Difference between Hamiltonian and Lagrangian Mechanics

[JTC]

Hello, I am trying to "integrate into my understanding" the difference between Hamiltonian and Lagrangian mechanics.

In a nutshell: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to the minimization principle?

YES; I KNOW about Hamilton's Second Principle with phase space and momenta, but I ask you to ignore that in this.

For example, I understand this:

"Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). Both formulations are equivalent by a Lengendre transformation on the generalized coordinates, velocities and momenta, therefore both contain the same information for describing the dynamics of a system."

HOWEVER!

I am also learning about The Calculus of Variations. And I am learning it is called Hamiltonian Dynamics (so I assume Hamilton's First principle involves this, BEFORE he gets to phase space with his other principle), but uses a Lagranian

And there you have my confusion: a mish-mosh soup of THREE terms and I am unable to state, concisely, the difference in these views with regard to using variational methods to extract the equations of motion for simple dynamical systems.

Can someone please distinguish these three words and the contributions of Hamilton vs Lagrange, simply, concisely (but under the aegis of ONLY variational methods, and not involving momentum/phase space)?

 

[vanhees71]

There is Hamilton's principle of least action (it's THE principle of all fundamental physics, by the way). It comes in a Lagrangian and a Hamiltonian formulation.

In the Lagrangian variant the trajectories of the varied curves are in configuration space, and the Lagrangian (usually) is a function of the generalized coordinates $q^j$ and the generalized velocities $\dot{q}^j$.

Then you define the conjugated momenta,
$$p_j=\frac{\partial L}{\partial \dot{q}^j},$$
and the Hamiltonian
$$H(q,p)=\sum_j \dot{q}^j p_j - L.$$
I have assumed here that all the $\dot{q}^j$ can be expressed as functions of the $q^j$ and $p_j$. Now it is easy to show that you can extent the variational principle to variations over trajectories in phase space, parametrized by the variables $(q^j,p_j)$. Now the equations of motion are given by the stationary points of the action when varying the generalized coordinates with fixed boundary conditions as well as independently without any restrictions the canonical momenta. This extended Hamilton principle gives rise to a large general symmetry, namely the symmetry under canonical transformations of phase-space variables. This is then the most elegant formulation of classical mechanics, which are the right starting point to introduce quantum theory (in the algebraic approach a la Dirac). You'll figure out that the phase space builds a socalled symplectic manifold, and with the Poisson brackets you have an easy handle of symmetries in terms of Lie groups and Lie algebras (the Poisson brackets realize a Lie algebra on the space of phase-space functions).

 

[JTC]

There is Hamilton's principle of least action (it's THE principle of all fundamental physics, by the way). It comes in a Lagrangian and a Hamiltonian formulation.

In the Lagrangian variant the trajectories of the varied curves are in configuration space, and the Lagrangian (usually) is a function of the generalized coordinates $q^j$ and the generalized velocities $\dot{q}^j$.

Then you define the conjugated momenta,
$$p_j=\frac{\partial L}{\partial \dot{q}^j},$$
and the Hamiltonian
$$H(q,p)=\sum_j \dot{q}^j p_j - L.$$
I have assumed here that all the $\dot{q}^j$ can be expressed as functions of the $q^j$ and $p_j$. Now it is easy to show that you can extent the variational principle to variations over trajectories in phase space, parametrized by the variables $(q^j,p_j)$. Now the equations of motion are given by the stationary points of the action when varying the generalized coordinates with fixed boundary conditions as well as independently without any restrictions the canonical momenta. This extended Hamilton principle gives rise to a large general symmetry, namely the symmetry under canonical transformations of phase-space variables. This is then the most elegant formulation of classical mechanics, which are the right starting point to introduce quantum theory (in the algebraic approach a la Dirac). You'll figure out that the phase space builds a socalled symplectic manifold, and with the Poisson brackets you have an easy handle of symmetries in terms of Lie groups and Lie algebras (the Poisson brackets realize a Lie algebra on the space of phase-space functions).

No, no... this is not what I am looking for. I do not want ANY mention of Momentum. At all. I understand what you are saying but that is not what I am looking for.

I want a SIMPLE CONCISE description that puts the following three in context (and IGNORE the fact that Hamilton went on to pose another principle. Just assume he stopped). I will try...

Hamilton and Lagrange both formulating a variational principle that relied on the Calculus of Variations.
Lagrange formulated the Lagrangian (see, now I can say what Lagrange did)
Hamilton incorporated it into the variational method (now I can say what Hamilton did)

Yes, I know Hamilton did another principle. I am not interested in that. I want these three underlined items categorized. But it bothers me because I cannot see the difference in what they did.

At the moment, it seems to me that Hamilton gets credit for what Lagrange did, but he still uses the name Lagrangian for L = T - V

That is all I am looking for. I am trying to figure out why the descriptions APPEAR to say they BOTH did a variational method, but one used a Lagrangian, when I see they BOTH used a Lagrangian.

I MEAN THIS: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to it?

 

[Andy Resnick]

Hello, I am trying to "integrate into my understanding" the difference between Hamiltonian and Lagrangian mechanics.

I'd be interested in answers to this as well. I have 2 'authoritative' books that cover both lagrangian and hamiltonian formulations (Arnold's 'mathematical methods of classical mechanics' and Weinberg's 'lectures in quantum mechanics') and they take opposite views.

Weinberg states that Hamiltonian mechanics is 'contained within' lagrangian mechanics: the lagrangian allows determination of conserved quantities based on symmetry principles, the Hamiltonian being one of the conserved quantities.

Arnold, on the other hand, states the opposite- the lagrangian is contained within hamiltonian mechanics as a special case.

So...? anyone?

 

[JTC]

I think what you want is slightly more than I am asking. I understand what you are asking and I am interested in that, too.

But now I think I figured out what I want to hear. So I am tossing it out.

"The Lagrangian APPROACH is one in which he formulated the Euler-Lagrange Equation. To do this, he used his Lagrangian. Hamilton, however, was able to develop the Euler-Lagrangian equation, NOT from a first principle, as Lagrange did, but from a minimization principle."

That is really all I want to hear. Something like that (if it is correct).

For, right now, books talk of the Lagrangian PRINCIPLE and the Hamilton PRINCIPLE and the LAGRANGIAN, and I cannot figure out what Hamilton did that was differnt from Lagrange. But now I htink it all comes down to the issue that the PRINCIPLE for LAGRANGE, was just him writing down the assume form of the Euler-Lagrange equation, while HAMILTON ensconced it within the framework of a minimization principle.

 

[vanhees71]

I MEAN THIS: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to it?

Because Hamilton invented the variational principle. I don't know, how Lagrange derived the Euler-Lagrange equations. Maybe from d'Alembert's principle?

 

[JTC]

Because Hamilton invented the variational principle. I don't know, how Lagrange derived the Euler-Lagrange equations. Maybe from d'Alembert's principle?

You have now hit the nail on the head. That is what is causing my problem...

HOW DID Lagrange derive the Euler-Lagrange equation without Hamilton's Principle that came later?

THAT will enable me to put this in order.

 

[vanhees71]

I've no clue! Perhaps one can find the original papers by Lagrange (provided one speaks French, which I don't). You can derive Lagrange's equations from d'Alembert's principle as well. That's found in many textbooks about analytical mechanics. Given that I never understood why one should bother with d'Alembert's principle to begin with, I've never been very fond of this derivation.

 

[Andy Resnick]

You have now hit the nail on the head. That is what is causing my problem...

HOW DID Lagrange derive the Euler-Lagrange equation without Hamilton's Principle that came later?

THAT will enable me to put this in order.

Ah- you are asking a 'historical question'. My copy of Dugas "A History of Mechanics" is at home, it's definitely covered and I'll take a look later tonight.

 

[JTC]

Ah- you are asking a 'historical question'. My copy of Dugas "A History of Mechanics" is at home, it's definitely covered and I'll take a look later tonight.

Actually, never mind.

I got it.

This whole issue was one of structuring the relationships between the principles. Once I saw that I needed to get the EL equations another way, it was straightforward.

I am fine now.

The problem for me, is that I became a victim of lazy textbooks that did not categorize. Sure they developed the equations OK, but I needed the categorization.

 

[Dr.D]

Ah- you are asking a 'historical question'. My copy of Dugas "A History of Mechanics" is at home, it's definitely covered and I'll take a look later tonight.

Even though JTC says, "Never mind, he's got it," I'd be interested in Dugas says, so please post your comments.

 

[JTC]

Actually, I would still like to hear what Dugas says, too.

And, in the meantime, to ensure I got this right, could someone critique my statement below?

The D’Alembert Principle is a statement of the fundamental classical laws of motion. It is the dynamic analogue of the Principle of Virtual work for applied forces in a static system.

Joseph-Louis Lagrange reformulated Newtonian mechanics into an energy-based mechanics. He based his work on the D’Alembert Principle.

From the D’Alembert Principle, Lagrange was able to develop the Euler-Lagrange equations. To do this, he considered that generalized forces are derivable from a potential energy function, which must be a function dependent only of the generalized coordinates.

William Rowan Hamilton generalized Lagrangian mechanics. Hamilton was able to confirm and derive the Euler-Lagrange equations, from another perspective. In this case, the same Lagrangian is the input to an extremal principle—known in mathematics as the Calculus of Variations—that may be used to solve for time evolution in dynamics.

Finally, a few points (just to make sure I got it):

· The Newton Equation is always applicable in dynamics.· D’Alembert’s principle or Lagrange equations are violated by forces such as sliding friction.· Hamilton’s Principle is generally the least applicable in that a generalized force might not have a potential function.

 

[Andy Resnick]

I'd be interested in Dugas says, so please post your comments.

Actually, I would still like to hear what Dugas says, too.

Dugas traces "principle of least action" in a very straight line, beginning with Descartes, then Fermat, Maupertius, Lagrange, ending finally with Hamilton. Interestingly, all these researchers began their approach to this principle in the context of geometrical optics.

Regarding Lagrange specifically, he developed his equations using Lagrangian multipliers (he invented those for statics), incorporated into the idea of 'virtual work' as applied to Newton's law ma + ∇V = 0. Perhaps most importantly, he was the first to realize the the 'principle of least action' is actually the 'principle of 'extremum action'.

Hamilton realized that nature, more often than not, chose to maximize the action rather than minimize. So, his insight was to propose that the correct property of the action was to be stationary (which is the extremal value in the calculus of variations). A consequence was the modern form of Lagrange's equation.

 

[Delta2]

My thoughts on this:

The statement that for every differentiable functional, its extremal points are stationary points, is a mathematical theorem. This theorem is for functionals, what Fermat's theorem(for stationary points) is for functions.

What nature and time evolution of physical systems has to do with this theorem? This is where Hamilton's principle comes into play, there is a certain functional assigned to each physical system, its action functional, and the system's evolution takes the path that minimizes or maximizes this action functional (and hence by the mathematical theorem) the stationary path of the action functional.

Lagrange knew this mathematical theorem and viewed some physical problems as optimization problems of functionals, but he didn't knew Hamilton's action principle.

 

[JTC]

My thoughts on this:

The statement that for every differentiable functional, its extremal points are stationary points, is a mathematical theorem. This theorem is for functionals, what Fermat's theorem(for stationary points) is for functions.

What nature and time evolution of physical systems has to do with this theorem? This is where Hamilton's principle comes into play, there is a certain functional assigned to each physical system, its action functional, and the system's evolution takes the path that minimizes or maximizes this action functional (and hence by the mathematical theorem) the stationary path of the action functional.

Lagrange knew this mathematical theorem and viewed some physical problems as optimization problems of functionals, but he didn't knew Hamilton's action principle.

Can I ask you to do one more thing? I have never read a clear concise definition PAIR that defines and distinguishes functional from function. I have seen functional used widely, so I assume this might be a laziness of speech. But you have made a clear cut usage in a near distinction. So, can you provide a definition of those two words? One that reflects off the other (and advise if if this definition is of use only in energy methods like Hamilton P.)

 

[vanhees71]

Actually, never mind.

I got it.

This whole issue was one of structuring the relationships between the principles. Once I saw that I needed to get the EL equations another way, it was straightforward.

I am fine now.

The problem for me, is that I became a victim of lazy textbooks that did not categorize. Sure they developed the equations OK, but I needed the categorization.

So now, I'd also like to know, how Lagrange derived the equations, now called EL equations!

 

[vanhees71]

Dugas traces "principle of least action" in a very straight line, beginning with Descartes, then Fermat, Maupertius, Lagrange, ending finally with Hamilton. Interestingly, all these researchers began their approach to this principle in the context of geometrical optics.

Regarding Lagrange specifically, he developed his equations using Lagrangian multipliers (he invented those for statics), incorporated into the idea of 'virtual work' as applied to Newton's law ma + ∇V = 0. Perhaps most importantly, he was the first to realize the the 'principle of least action' is actually the 'principle of 'extremum action'.

Hamilton realized that nature, more often than not, chose to maximize the action rather than minimize. So, his insight was to propose that the correct property of the action was to be stationary (which is the extremal value in the calculus of variations). A consequence was the modern form of Lagrange's equation.

Yes, and the analogy with geometrical optics played an important role for Schrödinger's heuristics to "derive" his equation of QT. Starting from L. de Broglie's idea of matter waves, he asked for a wave equation that in the leading-order eikonal approximation leads to classical mechanics of point particles. This is analogous to the relation between wave and ray optics. The latter can be derived from the Maxwell equations by the eikonal approximation (an expansion in terms of a typical wavelength $\lambda_0$ of the involved em. waves, for cases, where all other relevant length scales are large to $\lambda_0$).

 

[JTC]

So now, I'd also like to know, how Lagrange derived the equations, now called EL equations!

I found this online. Remember, I am more interested right now in scaffolding these all together. I will go back to read it carefully once I gather up everything I need. I skimmed it and it looks like what I need.

 

[JTC]

So now, I'd also like to know, how Lagrange derived the equations, now called EL equations!

I found this online. Remember, I am more interested right now in scaffolding these all together. I will go back to read it carefully once I gather up everything I need. I skimmed it and it looks like what I need.

 

[vanhees71]

Can I ask you to do one more thing? I have never read a clear concise definition PAIR that defines and distinguishes functional from function. I have seen functional used widely, so I assume this might be a laziness of speech. But you have made a clear cut usage in a near distinction. So, can you provide a definition of those two words? One that reflects off the other (and advise if if this definition is of use only in energy methods like Hamilton P.)

That's easy. "Function" is the general entity. It's a unique map of one set to another written as $f:M_1 \rightarrow M_2$, which maps any element of $M_1$ uniquely to an element of $M_2$ written as $x \mapsto f(x)$.

A functional is a function, where $M_1$ is some set of functions (e.g., the Hilbert space of square integrable functions $\mathbb{R}^n \rightarrow \mathbb{C}$) to the set of (real or complex) numbers, written as $A: \mathrm{L}^2(\mathbb{R}^n,\mathbb{C}) \rightarrow \mathbb{K}$ (with $K=\mathbb{R} \quad \text{or} \quad \mathbb{C}$). Usually it's written as $\psi \mapsto A[\psi]$.

 

[Delta2]

Can I ask you to do one more thing? I have never read a clear concise definition PAIR that defines and distinguishes functional from function. I have seen functional used widely, so I assume this might be a laziness of speech. But you have made a clear cut usage in a near distinction. So, can you provide a definition of those two words? One that reflects off the other (and advise if if this definition is of use only in energy methods like Hamilton P.)

Well you are right to some extent to confuse between the two.

A function is a mapping between two sets A and B. Sets A and B can be anything. However what we usually mean by simply "function" is when sets A and B are, loosely speaking, sets of scalars like say the set of real numbers R.

When set A and B are sets of (simple) functions then we call the function F:A->B an operator. Example :The operator of indefinite integral takes a function as its argument and return the antiderivative function as its output.

When set A is a set of functions and B is a set of scalars then we call the function F:A->B a functional. Example:The functional of definite integral with endpoints a and b takes a function at its input and returns a real number at its output which is the value of the definite integral of that function between two specified points a and b.