Energic vs. Mechanistic Worldview

[Xenophon]

I first became interested in the difference between the energic and the mechanistic worldview when I was studying Robotics last year. We were looking at different ways to solve the dynamics of a robotic system. We covered two different methods, the Euler-Newton method, and the Euler-Lagrange method.

For those who aren't familliar with these two ways of looking at a dynamics problem, the essential difference is that Newtonian methods look at the forces present in the system. By calculating all the forces that are present at any instant in time, you can solve the acceleration (and thus the movement) of the system. This method is very logical, and though it can become very complex with large systems, each little part is easy to understand. The Lagrangian method focuses on the total energy in the system. By subtracting the potential energy in the system from the kinetic energy, you come up with a term called the lagrangian, and once you have it, you can determine the output of the system to any input without having to solve the forces at all. While this method takes a lot more initial work, it greatly decreases the computation needed to solve the dynamical equations in any robot configuration when it is done.

Later I happened to be looking further into the work of Lagrange, and I found out that he generalized both Newtonian Mechanics and elementary calculus to Lagrangian Mechanics and the Calculus of Variations, of which the former are specific cases (sort of like Static analysis being a specific case of Dynamic analysis that happens when things aren't moving.) Though I am not a physicist, I have read that special relativity and quantum mechanics both use Lagrange's Calculus of Variations in their proofs. Though I don't understand this whole thing very well, my intuition tells me that there is something very important in this whole mess.

One of the problems that I have is that Newtonian mechanics just makes so much sense. If i had to say I had one strength in school, it would be that I can whip through Dynamics equations better than pretty much anybody that I know. However, learning that this is just a special case of reality makes me wonder what else that I have completely accepted as the be all and end all is just a some specific result from something greater that is beyond my comprehension.

I've ordered three textbooks on the Calculus of Variations and Feynmans Physics lecture series to help me come to grips with all this. When I have read those I'm sure I will have much more to say. But I was wondering what thoughts other people have about these ideas.

[Hypnos]

To be clear, Newtonian force-body mechanics and Lagrangian methods are two different ways to arrive at the same equations of motion. Over all of physics, Lagrangians _generate_ various objects of a particular nature: equations of motion, Feynman diagrams, and even non-linear phase changes.

The intuitive picture is that Lagrangians capture the gross energy minimization and symmetries of a system, and through the machinery of calculus of variations can generate all the interesting measurables in a system.

This is particularly useful in relativistic physics and/or time-dependent quantum mechanics, where the Hamiltonian can be thought of as just the momentum conjugate to time.

[Xenophon]

To be clear, Newtonian force-body mechanics and Lagrangian methods are two different ways to arrive at the same equations of motion. Over all of physics, Lagrangians _generate_ various objects of a particular nature: equations of motion, Feynman diagrams, and even non-linear phase changes.
This is what I find really interesting about Lagrangian methods. We have some mathematical model that is pretty much unobservable, yet in specific cases it can generate all the observable things that we take for granted every day. It makes me wonder about the validity of strictly empirical theories.


The intuitive picture is that Lagrangians capture the gross energy minimization and symmetries of a system, and through the machinery of calculus of variations can generate all the interesting measurables in a system.

This is particularly useful in relativistic physics and/or time-dependent quantum mechanics, where the Hamiltonian can be thought of as just the momentum conjugate to time.

I was following fine until you said: "the Hamiltonian can be thought of as just the momentum conjugate to time." What does that mean? I know what momentum is, and I know what a conjugate is, but I don't understand how they all string together like that. Also, I don't know really know anything about a Hamiltonian, is it somehow related to the Lagrangian? Sadly, in undergraduate engineering they tend to ignore high level mathematics like this.

[Lee]

Promise to self: oneday I will understand this.

[Claverhouse]

Promise to self: oneday I will understand this.
How many Angels can dance on the end of a pin may be slightly more important.



Claverhouse :ph34r:

[Hypnos]

This is what I find really interesting about Lagrangian methods. We have some mathematical model that is pretty much unobservable, yet in specific cases it can generate all the observable things that we take for granted every day. It makes me wonder about the validity of strictly empirical theories.
How can a theory be "strictly empirical?" You can have an empirical model that just reproduces the numbers you observe with some assumption of smoothness or some obvious symmetry; a theory, I would say, really reduces the information to a few degrees of freedom, and has high predictive value. This is the basis of science.


I was following fine until you said: "the Hamiltonian can be thought of as just the momentum conjugate to time." What does that mean? I know what momentum is, and I know what a conjugate is, but I don't understand how they all string together like that. Also, I don't know really know anything about a Hamiltonian, is it somehow related to the Lagrangian? Sadly, in undergraduate engineering they tend to ignore high level mathematics like this.
Sorry for being jargon-y. Consider any momentum, p, directed in the direction x. Then, p is considered to be conjugate to x, in the sense that p "generates" x. Similarly, angular momentum generates rotation, and the Hamiltonian (often equal to just the energy) generates time.

The Hamiltonian H is equal to the L - sum(p * dq/dt), where p is some momentum and q is the spatial, etc. degree of freedom to which it is conjugate.

[Xenophon]

How can a theory be "strictly empirical?" You can have an empirical model that just reproduces the numbers you observe with some assumption of smoothness or some obvious symmetry; a theory, I would say, really reduces the information to a few degrees of freedom, and has high predictive value. This is the basis of science.
That is true. A theory can't be strictly empirical. I suppose I am struggling with the distinction between knowing why something happens, and knowing how something happens. I get the feeling that science generally starts from the how and moves to the why. While I have a natural inclination to start with why, and then move to how. That's why I was both so amazed and frustrated when I discovered this entire avenue of scientific thought that was completely ignored in my undergraduate education. Of course, this is probably more of a complaint about engineering than science. We are the masters of knowing how without having any idea why.


Sorry for being jargon-y. Consider any momentum, p, directed in the direction x. Then, p is considered to be conjugate to x, in the sense that p "generates" x. Similarly, angular momentum generates rotation, and the Hamiltonian (often equal to just the energy) generates time.

The Hamiltonian H is equal to the L - sum(p * dq/dt), where p is some momentum and q is the spatial, etc. degree of freedom to which it is conjugate.
Ok, that sort of makes sense, but it is going to take some major pondering to sort out. But to see if I have understood clearly: since the lagrangian is the sum of kinetic energy - potential energy in each "direction", the hamiltonian is essentially the sum of the kinetic energy - the potential energy - the momentum in each direction?

This is completely out of left field, but does that mean that the kinetic energy of a photon is equal to the potential + the momentum?

[flan2dave]

My impression of calculus of variations is it takes some clever manipulation, but there's lots of freedom in designing the initial equation to be solved. Ideas in dynamic systems, physical phenomenon, or interesting math properties (or a linear combination of these *chuckles*) can be explored deeply. I get the feeling it's important too.


To be clear, Newtonian force-body mechanics and Lagrangian methods are two different ways to arrive at the same equations of motion. Over all of physics, Lagrangians _generate_ various objects of a particular nature: equations of motion, Feynman diagrams, and even non-linear phase changes.

The intuitive picture is that Lagrangians capture the gross energy minimization and symmetries of a system, and through the machinery of calculus of variations can generate all the interesting measurables in a system.

This is particularly useful in relativistic physics and/or time-dependent quantum mechanics, where the Hamiltonian can be thought of as just the momentum conjugate to time.

Sexy.

[Hypnos]

That is true. A theory can't be strictly empirical. I suppose I am struggling with the distinction between knowing why something happens, and knowing how something happens. I get the feeling that science generally starts from the how and moves to the why. While I have a natural inclination to start with why, and then move to how. That's why I was both so amazed and frustrated when I discovered this entire avenue of scientific thought that was completely ignored in my undergraduate education. Of course, this is probably more of a complaint about engineering than science. We are the masters of knowing how without having any idea why.
"Why" can mean different things: by what cause, or for what purpose? I think you hit the nail on the head: science picks apart "by what cause" in a slow march, and the "for what purpose" picture comes together over time in the form of deep physical principles. In engineering, you start with the purpose, and the "by what cause" elements come together with experience. Bottom-up versus top-down.



Ok, that sort of makes sense, but it is going to take some major pondering to sort out. But to see if I have understood clearly: since the lagrangian is the sum of kinetic energy - potential energy in each "direction", the hamiltonian is essentially the sum of the kinetic energy - the potential energy - the momentum in each direction?
First thing, I got typo'd the sign: H = sum(p_i*q_i) - L. This is a so called Legendre Transformation (http://mathworld.wolfram.com/LegendreTransformation.html). So in many instances, H is the total energy KE + PE; exceptions often occur when there is some additional symmetry that modifies how space and its derivatives behave.


This is completely out of left field, but does that mean that the kinetic energy of a photon is equal to the potential + the momentum?
Yeah, I'm not sure what that means :)

flan2dave: Ambiguity in how to formulate the equations of motion from an action (integral of the Lagrangian) is often an indication of the symmetries in the Lagrangian. For example, when quantizing the electromagnetic field in order to obtain photons on paper, one has to choose a gauge or deal with the gauge in the form of a parameter usually denoted by \ksi.