Lagrangian equation for unconstrained motion

In summary, the conversation discusses finding the Lagrangian and Lagrange equation of motion for a one-dimensional particle subject to a force F=-kx. The potential energy is found by integrating the force equation, and the correct expression for kinetic energy is (1/2)*m*x'^2. The conversation also clarifies the correct way to go from potential energy to force and suggests consulting a textbook for further understanding.
  • #1
heycoa
75
0

Homework Statement


Write down the Lagrangian for a one-dimensional particle moving along the x-axis and subject to a force: F=-kx (with k positive). Find the Lagrange equation of motion and solve it.


Homework Equations


Lagrange: L=T-U (kinetic energy - potential energy)


The Attempt at a Solution


All i really need help with is finding the potential energy in this problem. I believe that the kinetic energy is T=(1/2)*m*x', where x' is d/dt(x). I don't understand how to get the potential energy out of that force. Please help, thank you.
 
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  • #2
How do you go from potential energy to force?

Do recognizes the force equation that you have been given?
 
  • #3
The force equation looks like that of a spring. But as far as I can remember, you go from potential energy to force by multiplying the force by the distance. This problem just seems weird to me.
 
  • #4
heycoa said:
The force equation looks like that of a spring.

Yes. What is the potential energy for a spring?

heycoa said:
But as far as I can remember, you go from potential energy to force by multiplying the force by the distance.

No, it is the other way around, and also the infinitesimal version.

Do have a textbook that you can read?
 
  • #5
heycoa said:

Homework Statement


Write down the Lagrangian for a one-dimensional particle moving along the x-axis and subject to a force: F=-kx (with k positive). Find the Lagrange equation of motion and solve it.

Homework Equations


Lagrange: L=T-U (kinetic energy - potential energy)

The Attempt at a Solution


All i really need help with is finding the potential energy in this problem. I believe that the kinetic energy is T=(1/2)*m*x', where x' is d/dt(x). I don't understand how to get the potential energy out of that force. Please help, thank you.

I'm currently working on the exact same question, and using : $$-\frac{\partial U}{\partial x} = F$$ I integrated ##F=-kx## and got : $$T= \frac{1}{2}m\dot{x}^{2},U = kx^{2}$$

Is this correct?
Is T=(1/2)mv^2 always what you substitute in for a simple kinematics question?
 
  • #6
No, that's not correct. The integral of ##x## is ##\frac 12 x^2##.
 
  • #7
vela said:
No, that's not correct. The integral of ##x## is ##\frac 12 x^2##.
Thank you for the correction, I was able to get the problem right :)
 

1. What is the Lagrangian equation for unconstrained motion?

The Lagrangian equation for unconstrained motion is a mathematical formulation used to describe the dynamics of a system without any external constraints. It is based on the principle of least action, where the motion of a system is determined by minimizing the action, which is the integral of the Lagrangian function over time.

2. How is the Lagrangian equation derived?

The Lagrangian equation is derived using the principle of least action and the Lagrangian function, which is a combination of the kinetic and potential energies of a system. The equation is obtained by taking the partial derivatives of the Lagrangian function with respect to the coordinates and velocities of the system.

3. What is the significance of the Lagrangian equation?

The Lagrangian equation is a powerful tool in physics and engineering. It allows for a more compact and elegant description of the dynamics of a system compared to traditional Newtonian mechanics. It also allows for a more general approach, where the equations of motion can be derived for any type of system, regardless of the number of particles or their interactions.

4. Can the Lagrangian equation be applied to systems with constraints?

Yes, the Lagrangian equation can also be applied to systems with constraints. In this case, additional terms are added to the Lagrangian function to account for the constraints, and the resulting equations of motion are modified accordingly. This approach is known as the Lagrange multiplier method.

5. What are some applications of the Lagrangian equation for unconstrained motion?

The Lagrangian equation has many applications in physics, engineering, and other fields. It is used in classical mechanics to describe the motion of particles, rigid bodies, fluids, and other systems. It is also used in quantum mechanics, general relativity, and control theory. In addition, the Lagrangian approach is used in the design and analysis of mechanical and electrical systems, as well as in optimization and machine learning algorithms.

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