# Thread: Newtonian, LaGrangian, Hamiltonian ?

1. ## Newtonian, LaGrangian, Hamiltonian ?

Can someone please elaborate on the differences between these? Newtonian is classical, and Hamiltonian relates to the mathematical manipulation of Energy and used in Quantum. How about LaGrangian?

Can someone please sum this up?

3. ## Re: Newtonian, LaGrangian, Hamiltonian ?

Originally Posted by tom
Can someone please elaborate on the differences between these? Newtonian is classical, and Hamiltonian relates to the mathematical manipulation of Energy and used in Quantum. How about LaGrangian?

Can someone please sum this up?
The LaGrangian is called the "action" and is the difference between the kinetic and the potential energy:

L = T(m.v) - V(x,t). In QM, this quantity is Planck's constant (for conserved photo-electron mass and other forms via the DeBroglie relation) For STR, V(x,y) = 0 (the mass/energy does not depend on coordinates for a conserved "inertial" system) the "rest" and "final" Lagrangians (ct,ct') are constants.

For QM, the wave equation is given by:

where Px and Et are scaled relative to h around a circle (in 2 directions) with the source and sink (Et) at the same "position" (a single point). If nothing happens during the round trip, then Px-Et = 0, and

The Hamiltonian is then the total energy of the system, as created and destroyed at the source/sink.

The wave equation can be "analytically continued" by imagining consecutive cycles, but this is a classical concept - a "momentum" exists first order at the boundaries of 0 and 2pi, whereas the system is total described (but not observed) by the above Et=Px.

STR is the equivalent of Newton's laws of motion, but relates an initial condition to a final condition via a linear perturbation:

E=mc2
(Relativity)

v=x/t, k.e. = mv2 (Newton)

(There is more to this story....

IMO, YMMV....

4. ## Re: Newtonian, LaGrangian, Hamiltonian ?

Originally Posted by BuleriaChk
The LaGrangian is called the "action" and is the difference between the kinetic and the potential energy:
The action is the integral of the Lagrangian over time, its units are energy x time
L = T(m.v) - V(x,t). In QM, this quantity is Planck's constant (for conserved photo-electron mass and other forms via the DeBroglie relation) For STR, V(x,y) = 0 (the mass/energy does not depend on coordinates for a conserved "inertial" system) the "rest" and "final" Lagrangians (ct,ct') are constants.

For QM, the wave equation is given by:

where Px and Et are scaled relative to h around a circle (in 2 directions) with the source and sink (Et) at the same "position" (a single point). If nothing happens during the round trip, then Px-Et = 0, and

The Hamiltonian is then the total energy of the system, as created and destroyed at the source/sink.

The wave equation can be "analytically continued" by imagining consecutive cycles, but this is a classical concept - a "momentum" exists first order at the boundaries of 0 and 2pi, whereas the system is total described (but not observed) by the above Et=Px.

STR is the equivalent of Newton's laws of motion, but relates an initial condition to a final condition via a linear perturbation:

E=mc2
(Relativity)

v=x/t, k.e. = mv2 (Newton)

(There is more to this story....

IMO, YMMV....

5. ## Re: Newtonian, LaGrangian, Hamiltonian ?

(in QM and Special Relativity, the action h IS the Lagrangian, since coordinates are unobservable (there is no V(x), only the x in Px); Et/h is dimensionally consistent, t is just a scaling factor on the Energy E, since one takes the derivative of the wave equation w.r.t. t OR x and multiplies the result by h/(2) to remove the coordinate (angular phase of the round trip) to calculate the probability of an interaction as an observable (if no interaction, then the factor is 0 so "probability" is 1 - the wave formulation is irrelevant)

t'/t-= ,

For the factor (Px-Et), use xc=ct so: t(Pc-E) is the factor in the wave equation, Pc is the relativistic momentum, E is the source/sink energy, and any change (observable) is extracted by taking the derivative of the wave equation with respect to t. (if Pc = E), the derivative is 0 (no interaction) for any t, and the derivative is irrelevant (there is no observable).

If the variables are separable, one can take the derivative of EITHER Px or Et and extract the "momentum" and "energy" independently, but further discussion is outside the scope of my interest in this thread).

(On a personal note, Ableton Live just came out with version 9.5 and Push 2, and Maschine upgraded to version 2.4 on the same day (Monday), so I'll be busy with that for awhile... and my Push 2 arrives on Friday)...

6. ## Re: Newtonian, LaGrangian, Hamiltonian ?

PS: It really helps to imagine "x" in QM as a radius, even though it is factored out when one takes the derivative, so that Px = Pr (one can then assume that the dot product is taken: P.r; if a cross product is taken, a new particle is formed.: (P(x)r

Substitute r=ct for fun, relativistically speaking, and use relativistic definitions of P and E ....

Then the observable is in a relativistic "inertial frame", independent of coordinates... (unless it itself is a function of coordinates, in which the derivative gets way more complicated..

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