hamiltonian-lagrangian

[AntonioLao]

the Hamiltonian is defined as the sum of kinetic energy (T) and potential energy (V) and for time independent systems, the Hamiltonian is a constant equals the total energy of the system. On the other hand, the Lagrangian is defined as the difference between kinetic energy and potential energy.

Can a square of energy defined as the product of the Hamiltonian and Lagrangian as follow?

[math]E^2=H*L=(T+V)(T-V)=T^2-V^2[/math]

[Guille]

Can a square of energy defined as the product of the Hamiltonian and Lagrangian as follow?

[math]E^2=H*L=(T+V)(T-V)=T^2-V^2[/math]

hummm.....

I was about to post an important and long reply, but I just wanted to ask you about what do you mean by the sqaure of energy? the square of the sum of the energies of a system (ke+pe)? or what?

[AntonioLao]

In special relativity, the square of energy is given by

[math]E^2=c^2p^2+m^2c^4[/math]

Dirac used it to theorize the existence of antiparticles.

[Guille]

In special relativity, the square of energy is given by

[math]E^2=c^2p^2+m^2c^4[/math]

Dirac used it to theorize the existence of antiparticles.

ok, now I see the conection between so many posts and threads of yours...

given this, then [math]E^2=H*L=(T+V)(T-V)=T^2-V^2[/math]

if we asume these two equations then we get H*L=c^2p^2+m^2c^4

and also because your quantized space is E^2 then quantized space=H*L

did you MEAN this two?

note: in the equationa I wrote quantized space because you represent it with (H) which is the same letter/signal as the hamilton (H).

[AntonioLao]

did you MEAN this two?

the confusion was not intended. The H of quatized space stands for Hadamard matrices (nonlinear energy functions), while the H in classical and quantum mechanics stand for the linear energy functions. It is the total energy of time independent classical systems and as the complex energy operator in quantum mechanics. For more concise descriptions in quantum mechanics, the Hamiltonian operators have been replaced in almost all cases by the Lagrangian operators. But both Hamiltonian and Lagrangian are linear energy operators, while the H in quantized space is a nonlinear operator if only a differential equation exists. I am searching for this differential equation. It might be a hyperbolic type due to the difference of square functions.

[Guille]

[math]E^2=H*L=(T+V)(T-V)=T^2-V^2[/math]

can you think of a number that when it is squared it equals the difference between two numbers times the sum of those two numbers?

although in physics it does make sense, and I think you can expres it like this. if you notice, getting the begining and ending parts, you get a sort of pythagoras theorem:

E^2+V^2=T^2

I am stil thinking on it, trying to get more things to say.

[AntonioLao]

additional thoughts:

From our existential perspective - matter is four dimensional, energy is three dimensional, continuous space is two dimensional, quantized space is one dimensional, the false vacuum is of fractal dimensions, and lastly, the true vacuum is zero dimension.

From a higher LOE (level of existence) - our universe is zero dimension.

[Guille]

in extent to my other posts, if
[math]E^2=H*L=(T+V)(T-V)=T^2-V^2[/math]
then we get that:

T^2-V^2=H*L

so would yopu say that the difference between kinetic and potential energy is equal to to total energy? thus, potential or maybe kinetical energy is the opposite to the other energy?

[AntonioLao]

thus, potential or maybe kinetical energy is the opposite to the other energy?

In thermodynamics, the virial theorem of Clausius states that in a time average configuration of the 1st order, the potential energy is exactly equal to 1/2 the kinetic energy.

[Guille]

thanks for that.

I want to get more knwoledge on it, so, what is actually potential energy?
please...


(I know enough about KE, but if you want to also explain me, thanks)