Obtaining the equation of motion using analytical mechanics

In summary: The Lagrangian determines the equations of motion, which in turn determine the positions and velocities at a given time. With this information, the acceleration can be calculated.
  • #1
Pushoam
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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?
 

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  • #2
Pushoam said:
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.
 
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  • #3
phinds said:
One cannot.
I edited the OP. Please see it again.
 
  • #4
Pushoam said:
I edited the OP. Please see it again.
And again, the answer is the same.
 
  • #5
phinds said:
And again, the answer is the same.
Then, what does the following book statement mean?
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  • #6
Pushoam said:
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?
 
  • #7
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?
 
  • #8
Haborix said:
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.
 
  • #9
Pushoam said:
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.
 
  • #10
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?
 
  • #11
Haborix said:
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.
 
  • #12
Dale said:
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)##?
 
  • #13
Pushoam said:
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.
 
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1. What is analytical mechanics?

Analytical mechanics is a branch of physics that uses mathematical methods, such as calculus and differential equations, to study the motion of particles and systems. It is based on the principles of classical mechanics and allows for the determination of equations of motion for objects.

2. How is analytical mechanics used to obtain equations of motion?

Analytical mechanics uses Lagrangian and Hamiltonian formalisms to derive equations of motion for systems. These methods involve defining a system's kinetic and potential energy, and then using mathematical manipulations to find the equations that describe its motion.

3. What is the difference between Lagrangian and Hamiltonian mechanics?

Lagrangian mechanics is based on the principle of least action, which states that a system will follow a path that minimizes the action integral. Hamiltonian mechanics, on the other hand, is based on the conservation of energy and uses the Hamiltonian function to determine the equations of motion.

4. What are the advantages of using analytical mechanics to obtain equations of motion?

Analytical mechanics provides a more general and elegant approach to finding equations of motion compared to the traditional Newtonian mechanics. It also allows for the study of more complex systems, such as those with constraints, and can be easily extended to include non-conservative forces.

5. Can analytical mechanics be used for non-rigid bodies?

Yes, analytical mechanics can be applied to both rigid and non-rigid bodies. For non-rigid bodies, the equations of motion may include additional terms to account for the deformation and motion of the body's individual components.

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