# Lagrangian mechanics -- Initial questions

• Umaxo
In summary: The lagrangian is actually quite indeterminate, so there comes a point where you can interpret its terms as you please as long as you get the correct equations of motion out of it.When you get to non-inertial frames of reference, you will learn about Coriolis force, which is velocity-dependent and can be derivated from a potential where v appears explicitly, and there is no finite popagation velocity to speak about.That's correct. The book assumes that the Lagrangian is always in the form of the kinetic term plus an interaction term, but this is only an approximation. In reality, the Lagrangian could be more general, depending on the forces between the particles
Umaxo
Hi,

I am reading Landau-Lifshitz course in theoretical physics 1. volume, mechanics. The mechanics is derived using variatonal principle from the start.

At first they start with point particles, that do not interact with each other. Thus the equations of motions must be independent for the particles and therefore lagrangian might be written as sum of lagrangians of individual particles.

Now, if we assume homogeneity and isotropy of space and homogenity of time, then lagrangian of free particle cannot depend on spatial coordinates, time coordinate, nor can it depend on direction of velocity (independence of lagrangian on higher derivatives is assumed as experimental fact):
$$L=L\left(v^2\right)$$

Then if we use galileo relativity principle we can compare lagrangian in two systems moving relative to each other and we get usual kinetic term.

If we want mechanical system of n noninteracting free particles, we just use additivity of lagrangian. However, if we want particles to be interacting, in the book they subtract term: $$U=U\left(\vec{r}_1...\vec{r}_n\right)$$
This form is good, because it is evident that propagation velocity of interactions is infinite which is needed because of absolutness of time and galileo relativity principle.
As the book goes on, one finds out that explicit dependence on time coordinate means there is some kind of external field, or at least one can interpret it as such.

Now, I have two questions:

1.) Does dependence of interacting term on velocities means there is finite propagation velocity of interactions? If not, what would including such dependence mean?

2) The lagrangian in the form of kinetic (free particles) term + interaction term is in the book assumed. Obviously, if interaction term could depend on velocities, one can always write lagrangian in this form. But if not, does it make sense to have lagrangian with more general kinetic term than just free particle lagrangian?

Thanks :)

Umaxo said:
Now, I have two questions:

1.) Does dependence of interacting term on velocities means there is finite propagation velocity of interactions? If not, what would including such dependence mean?

That doesn't necessarily mean that interactions propagate at a finite velocity. For now you could interpret such dependence as the presence of a force which is stronger or weaker depending of the particles' velocity. When you get to non-inertial frames of reference, you will learn about Coriolis force, which is velocity-dependent and can be derivated from a potential where v appears explicitly, and there is no finite popagation velocity to speak about.

Umaxo said:
2) The lagrangian in the form of kinetic (free particles) term + interaction term is in the book assumed. Obviously, if interaction term could depend on velocities, one can always write lagrangian in this form. But if not, does it make sense to have lagrangian with more general kinetic term than just free particle lagrangian?
The lagrangian is actually quite indetermined, so there comes a point where you can interpret its terms as you please as long as you get the correct equations of motion out of it.

Umaxo
angrystudent said:
When you get to non-inertial frames of reference, you will learn about Coriolis force, which is velocity-dependent and can be derivated from a potential where v appears explicitly, and there is no finite popagation velocity to speak about.

Make sense... But in the book, they assume to be in frame of reference in which space and time are homogeneous and isotropic. Doesnt this have any influence on your discussion?

Umaxo said:
But in the book, they assume to be in frame of reference in which space and time are homogeneous and isotropic. Doesnt this have any influence on your discussion?

Ok i thought about it and this question is no longer relevant for me. However, another quetion arose in me. Sadly i don't have a lot of time right now. but i will be back soon:)

## 1. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework used to describe the motion of particles and systems in physics. It is based on the principle of least action, which states that the true path of a system is the one that minimizes the action, a quantity related to the energy of the system.

## 2. How is Lagrangian mechanics different from Newtonian mechanics?

In Newtonian mechanics, the motion of a system is described by Newton's laws of motion, which are based on the concept of forces. In Lagrangian mechanics, the motion of a system is described by the Lagrangian function, which is a combination of the kinetic and potential energies of the system. This approach is more general and can be applied to systems with complex forces and constraints.

## 3. What is the role of the Lagrangian in this framework?

The Lagrangian function is used to determine the equations of motion of a system. It is a function of the generalized coordinates and velocities of the system, and it encapsulates all the information about the system's energy and constraints. By finding the stationary point of the action with respect to the generalized coordinates, we can obtain the equations of motion.

## 4. What are generalized coordinates in Lagrangian mechanics?

Generalized coordinates are a set of independent variables used to describe the configuration of a system. They can be chosen to be any variables that fully specify the position and orientation of the system. Unlike in Newtonian mechanics, where the coordinates are typically Cartesian coordinates, generalized coordinates can be more convenient for describing complex systems.

## 5. How is Lagrangian mechanics applied in real-world scenarios?

Lagrangian mechanics has been applied in various areas of physics, such as classical mechanics, electromagnetism, and quantum mechanics. It is used to study the motion of particles, fluids, and rigid bodies, as well as the behavior of systems with elastic and dissipative forces. It is also used in engineering and robotics to model and control the motion of mechanical systems.

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