A Treating Velocity as Independent of position till the end in Lagrangian Mechanics

[LightPhoton]

TL;DR Summary

Identification of relation between coordinates in phase space is only made after the very end of calculation via Euler-Lagrange equations of motion, why is that so?

Why do we treat velocity and coordinates as independent variables until the very end, where we then assume the dependence of velocity on coordinates via a time derivative? That is, let the Lagrangian of a given system be simply

$$\mathcal L=\frac12mv^2$$

Now, plugging this into the Euler-Lagrange equation, we get

$$\frac\partial{\partial x}\mathcal L-\frac d{dt}\frac\partial{\partial v}\mathcal L=0$$

$$\frac12m\bigg(\frac\partial{\partial x}v^2\bigg)-\frac12m\bigg(\frac d{dt}\frac\partial{\partial v}v^2\bigg)=0\tag1$$

$$\frac d{dt}v=0\rightarrow v= \text{constant wrt time}$$

So far, this calculation has shown that in phase space, $v$ is a constant in time.

Now, how can we justify the identification $v=dx/dt$ when we initially treated $v$ as independent and neglected this definition in the first term of (1)?

 

[Ibix]

Fundamentally, velocity and position are unrelated. There is no physical law that says that if your $x$ position is $x_0$ then your velocity must be $v_0$. You can be anywhere with any velocity.

Cranking through the Euler-Lagrange equations as you did gets you to $v=\mathrm{const}$, and still you can still have any velocity at any position. But the equation you derived tells you how those things will evolve, so given an initial velocity and position it tells you how they change given the action you specified.

So we are fine with the velocity and position being initially unrelated, because what we're getting out of this process is what constraints the action imposes on them. Note also that you are ising the time independent form of the Euler-Lagrange equations, so explicitly assuming that how something with a given position and velocity behaves is independent of time. You will get messier behaviour from the time dependence of the action if it has any.

 

[PeroK]

Now, how can we justify the identification $v=dx/dt$ when we initially treated $v$ as independent and neglected this definition in the first term of (1)?

This is a good question. We start with the Lagrangian, which is the kinetic energy minus the potential energy for the system.

At this point, however, we recognise that the Lagrangian has the form of a function of several variables - let's say position and velocity. The brilliant idea of Lagrangian mechanics is to study this function. The key point isn't really to treat velocity as independent of position and time (although that's what people often say), but to study the functional form of the Lagrangian.

In particular, we consider how we would mininise the integral of the Lagrangian along a path through the space defined by its variables. This in principle is a purely abstract, mathematical exercise. In any case, the result is the Euler-Lagrange equations.

The second brillaint idea is to suggest that the path defined by the E-L equations has some physical significance. If we go back to viewing the abstract variables in the Lagrangian as the position and velocity we find that the path defined by the Euler-Lagrange equations is equivalent to Newton's laws of motion!

From this point of view, no justification is needed. Using some abstract mathematics, we found an entirely different way to encode Newton's laws of motion. That said, you may well still ask why on Earth that worked? One answer is that the Euler-Lagrange equations represent a deeper insight into the laws of physics. This technique is generalised far beyond the initial case of Newtonian mechanics. So, you could see this whole exercise as justifying Newton's laws in the first place. I.e. the Euler-Lagrange equations are really the underlying laws of physics.

PS Newton once said he felt like he was a boy on the beach, picking up the odd brightly coloured stone, while the vast ocean of knowldege lay untouched. You could view Newton's laws as those coloured stones. And, that the ocean of knowledge, beside which those stones were washed up, included Lagrangian mechanics.

 

[Demystifier]

TL;DR Summary: Identification of relation between coordinates in phase space is only made after the very end of calculation via Euler-Lagrange equations of motion, why is that so?

Why do we treat velocity and coordinates as independent variables until the very end, where we then assume the dependence of velocity on coordinates via a time derivative? That is, let the Lagrangian of a given system be simply

$$\mathcal L=\frac12mv^2$$

Now, plugging this into the Euler-Lagrange equation, we get

$$\frac\partial{\partial x}\mathcal L-\frac d{dt}\frac\partial{\partial v}\mathcal L=0$$

$$\frac12m\bigg(\frac\partial{\partial x}v^2\bigg)-\frac12m\bigg(\frac d{dt}\frac\partial{\partial v}v^2\bigg)=0\tag1$$

$$\frac d{dt}v=0\rightarrow v= \text{constant wrt time}$$

So far, this calculation has shown that in phase space, $v$ is a constant in time.

Now, how can we justify the identification $v=dx/dt$ when we initially treated $v$ as independent and neglected this definition in the first term of (1)?

Where did you see Lagrangian mechanics formulated that way? I don't think it's the correct way to formulate it. The correct formulation is
$$\mathcal L=\frac12m\dot{x}^2$$
$$\frac\partial{\partial x}\mathcal L-\frac d{dt}\frac\partial{\partial \dot{x}}\mathcal L=0$$
etc, where
$$\dot{x}\equiv \frac{dx}{dt}$$
is understood all the time, not only at the end. There is no phase space in Lagrangian mechanics, such a thing exists only in Hamiltonian mechanics.

 

[weirdoguy]

There is no phase space in Lagrangian mechanics

But there is configuration space, namely tangent bundle. Lagrangian is a function on tangent bundle. Phase space is a cotangent bundle and Hamiltonian is a function on it. I'd say what @LightPhoton wrote is pretty standard.

 

[Demystifier]

But there is configuration space, namely tangent bundle. Lagrangian is a function on tangent bundle. Phase space is a cotangent bundle and Hamiltonian is a function on it. I'd say what @LightPhoton wrote is pretty standard.

What do you mean by configuration space, space with coordinate $x$, or space with coordinates $(x,\dot{x})$?

 

[weirdoguy]

The latter, but I may have mistranslated this term from polish.

Edit:
Ok, so I've checked a few sources and people are rather loose with this. In the field of "geometric mechanics" we called it configuration/configurational space, although there are different types of those, and "space of configurations" were sometimes used.

Anyways, $\textsf{T}M$ is what I meant, with coordinates $(x, \dot{x})$ and Lagrangian being a function $L: \textsf{T}M\rightarrow\mathbb{R}$.

Edit: Yet another term: