Generality of the Euler Lagrange equations

[Gavroy]

Hi

I wanted to know for which cases the Euler Lagrange equations are applicable?

1.) Imagine that we have a kinetic Energie T(q,q') and a potential that also depends on velocity V(q,q'). As far as i know the Euler Lagrange equations for a particle still hold in this case, is that true?
Now:
2.) Imagine that I have two particles with velocities v1 and v2 and imagine that the non holonomic constraint v1-v2=0 holds.
and they are both placed in an external potential that depends on both, the velocities and positions of the particles. how would the euler lagrange equations look like?

 

[WannabeNewton]

The Euler-Lagrange equations come out of first variation of an action $S(f) = \int_{x_1}^{x_2}L(x,f(x),f'(x))dx$. As you can see, as long as $L = L(x,f(x),f'(x))$, we can take a first variation and get the Euler-Lagrange equations. This is much more general than the specific case where $L$ is a lagrangian.

Anyways, consider the lagrangian for a charged particle in an electromagnetic field which is given by $L = -mc^{2}\sqrt{1 - \frac{v^{2}}{c^{2}}} + \frac{e}{c}\textbf{A}\cdot \textbf{v} -e\varphi$. The term $\frac{e}{c}\textbf{A}\cdot \textbf{v} -e\varphi$ describes the interaction of the charged particle with the electromagnetic field (here $\textbf{A}$ is the vector potential and $\varphi$ is the scalar potential). The Euler-Lagrange equations certainly hold true here and they will just give you the Lorentz force.

 

[Jano L.]

1.) Imagine that we have a kinetic Energie T(q,q') and a potential that also depends on velocity V(q,q'). As far as i know the Euler Lagrange equations for a particle still hold in this case, is that true?

That depends on the nature of the system, i.e. equations of motion or some other specification like variational principle. For the latter, if the action is given by
$$
S(t_1,t_2) = \int_{t_1}^{t_2} T -V dt,
$$
then you can use Lagrangian T - V and the E-L equations. But if you do not know that there is this variational principle, then they may not hold - there may be some additional friction forces which are not contained in T or V and then the Euler-Lagrange equations do not hold in their standard form.

2.) Imagine that I have two particles with velocities v1 and v2 and imagine that the non holonomic constraint v1-v2=0 holds.
and they are both placed in an external potential that depends on both, the velocities and positions of the particles. how would the euler lagrange equations look like?

What you are looking for is generalized Hamilton's principle for "semi-holonomic" constraints. Some explanation is given in Goldstein's book, section 2.4 of the 3rd edition. It should be possible to use the standard procedure, with the Lagrangian containing new term due to the constraint:

$$
\boldsymbol{\lambda}\cdot (\mathbf v_1 - \mathbf v_2).
$$

 

[espen180]

They are very general. Maybe the most general case is for uncountably many degrees of freedom, which is the case for a field theory, and you may also be able to generalize it to noncommutative variables, or variables with torsion (x^k=0 for some k), but I am not 100% sure about this.

 

[PJC01]

Hi there,

The Euler Lagrange equations are a fundamental tool in classical mechanics, and they are applicable in a wide range of cases. They can be used to derive the equations of motion for a single particle, as well as for systems with multiple particles and constraints.

To answer your first question, yes, the Euler Lagrange equations can still be applied when there is a potential that depends on velocity. This is known as a non-conservative potential and it can be accounted for in the equations.

For your second question, the Euler Lagrange equations can be used to find the equations of motion for a system of particles with constraints. In this case, the constraint would be incorporated into the equations as a Lagrange multiplier, which helps to enforce the constraint during the derivation of the equations of motion.

In general, the Euler Lagrange equations can be applied to systems with any number of particles, any type of constraints, and any type of potential (as long as it is a function of the relevant variables). They are a powerful tool for analyzing and understanding the motion of physical systems.

I hope this helps clarify the generality of the Euler Lagrange equations. Let me know if you have any further questions.

Best,