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In summary, the conversation discusses the principle of least action, which states that a particle will take the path of least Lagrangian. This is known as Hamilton's principle in classical mechanics. It is noted that this principle can be derived from Newton's laws and ideal and holonomic constraints. However, it can also be used to derive equations of motion for nonholonomic constraints. The discussion then delves into the use of Lagrange equations and the Hamilton variational principle for nonholonomic constraints, as well as the distinction between holonomic and nonholonomic constraints. The conversation ends with a request for a citation of a book that contains a statement about the Hamilton variational principle for nonholonomic constraints.

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Oh the Lagrangian does not necessarily be T-U

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Hamilton's principle is derived from Newton's laws + hypothesis on constrains they must be ideal and holonomic

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Let's clarify it by a simplest example.Assume we have a Lagrangian

$$L=\frac{1}{2}g_{ij}\dot x^i\dot x^j,\quad i,j=1,\ldots, m.$$ The coefficients ##g_{ij}## form a Riemann metric; for simplicity we assume that all ##g_{ij}## are constants.

Consider** a variational problem **of finding extremums of a functional

$$ x(\cdot)\mapsto \int_{t_1}^{t_2}Ldt$$ in the class of smooth functions ##x(t)=(x^1,\ldots,x^m)(t)## such that ##x(t_i),\quad i=1,2## are fixed and $$a_i(x)\dot x^i=0,\quad t\in[t_1,t_2].\qquad (*)$$

From Calculus of Variations [see for example Bruce van Brunt: The Calculus of Variations. Springer 2004] it is known that the extremum to this problem is a solution to the Lagrange equations with the following Lagrangian ##\mathcal L(x,\dot x)=L-\lambda a_k\dot x^k##. Simultaneously the multiplier ##\lambda=\lambda(t)## is chosen to fulfill equations (*).

The system with Lagrangian ##\mathcal L## is written as follows

$$g_{si}\ddot x^i-\dot\lambda a_s-\lambda\dot x^p\Big(\frac{\partial a_s}{\partial x^p}-\frac{\partial a_p}{\partial x^s}\Big)=0. \qquad (**)$$

To express ##\dot\lambda## let us differentiate formula (*) in ##t##:

$$\frac{\partial a_k}{\partial x^n}\dot x^k\dot x^n+a_k\ddot x^k=0.$$

Substituting here ##\ddot x## from (**) we finally yield

$$\dot\lambda=-\frac{1}{g^{ij}a_ia_j}\Big(\frac{\partial a_i}{\partial x^n}\dot x^n\dot x^i+\lambda a_i\dot x^sg^{ik}\Big(\frac{\partial a_k}{\partial x^s}-\frac{\partial a_s}{\partial x^k}\Big)\Big) .\qquad (***)$$

**Generally equations (**)-(***) describe the vaconomic mechanics of system with Lagrangian ##L##. To obtain a solution to this system we must impose the initial conditions: ##x(0),\dot x(0),\lambda(0).## For the same ##x(0),\dot x(0)## we can take different ##\lambda(0)## and obtain different solutions ##x(t)##. This means that from the "classical" standpoint when we trace only the function ##x(t)##, system (**)-(***) does not possesses the uniqueness property. Thus in general it can not be a system of classical mechanics.**

We point just a simplest case when system (**)-(***) turns to be a system of classical mechanics. Assume that $$\frac{\partial a_s}{\partial x^p}-\frac{\partial a_p}{\partial x^s}=0.\qquad (!)$$ Then substituting ##\dot\lambda## from (***) to (**) we get

$$g_{si}\ddot x^i+\frac{a_s}{g^{ij}a_ia_j}\frac{\partial a_i}{\partial x^n}\dot x^n\dot x^i=0.$$

It is not hard to show that these equations describe the classical Lagrangian system with Lagrangian ##L## and ideal constraints (*).

**Introduce a differential form ##\omega=a_idx^i##. Then the condition (!) implies that locally there exists a function ##f=f(x)## such that ##df=\omega## i.e. the constraint (*) is integrable and our system is holonomic. In this case the described above variational problem tuns to be the Hamilton principle. Generally, necessary and sufficient condition of integrability of constraint (*) is as follows ##\omega\wedge d\omega=0##. In case of nonholonomic constraint the formulated above variational problem does not give equations of classical mechanics.**

Thus it important not to mix up Hamilton's variational principle and the Lagrange-d'Alembert principle

$$L=\frac{1}{2}g_{ij}\dot x^i\dot x^j,\quad i,j=1,\ldots, m.$$ The coefficients ##g_{ij}## form a Riemann metric; for simplicity we assume that all ##g_{ij}## are constants.

Consider

$$ x(\cdot)\mapsto \int_{t_1}^{t_2}Ldt$$ in the class of smooth functions ##x(t)=(x^1,\ldots,x^m)(t)## such that ##x(t_i),\quad i=1,2## are fixed and $$a_i(x)\dot x^i=0,\quad t\in[t_1,t_2].\qquad (*)$$

From Calculus of Variations [see for example Bruce van Brunt: The Calculus of Variations. Springer 2004] it is known that the extremum to this problem is a solution to the Lagrange equations with the following Lagrangian ##\mathcal L(x,\dot x)=L-\lambda a_k\dot x^k##. Simultaneously the multiplier ##\lambda=\lambda(t)## is chosen to fulfill equations (*).

The system with Lagrangian ##\mathcal L## is written as follows

$$g_{si}\ddot x^i-\dot\lambda a_s-\lambda\dot x^p\Big(\frac{\partial a_s}{\partial x^p}-\frac{\partial a_p}{\partial x^s}\Big)=0. \qquad (**)$$

To express ##\dot\lambda## let us differentiate formula (*) in ##t##:

$$\frac{\partial a_k}{\partial x^n}\dot x^k\dot x^n+a_k\ddot x^k=0.$$

Substituting here ##\ddot x## from (**) we finally yield

$$\dot\lambda=-\frac{1}{g^{ij}a_ia_j}\Big(\frac{\partial a_i}{\partial x^n}\dot x^n\dot x^i+\lambda a_i\dot x^sg^{ik}\Big(\frac{\partial a_k}{\partial x^s}-\frac{\partial a_s}{\partial x^k}\Big)\Big) .\qquad (***)$$

We point just a simplest case when system (**)-(***) turns to be a system of classical mechanics. Assume that $$\frac{\partial a_s}{\partial x^p}-\frac{\partial a_p}{\partial x^s}=0.\qquad (!)$$ Then substituting ##\dot\lambda## from (***) to (**) we get

$$g_{si}\ddot x^i+\frac{a_s}{g^{ij}a_ia_j}\frac{\partial a_i}{\partial x^n}\dot x^n\dot x^i=0.$$

It is not hard to show that these equations describe the classical Lagrangian system with Lagrangian ##L## and ideal constraints (*).

Thus it important not to mix up Hamilton's variational principle and the Lagrange-d'Alembert principle

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$$A[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t)$$

for all variations around the stationary point ##q(t)## which are subject to the constraints

$$a_{ik}(q,t) \delta q^{k}=0.$$

To implement them you take the variation of ##A## and add the usual Lagrange-multiplier terms:

$$\delta A[q] = \int_{t_1}^{t_2} \mathrm{d} t \delta q^{k} \left [\frac{\partial L}{\partial q^k}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}^k} - \lambda^i a_{ik} \right] \stackrel{!}{=}0,$$

which leads to the equations of motion

$$\frac{\partial L}{\partial q^k}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}^k}=\lambda^i a_{ik}.$$

Now, for the "true trajectory" you can write the non-holonomic contraints as

$$a_{ik} \dot{q}^k=0.$$

Note that the vaskonomic dynamics and the correct d'Alembert dynamics coincide if and only if the an-holonomic constraints are integrable, i.e., can be implemented as holonomic constraints.

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ok, then cite please a book which contains such a statement:vanhees71 said:We should not repeat this superfluous discussion again

and this statement is called the Hamilton variational principle (for nonholonomic constraints, you know).vanhees71 said:This means you have to minimize

A[q]=∫t2t1dtL(q,˙q,t)A[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t)

for all variations around the stationary point q(t)q(t) which are subject to the constraints

aik(q,t)δqk=0.

It would be great if you cut the corresponding text from that book and post the scan here

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Because this is not a variational problem. Variational problem is a problem about extremums of a functional which is defined on some manifold with constraints. When you solve a variational problem you can not choose variations as you want. The constraints determine the space of variations.vanhees71 said:But, I have derived the nonholonomic equations of motion from the variational principle in the way shown in my previous posting. So why are you saying that's wrong?

So the phrase:

is meaningless in the mathematical sense. The constraints ##a_i\dot x^i=0## imply the following space of variations ##\{\delta x^j\mid a_i\frac{d}{dt}(\delta x^i)+\frac{\partial a_i}{\partial x^k}\dot x^i\delta x^k=0\}##. This corresponds to the vaconomic mechanics. But the Largange-D'Alembert equations do not correspond to conditional extremum of any functional in nonholonomic case.vanhees71 said:This means you have to minimize

A[q]=∫t2t1dtL(q,˙q,t)A[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t)

for all variations around the stationary point q(t)q(t) which are subject to the constraints

aik(q,t)δqk=0.

And please never tell to specialists in dynamical systems that Landau&Lifshitz vol. 1 is a good book they will die laughing

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I don't bother, how you call what I did. I find this very straight forward, and I never understood, why non-holnomic constraints are more complicated than holonomic ones. Again the logic is the following. One defines the action functional as usualwrobel said:Because this is not a variational problem. Variational problem is a problem about extremums of a functional which is defined on some manifold with constraints. When you solve a variational problem you can not choose variations as you want. The constraints determine the space of variations.

So the phrase:

is meaningless in the mathematical sense. The constraints ##a_i\dot x^i=0## imply the following space of variations ##\{\delta x^j\mid a_i\frac{d}{dt}(\delta x^i)+\frac{\partial a_i}{\partial x^k}\dot x^k\delta x^i=0\}##. This corresponds to the vaconomic mechanics. But the Largange-D'Alembert equations do not correspond to conditional extremum of any functional in nonholonomic case.

And please never tell to specialists in dynamical systems that Landau&Lifshitz vol. 1 is a good book they will die laughing

$$A[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t).$$

Then one defines the (in general nonholonomic) constraints as constraints on the allowed variations

$$a_{ik}(q,t) \delta q^k=0.$$

These have to be included by the introduction of Lagrange multipliers in the usual way, leading to

$$\delta A[q] = \int_{t_1}^{t_2} \mathrm{d} t [\delta L-\lambda^{i} a_{ik} \delta q^k] \stackrel{!}{=}0.$$

Thanks to the Lagrange multipliers now we can vary the ##\delta q^k## independently and get the usual equations of motion as from d'Alembert's principle,

$$\frac{\partial L}{\partial q^k}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\dot{q}^k}=\lambda^{i} a_{ik}.$$

That's, how it's derived in Landau&Lifshitz vol. I, Section 38 too, as I've just checked.

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I apologize for the harshness

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No problem; physics discussions sometimes get excited :-)).

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O, it seems I have found proper words.

Logic of variational problem is as follows. (little bit informally) What is ##\delta A##? To calculate it we do the following procedure:

##\delta A=\frac{d}{ds}\Big|_{s=0}\int_{t_1}^{t_2}L\Big(x+s\delta x,\frac{d}{dt}\Big( x+s\delta x\Big)\Big)dt.##That is we differentiate the functional ##A## on the set ##\{ x+s\delta x\}## with ##x(t)## fixed. To determine the space of variations ##\delta x## we must do the same for the equation of constraint:

$$a_i(x)\dot x^i=0\Longrightarrow\frac{d}{ds}\Big|_{s=0}\Big[a_i(x+s\delta x)\frac{d}{dt}\Big(x^i+s\delta x^i\Big)\Big]=0.$$

It is exactly the same as when we look for conditional minimum for example for the function ##f(u,v)=u-v## under constraint ##u^2+v^2=1##.

Logic of variational problem is as follows. (little bit informally) What is ##\delta A##? To calculate it we do the following procedure:

##\delta A=\frac{d}{ds}\Big|_{s=0}\int_{t_1}^{t_2}L\Big(x+s\delta x,\frac{d}{dt}\Big( x+s\delta x\Big)\Big)dt.##That is we differentiate the functional ##A## on the set ##\{ x+s\delta x\}## with ##x(t)## fixed. To determine the space of variations ##\delta x## we must do the same for the equation of constraint:

$$a_i(x)\dot x^i=0\Longrightarrow\frac{d}{ds}\Big|_{s=0}\Big[a_i(x+s\delta x)\frac{d}{dt}\Big(x^i+s\delta x^i\Big)\Big]=0.$$

It is exactly the same as when we look for conditional minimum for example for the function ##f(u,v)=u-v## under constraint ##u^2+v^2=1##.

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$$a_i \delta x^i=0.$$

In your description the contraint rather reads

$$\delta x^k \partial_k a_i(x) \dot{x}^i+a_i \frac{\mathrm{d}}{\mathrm{d} t} \delta x^i=0.$$

This is something different than the correct d'Alembertian virtual displacement constraint given above.

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Sure! it is vaconomic againvanhees71 said:Hm, but aren't you then lead back to the vakonomic dynamics again

Sure, but this equation must be imposed if we want to minimize the functional ##A## on the space of functions ##\{x(t)\mid a_i(x(t))\dot x^i=0\}##. It is obviously not the same as ##a_i\delta x^i=0##. It is the main thing I am trying to say: the Lagrange-d'Alembert equations are not generated by the problem of minimization.vanhees71 said:In your description the contraint rather reads

δxk∂kai(x)˙xi+aiddtδxi=0.

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what exactly is wrong? Just cite my phrasevanhees71 said:Yes, and IT IS WRONG!

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I have never assumed that. Please read cearfully what I have writtenvanhees71 said:To assume that you can derive the correct equations by treating it as you say

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1) The problem of minimization of the functional ##A## on the space ##\{x(t)\mid a_i(x(t))\dot x^i=0\}## leads to the vaconomic equations. It is a fact from variational calculus, I have already cited the textbook.

2) In general the Lagrange-D'Alembert equations do not follow from the variational principle formulated in 1).

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So I guess, we have to leave it at that and agree to disagree.

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it is an interesting storyvanhees71 said:I don't understand, how one came to the idea to use (1)

and from the Preface of Bloch's book:

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Hamilton's principle, also known as the principle of least action, is a fundamental principle in classical mechanics that states that the motion of a physical system can be determined by minimizing the action, which is the integral of the Lagrangian over time.

No, Hamilton's principle is not an empirical statement. It is a mathematical principle that is used to derive the equations of motion for a physical system. It is based on the assumption that the actual path of a system is the one that minimizes the action, and this has been shown to be true in numerous experiments.

Hamilton's principle is used to derive the equations of motion for a physical system by finding the path that minimizes the action. This is done by setting up the Lagrangian, which is a function that describes the system's kinetic and potential energies, and then using the Euler-Lagrange equations to find the path that minimizes the action.

Yes, Hamilton's principle is applicable to all physical systems, as long as they can be described using the Lagrangian formalism. This includes systems such as particles, fluids, and even quantum systems.

Hamilton's principle is a powerful tool in classical mechanics, but it does have some limitations. It assumes that the system is conservative, meaning that it does not dissipate energy. It also does not take into account any external forces or non-conservative forces acting on the system, so it may not accurately describe certain systems such as those with friction or air resistance.

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