Lagrangian equation for motion

[Avichal]

A few doubts regarding lagrnagian method to deal with motion of particles:

1) It seems like a heuristic method of solving for motion of a particle. In Newtonian mechanics, you carefully consider all the forces and find out the particle's motion. In this, based on intuition you guess the paticle will move in such and such trajectory, decide the generalised coordinates and solve it.
Am I right?

2) I was studying its proof when it said that ∂T / ∂x = 0. I don't understand this.
T is kinetic energy which of course is in terms of velocity. Velocity is dx / dt. Then how come it is zero?

 

[ehild]

The Lagrangian is function of the general coordinates an momentum components. If the expression of the kinetic energy contains the momentum components only, its partial derivative with respect any coordinate is zero.

As an example, consider the vertical motion of a ball thrown up. Its coordinate is x, the momentum is p=mdx/dt, the potential energy is V=mgx, kinetic energy is T=p²/(2m). The partial derivative ∂T/∂x is clearly zero, as T does not depend explicitly on x. In case of functions of two variables p and x, and p also function of x, then the total derivative of a function f(p,x) is df/dx=∂f/∂x+∂f/∂dp dp/dx. If f does not depend explicitely on x, ∂f/∂x=0, but the total derivative is different from zero.

ehild

 

[WannabeNewton]

You don't guess the trajectory. The trajectory is exactly what you are solving for. What you do is write down the lagrangian in some generalized coordinates, vary the action looking for a stationary point (i.e. ##\frac{\delta S}{\delta q} = 0##) which then results in the equations of motion (Lagrange's equations). Of course in practice you would just write down the lagrangian in some generalized coordinates and then plug it straight into Lagrange's equations.

 

[voko]

A few doubts regarding lagrnagian method to deal with motion of particles:

1) It seems like a heuristic method of solving for motion of a particle. In Newtonian mechanics, you carefully consider all the forces and find out the particle's motion. In this, based on intuition you guess the paticle will move in such and such trajectory, decide the generalised coordinates and solve it.
Am I right?

What you mean by "guessing" is taking into account the constraints of the system. For example, a mass swinging on a light rod is constrained to move in a circle, so its has only one degree of freedom, so you need only one coordinate to describe its motion. Up to this point, there is no guessing involved. Then you can be creative, because the coordinate can be anything that describes the motion unambiguously, the angle with the vertical being a particularly convenient choice.

2) I was studying its proof when it said that ∂T / ∂x = 0. I don't understand this.
T is kinetic energy which of course is in terms of velocity. Velocity is dx / dt. Then how come it is zero?

You need to show the proof to discuss this meaningfully.

 

[voko]

What you do is write down the lagrangian in some generalized coordinates, vary the action looking for a stationary point (i.e. ##\frac{\delta S}{\delta q} = 0##) which then results in the equations of motion (Lagrange's equations).

That's not the only way to obtain Lagrange's equations. In fact, Lagrange himself, who was a pioneer of the calculus of variations, did not use this approach in his Mécanique analytique to obtain the equations. He did use variations, though, in ways that will make a contemporary mathematician cringe, to obtain them from an equivalent of Newton's second law. That method, however, can be carried out successfully using modern conventions.

 

[WannabeNewton]

That's quite interesting. I have to say I have never seen Lagrange's approach myself but I will certainly search for translations of his original manuscript on google, thanks voko!

 

[voko]

What he does is briefly this:

1. He obtains what we now call D'Alembert's principle (which he calls the general formula of dynamics): ## \sum_i (m\vec{a}_i - \vec{F}_i) \cdot \delta \vec {r}_i = 0 ## (he did not use vectors, of course). He does that by accepting that a constant force changes velocity linearly with time.

2. He does some highly formal symbolic manipulation on ## \sum_i m\vec{a}_i \cdot \delta \vec {r}_i ## and shows that it is equal to ## \sum_i (\frac {d} {dt} \frac {\partial T} {\partial \dot{q_i} } - \frac {\partial T} {\partial q_i }) \delta q_i ##.

3. He observes that all forces in nature are such that ## \sum_i \vec{F}_i \cdot \delta \vec {r}_i ## is integrable, thus there is some ## V ##, the partial derivatives of which give the (minus) forces.

4. So he ends up with ## \sum_i (\frac {d} {dt} \frac {\partial T} {\partial \dot{q_i} } - \frac {\partial T} {\partial q_i } + \frac {\partial V} {\partial q_i }) \delta q_i = 0 ## and observes that if the coordinates are chosen so that ## \delta q_i ## are independent, the equation yields a system of ##\frac {d} {dt} \frac {\partial T} {\partial \dot{q_i} } - \frac {\partial T} {\partial q_i } + \frac {\partial V} {\partial q_i } = 0 ##.

5. If the variations are not independent, he suggests the method of multipliers could be used (which he introduced earlier, in Statics).

 

[Avichal]

Okay, I got it wrong. But there is obviously a change in approach which I am unable to express it clearly.
Can someone explain what is the change in method for solving the motion of a body compared to Newtonian mechanics?

 

[vanhees71]

The Hamilton principle is more general than just Newtonian mechanics. Its the basis of all fundamental physics in the sense that all fundamental natural laws are formulated in terms of the Hamilton principle of least action.

Applied to Newtonian mechanics it's of course equivalent to Newton's postulates. In more complicated applications usually the Hamilton principle is an elegant tool to derive the equations of motion, to analyze their symmetry properties, finding conservation laws and then finally to solve the equations of motion or at least get a qualitative understanding of the motion.

 

[voko]

Okay, I got it wrong. But there is obviously a change in approach which I am unable to express it clearly.
Can someone explain what is the change in method for solving the motion of a body compared to Newtonian mechanics?

There are two major differences. First, you don't care about forces, you use energies. Second, you especially don't care about the normal forces due to constraints - instead, you choose coordinates that eliminate motions impossible due to constraints.

This latter bit is probably what you confuse with "guessing".

 

[Avichal]

There are two major differences. First, you don't care about forces, you use energies. Second, you especially don't care about the normal forces due to constraints - instead, you choose coordinates that eliminate motions impossible due to constraints.

This latter bit is probably what you confuse with "guessing".

Thanks!
I am still learning lagrangian mechanics and so far some of the problems seem to be easily solved using this. Why isn't this taught in high school? Does it have some deeper concept hidden behind it that I am not getting, because this can be easily taught at high school level.

 

[voko]

Thanks!
I am still learning lagrangian mechanics and so far some of the problems seem to be easily solved using this. Why isn't this taught in high school? Does it have some deeper concept hidden behind it that I am not getting, because this can be easily taught at high school level.

Well, I am not sure about you, but my physical education started when I had zero knowledge of calculus, and I do not think I ever came across partial derivatives in high school. Lagrangian mechanics makes heavy use of these, so covering it in high school would seem quite challenging to me. Grasping how Lagrangian mechanics emerges from Newtonian mechanics, or some variational principle requires quite a bit of skill. Applying it to problems is much easier, yet a firm technique in calculus is still requisite.

 

[Avichal]

How did this idea of lagrangian equation come up?
If you don't want to worry about forces and solve directly you rely on the idea that energy remains constant. So the obvious equation that you come up with is dT/dt = 0 where T is the total energy.
So is the lagrangian equation just an advanced version of conservation of energy?

 

[voko]

Conservation of energy gives you only one equation. If the number of degrees of freedom is one, that is enough to describe its motion. For example, a mass on a spring: total energy is ##m\dot{x}^2/2 + kx^2/2 = E_0 ##. Using the Lagrangian approach, we would get ## m\ddot{x} + kx = 0 ##. It can be seen that the latter is simply the former differentiated with respect to time, so they both yield the same solution.

If you have more degrees of freedom, then conservation of energy alone is not enough. Conservation of energy gives just one equation, but there are more than one unknown.

 

[Avichal]

Conservation of energy gives you only one equation. If the number of degrees of freedom is one, that is enough to describe its motion. For example, a mass on a spring: total energy is ##m\dot{x}^2/2 + kx^2/2 = E_0 ##. Using the Lagrangian approach, we would get ## m\ddot{x} + kx = 0 ##. It can be seen that the latter is simply the former differentiated with respect to time, so they both yield the same solution.

If you have more degrees of freedom, then conservation of energy alone is not enough. Conservation of energy gives just one equation, but there are more than one unknown.

So, can I view Lagrangian equation has some advanced equation for conservation of energy?
Basically I'm looking for an intuition behind the Lagrangian equation - how it is true and why it works? Also the proof does not help. It just looks like some algebraic manipulation of Newton's laws to get to this equation.

 

[voko]

So, can I view Lagrangian equation has some advanced equation for conservation of energy?

No, that would not be correct.

Basically I'm looking for an intuition behind the Lagrangian equation - how it is true and why it works? Also the proof does not help. It just looks like some algebraic manipulation of Newton's laws to get to this equation.

Fundamentally, Lagrange's equations are equivalent to Newton's laws, so you could equally ask "why it works" about the latter. It works because we know it does. There are a few equivalent formulations of mechanics and you cannot really say that one is more fundamental than some other. The conservation laws are even more fundamental, and we expect that any given formulation must have them, but they alone do not give a full description (except in very simple cases). You need something else, and that something else is given in Newton's laws, Lagrange's equations, the principle of least action, etc.

 

[Avichal]

Fundamentally, Lagrange's equations are equivalent to Newton's laws, so you could equally ask "why it works" about the latter. It works because we know it does. There are a few equivalent formulations of mechanics and you cannot really say that one is more fundamental than some other. The conservation laws are even more fundamental, and we expect that any given formulation must have them, but they alone do not give a full description (except in very simple cases). You need something else, and that something else is given in Newton's laws, Lagrange's equations, the principle of least action, etc.

I'm not asking why it works (perhaps it seemed so from my previous post).
For me Newton's laws seem very intuitive (maybe because I have been learning about it). I can directly relate it to the experiments performed by Galileo and other people.
But Lagrangian equation seems to come out of nowhere. How do I get an intuitive understanding of it?

 

[vanhees71]

For me, it's the opposite. I find Newton's equations hard to use for deriving equations for a concrete application. The Hamilton principle is much simpler, because I can choose the approrpiate coordinates right away and analyze the symmetries of the problem, simplifying the solution significantly. The only way you get used to that is to just use it in solving problems!

 

[voko]

Lagrange's equations are not essentially different from Newton's laws. If one uses Cartesian coordinates, then one obtains Newton's laws as Lagrange's equations. The distinctive general form of Lagrange's equations is due to the use of generalized coordinates and generalized forces (or potential energy).

For any generalized coordinates, one can obtain equations similar to Lagrange's equations from Newton's laws. But this is a "less automatic" process.

 

[zerokool]

Avichal,

I think the derivation you are talking about in your second question is related to the Lagrangian of a free particle. What the math is saying is the kinetic energy does not change with or is independent of position. The interpretation of this is the kinetic energy, i.e. velocity, of a particle in empty space does not change with position, i. e. a particle in motion will remain in motion unless acted upon by an external force. The example you bring up is a really beautiful way to prove Newton's first law. What I've written here is from Landau's Mechanics chapter 1 section 3.

 

[Avichal]

Well, afte reading the Wikipedia article it seems as if I need to know the concept of action in physics. I directly jumped from Newtonian mechanics to Lagranigan mechanics and it turns out that the motivation of Lagrangian mechanics was from the principle of least action.
I'm very surprised that I'm hearing all these concepts now in college. I never heard them in high school!

So are all these concepts - action, lagrangian, hamiltonian just to find an alternative way to describe mechanics? What was the motivation and what is it's use?

 

[vanhees71]

No! The Hamilton principle of least action is much more general than just mechanics. It governs all fundamental theories of physics, including quantum theory. Why the action must be stationary in the classical limit is only understandable from (the path-integral formalism) of quantum theory, but of course you can formulate it without this derivation from the more fundamental quantum theory.

The reason that this is not covered in high school is that you don't have the needed mathematical tools at hand, because you need functional analysis (variational calculus) to formulate and use it.

 

[voko]

Well, afte reading the Wikipedia article it seems as if I need to know the concept of action in physics.

This is not as clear cut as it may seem. The principle of least action is another principle on which mechanics can be built. It is not required to derive Lagrangian equations, although it is frequently used as such, because many people believe it is the cleanest approach.

I directly jumped from Newtonian mechanics to Lagranigan mechanics and it turns out that the motivation of Lagrangian mechanics was from the principle of least action.

Lagrange definitely knew that he could derive his equations from this principle. Yet in his book he did not do that. We could argue forever how he himself was motivated. Unfortunately, there does not seem to be any historical record.

I'm very surprised that I'm hearing all these concepts now in college. I never heard them in high school!

As I said, you need quite a bit of calculus to be able to understand those things.

So are all these concepts - action, lagrangian, hamiltonian just to find an alternative way to describe mechanics? What was the motivation and what is it's use?

Lagrange's stated intent was to make mechanics easier in applications. This is still valid, but these days we also regard the Lagrangian formulation as a fundamental principle - more fundamental than Newton's laws.

 

[Avichal]

I found a thread which addresses the question that I'm looking for. Here is the link.

I didn't find the answer there nor I have the answer here.
Can you explain how you understand the Lagrangian?

 

[UltrafastPED]

If you would like to see the details carried out slowly see "Notes on Analytical Mechanics", from a seminar I gave a few years ago: http://www.ebyte.it/library/StansPhysicsLinks.html

There is also a brief review of the chain rule, and quite a bit on partial derivatives which are required for the derivations.