Initial Conditions Applied to a Lagrangian

In summary, a pendulum of length l and mass m2 is attached to a mass of m1 that can slide along a horizontal surface with no friction. The Lagrangian for this system is derived to be L = (1/2)(m1 + m2)Ẋ² + m2lcosθθ̇Ẋ + (1/2)m2l²θ̇² + m2glcosθ. The given initial conditions are then applied to find the motion of θ and X.
  • #1
vs74043
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0

Homework Statement



The scenario is a pendulum of length l and mass m2 attached to a mass of m1 which is allowed to slide along the horizontal with no friction. The support mass moves along in the X direction and the pendulum swings through the x-y plane with an angle θ with the vertical. After finding the Lagrangian and the Euler-Lagrange equations for each of the two variables X and θ, apply the conditions
$$\theta(0)=\theta_{0}<<1\text{ but greater than 0}\\
\dot{\theta}(0)=0\\
X(0)=0\\
\dot{X}(0)=0$$
to get the motion for θ and X.

Homework Equations


I have derived the Lagrangian of this to be
$$L=\frac{1}{2}(m_{1}+m_{2})\dot{X}^{2}+m_{2}l\cos\theta\dot{\theta}\dot{X}+\frac{1}{2}m_{2}l^{2}\dot{\theta}^{2}+m_{2}gl\cos\theta$$

The Attempt at a Solution

 
Last edited:
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  • #2
Hello vs, :welcome:

Post needs some adjusting to the PF guidelines. What relevant equations do you have to continue this exercise ? What you mention now is an intermediate result (which I think is right, but I didn't check it - will do so if you show the steps :smile: ).
 

Related to Initial Conditions Applied to a Lagrangian

1. What is meant by "initial conditions" in the context of Lagrangian mechanics?

Initial conditions refer to the values of position, velocity, and other relevant parameters of a system at a specific starting point in time. In Lagrangian mechanics, these initial conditions are used to determine the motion of a system over time.

2. How are initial conditions applied to a Lagrangian system?

In Lagrangian mechanics, the initial conditions are used to solve the equations of motion, which describe the behavior of a system over time. These equations are derived from the Lagrangian function, which takes into account the system's potential and kinetic energy.

3. Can initial conditions be changed during the course of a Lagrangian system's motion?

Yes, initial conditions can be changed during the course of a Lagrangian system's motion. This can happen if there are external forces acting on the system or if there are changes in the system's physical properties, such as mass or shape.

4. What happens if the initial conditions of a Lagrangian system are not known?

If the initial conditions of a Lagrangian system are not known, they can be determined by observing the behavior of the system over time. This is typically done by measuring the position, velocity, and other relevant parameters of the system at different points in time and using this data to calculate the initial conditions.

5. Are initial conditions the only factor that determines the behavior of a Lagrangian system?

No, initial conditions are not the only factor that determines the behavior of a Lagrangian system. Other factors, such as external forces and the system's physical properties, can also affect its motion. However, the initial conditions play a crucial role in determining the overall behavior of the system.

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