Can we treat non-conservative forces in the Lagrangian formulation?

[MuIotaTau]

In Lagrangian mechanics, the Euler-Lagrange equations take the form $$\frac{\partial L}{\partial x} = \frac{\mathbb{d}}{\mathbb{d}t}\frac{\partial L}{\partial \dot{x}}$$ From this, we can define the left side of the equation as force, and by carrying out the actual derivative, we get $$F = -\frac{\partial V}{\partial x}$$ But by definition, this is only true for conservative forces; in other words, there exist forces that cannot be expressed in this form, such as friction or drag. So are these forces simply inexpressible in Lagrangian mechanics without recourse to the Newtonian formulation, or is there simply something I'm not seeing?

 

[arildno]

1. Well, in some specific cases, with for example the well known Rayleigh dissipation function, you may include dissipative effects in the Lagrangian (it requires that the non-conservative elements are sufficiently "nice").
But, in general, when you confine yourself to "normal" kinematics on the macro-scale, there is, indeed, a limitation here relative to Newton. But, so what, really?

2. Remember that Energy is a much fuller concept that those types we "ordinarily" work with, kinetic energy, and those potential energies directly included in the mechanical energy budget balance! The whole of thermodynamics, for example, where energy in the form of heat is included, is one such example.

And, when you therefore work within, for example, a framework and system in which ALL energy is CONSERVED, more advanced variational techniques can be developed here to work as well, but it won't "look like" our ordinary kinematics since those are not the primary variables we work with.

3. In effect, such as Hamiltonian approaches are those that show themselves most amenable to mathematical generalizations, for a vast array of problems, going way beyond "Newtonian" fields of applications.
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Another perspective is the following:
When we are dealing with a complex macro-system, it is in practice IMPOSSIBLE to calculate on a theoretical basis all the subtle effects that go into what we call, say "viscosity", or the local "geometry" of the pipe a fluid, say, flows through.
We use empirically derived APPROXIMATIONS here, and by default, we necessarily have to deal with systems that cannot be given a theoretically "proper" variational/lagrangian formulation (not because it doesn't exist any such, but we're unable to formulate it!). We use Newton by default.

For those particle systems PHYSICISTS look on, what we could call the "core systems of reality", then ALL variables are to be accounted for by means of the truly fundamental laws, and those CAN then be recast in the best apparatus to study them under.

 

[UltrafastPED]

In the form presented, no, but the Euler-Lagrange equations can also be written with generalized forces that need not be conservative.

For example, see: http://planning.cs.uiuc.edu/node706.html

Note that there is still a Lagrangian, but the RHS is not zero. This system cannot be transformed to a Hamiltonian.

 

[MuIotaTau]

1. Well, in some specific cases, with for example the well known Rayleigh dissipation function, you may include dissipative effects in the Lagrangian (it requires that the non-conservative elements are sufficiently "nice").
But, in general, when you confine yourself to "normal" kinematics on the macro-scale, there is, indeed, a limitation here relative to Newton. But, so what, really?

That's true, and after all, different formulations are suited to different purposes.

2. Remember that Energy is a much fuller concept that those types we "ordinarily" work with, kinetic energy, and those potential energies directly included in the mechanical energy budget balance! The whole of thermodynamics, for example, where energy in the form of heat is included, is one such example.

And, when you therefore work within, for example, a framework and system in which ALL energy is CONSERVED, more advanced variational techniques can be developed here to work as well, but it won't "look like" our ordinary kinematics since those are not the primary variables we work with.

3. In effect, such as Hamiltonian approaches are those that show themselves most amenable to mathematical generalizations, for a vast array of problems, going way beyond "Newtonian" fields of applications.
-----
Another perspective is the following:
When we are dealing with a complex macro-system, it is in practice IMPOSSIBLE to calculate on a theoretical basis all the subtle effects that go into what we call, say "viscosity", or the local "geometry" of the pipe a fluid, say, flows through.
We use empirically derived APPROXIMATIONS here, and by default, we necessarily have to deal with systems that cannot be given a theoretically "proper" variational/lagrangian formulation (not because it doesn't exist any such, but we're unable to formulate it!). We use Newton by default.

For those particle systems PHYSICISTS look on, what we could call the "core systems of reality", then ALL variables are to be accounted for by means of the truly fundamental laws, and those CAN then be recast in the best apparatus to study them under.

Thank you, I think that all puts things into perspective quite well!

In the form presented, no, but the Euler-Lagrange equations can also be written with generalized forces that need not be conservative.

For example, see: http://planning.cs.uiuc.edu/node706.html

Note that there is still a Lagrangian, but the RHS is not zero. This system cannot be transformed to a Hamiltonian.

Ahh, okay. But this is still resorting to a Newtonian point of view, right? Not that there's anything wrong with that; the physics still works, so of course it's still valid, so I guess I'm just talking philosophy at this point.

 

[UltrafastPED]

The Euler-Lagrange equations are equivalent to Newton ... just a more convenient form for some things.

 

[arildno]

The Euler-Lagrange equations are equivalent to Newton ... just a more convenient form for some things.

Well, well.
You can't set up a variational principle to derive, for example, the Navier-Stokes equations, which is a Newtonian (approximation) to viscous fluid flow.

Or, if you can do that, you will be accoladed with prizes.
(there exist, though, variational formulations for specific cases of N-S equations)

 

[UltrafastPED]

What about this report?

http://www.wseas.us/e-library/conferences/2007australia/papers/550-305.pdf

If the differential equation is self-adjoint then there should be a variational principle.
But I don't pretend to any expertise with fluid mechanics; it is far outside my field - optics/lasers.

 

[arildno]

What about this report?

http://www.wseas.us/e-library/conferences/2007australia/papers/550-305.pdf

If the differential equation is self-adjoint then there should be a variational principle.
But I don't pretend to any expertise with fluid mechanics; it is far outside my field - optics/lasers.

Well, that is...JUST FANTASTIC!
(And what I've been hoping for!)
I haven't read your link yet, but this is certainly something I haven't been updated on since MY studies (late 1990s-early 2000s), in which a FULLY general variational principle had NOT been established for the N-S equations. Your link is from a paper in 2007, so this would be definite updating relative to MY time of following current research.
--
However, it is quite something different to say that a VARIATIONAL principle exists, than that the Euler-Lagrangian equations exist.
The latter ARE derived by presuming all (locally acting) forces are GRADIENT fields, and that is certainly not the case in N-S.
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E-L equations constitute a SUB-class of variational methods.

 

[arildno]

If the differential equation is self-adjoint then there should be a variational principle.

According to your link, an N-S+its adjoint has already been formulated, but at the terrible cost of doubling the functions.

This has made THAT strategy not viable for practical purposes (N-S by itself is EXTREMELY computationally heavy).

 

[UltrafastPED]

First you find a theoretical formulation, then you look for improved solutions.

If you base your numerical integration on the variational principle it should be optimal ... I took a graduate seminar course in optimal systems based on application of Lie algebras to variational principle problems. Something like this:
http://gmcnet.webs.ull.es/Summer_School/documentos/leok/mleok_scicade07.pdf

We mostly solved orbital problems - it was an aerospace engineering course - and were able to generate extremely stable orbital calculations, as well as fast routines for elastic systems. But you had to calculate the integrator for each problem.

 

[arildno]

I suggest you actually read your previous links.
The chase for a good VP.principle for the N-S equations has been around for the last century or so, it is damn difficult, but it seems this article might be a great breakthrough.