \frac{1}{2}m(\vec{v}\cdot\vec{v})$ i can not see velocity square of that mass for the same system. This is the problem.

Sorry I forgot emojis were not allowed :P

Not sure what you mean @123

@Charlie I've seen people write polar coordinates as $x^r, x^\theta, x^\phi$ :P

@ACuriousMind Polar coordinates is for kids only

then they're nerds!

Relativistic coordinate systems is pro!

Anyone uses Homogeneous coordinate system?

It looks coolest!

I don't know what a "relativistic coordinate system" is, nor why you couldn't choose the spatial part to be polar.

Only spatial parts are polar

That's the proest!

@ACuriousMind In linear track if object having mass m and velocity v. I can write momentum as quantity of motion any one parameter change its effect can be experience be stopping it. But for the same system i write $KE = \frac{1}{2}mv^2$ there is no velocity square in that system

Hey what are your Halloween ideas? Gimme som ideas!

@123 I think the problem might be that you are trying to see it. You can't really see momentum either. You can see the velocity, and determine the mass; but just watching the object move doesn't really tell you the momentum or kinetic energy; and the same two measurements will tell you both.

@123 I don't know what " there is no velocity square in that system" is supposed to mean

Sorry guys for long discussion. Hope after this discussion with you guys i will come to the very good explanation.

if you can measure velocity, you can measure its square

not every quantity in our equations has to exist "visibly" in the system for it to be meaningful

what i am looking object is moving with velocity not with velocity square

sure, because the square of velocity is not a velocity :P what's the problem with that?

@ACuriousMind Ookay. It means it is just for calculation purposes.

And the useful quantity "energy" depends on velocity squared, so we calculate that when using energy because it actually works, and still only requires only the same two measurements of mass and velocity

@JMac Yes this was my question. Why it is more important to learn Energy , KE, PE , work-done all the way in physics. If we have other quantities to measure e.g: distance, speed, temperature etc..

.

In projectile motion and vertical motion the amount of work is same???

If an object achieve same height in projectile as in vertical motion. What is the amount of work it is same or different?

the amount of work done by what force?

Only force due to gravity.

then sure, that force only cares about vertical portions of the displacement

@ACuriousMind Thanks. mean work only cares about in the direction of force. The other component of motion is neglected.

It means both motions have same amount of work.

@123 Temperature is a great example of why energy is important in physics. To figure out how a system changes temperature, you typically measure the heat transfer between two systems. This heat transfer is the exchange of thermal energy between the two systems.

@JMac Great Now i think with this example i can find the importance of work.

That's i have given the example of temperature. Thermodynamics always uses idea of energy.

Thermodynamics also gives good examples of other types of energy turning into macroscopic work, or vice versa.

@JMac It is better to understand. Energy gives us an idea how temperature changes between objects.

hi all, a quick question

...and are you going to actually ask the question? :P

in this passage the intertwining linear map it is considered as an operator that modify the Hilbert space. T instead here is a generator of a Lie algebra. To me this seems as a general case respect the one most known $F=H $ (hamiltonian) with $[H,T]=0$ imply conservation of T.

I recognised I expressed myself badly

@ACuriousMind it is taking time ahahah

no worries

what i Mean is, I can express a qunatity conservation under the temporal evolution of the states with [H,T]=0, but I can even consider a more general case of quantity conserved under a general transofrmation of the Hilbert States given by F, with [F,T]=0.

arxiv.org/abs/0904.1556, here is the paper if anyone is interested

...and the question is? Until now you've more or less rephrased what's written in the picture you posted

this is it, i just wanted be sure there was this matching with QM commutators, and intertwining maps. I am having more or less a dffuclt time understanding the concept of intertwining map as a physical process

"intertwiner" is just another name for something that commutes with the group/algebra action

note that they are careful to talk about two different spaces $V$ and $W$, but in the cases you've mentioned, $V=W$ is just the single physical Hilbert space of states and the intertwiner is therefore just an operator on $V=W$.

yes you are right, I didn't notice. So indeed it is even a more generalization than what I thought, because it can relates even different Hilbert spaces.

this actually helps me understand the next part, as i think I can see this kind of map as the one that relates a one particle Hilbert space and a two particle Hilbert space, like in $\pi^- + p \rightarrow n$

And the intertwining map being linear respect SU(2) gives me conservation of isospin for the process

If you can't beat them join them

I have just seen the conversation, actually I thought it was a term used mainly in mathematics ahaha