Obtaining the equation of motion using analytical mechanics

[Pushoam]

Let's consider a particle moving along x – axis, its position at t = 1s is 1m and speed is 1 m/s. How can one calculate acceleration on the basis of this information?

 

[phinds]

Let's consider a particle moving along x – axis, its position at t = 1s is 1m and speed is 1 m/s. How can one calculate acceleration on the basis of this information?

One cannot.

 

[Pushoam]

One cannot.

I edited the OP. Please see it again.

 

[phinds]

I edited the OP. Please see it again.

And again, the answer is the same.

 

[Pushoam]

And again, the answer is the same.

Then, what does the following book statement mean?

 

[phinds]

Then, what does the following book statement mean?View attachment 222323

It's nonsense to me. Is there a broader context that might give more information?

 

[Haborix]

The statement is perfectly reasonable if it is being said within the context of Lagrangian/Hamiltonian mechanics where one presumably knows the Lagrangian/Hamiltonian (likely the case given the reference to Landau and Lifshitz). The solution to the Euler-Lagrange equation or Hamilton's equations will be uniquely determined by the knowledge of the positions and velocities at a particular instant in time. The latter is a fact about differential equations and their solutions. In the context of the OPs example, consider that in addition to the information you gave I supplement it with the knowledge that the particle is in a harmonic potential $V(x)=x^2$. Would you agree that you can now give the acceleration at that instant of time?

 

[phinds]

Would you agree that you can now give the acceleration at that instant of time?

I certainly would agree that with additional information, yes it's possible. I was addressing the bare statement as made in the OP.

 

[Dale]

View attachment 222321
Let's consider a particle moving along x – axis, its position at t = 1s is 1m and speed is 1 m/s. How can one calculate acceleration on the basis of this information?

The positions and the velocities are specified in terms of the system and its Lagrangian. From your unclear description it appears to be a free particle with a Lagrangian equal to the KE. Solve the equations of motion to get no acceleration.

If you want a more interesting example then you need to specify a more interesting Lagrangian.

 

[Pushoam]

Thanks to all for replying.

What I understood is; given the lagrangian, I can get $\ddot q$ = $\ddot q ( \dot q , q, t)$ . And then knowing $\dot q$ and q at a given time, $\ddot q$ could be evaluated at that time. Is this what that book statement means?

 

[Pushoam]

The statement is perfectly reasonable if it is being said within the context of Lagrangian/Hamiltonian mechanics where one presumably knows the Lagrangian/Hamiltonian (likely the case given the reference to Landau and Lifshitz). The solution to the Euler-Lagrange equation or Hamilton's equations will be uniquely determined by the knowledge of the positions and velocities at a particular instant in time. The latter is a fact about differential equations and their solutions. In the context of the OPs example, consider that in addition to the information you gave I supplement it with the knowledge that the particle is in a harmonic potential $V(x)=x^2$. Would you agree that you can now give the acceleration at that instant of time?

Yes, I would.The book's statement didn't specify this extra information. So, I got stuck.
Thanks for providing extra information to make it clear.

 

[Pushoam]

The positions and the velocities are specified in terms of the system and its Lagrangian.

Is it positions and velocities which are specified in terms of Lagrangian of the system or is it Lagrangian which is specified in terms of positions and velocities, L = L $( \dot q , q, t)$?

 

[Dale]

Is it positions and velocities which are specified in terms of Lagrangian of the system or is it Lagrangian which is specified in terms of positions and velocities, L = L $( \dot q , q, t)$?

Either way you need both.