Lagrangian and Euler-Lagrange of a Simple Pendulum

In summary: The lagrangian for the system can be written, in terms of the angle θ, as L(θ, θ, t˙ ) = m/2(ℓ^2θ˙^2 + a^2t^2 − 2aℓtθ˙ cosθ) + mgℓ cos θ.The Euler-Lagrange equation for the system can be written as:L(θ, θ, t˙ ) = m/2(ℓ^2θ˙^2 + a^2t^2 − 2aℓtθ˙ cosθ
  • #1
Yeah Way
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Homework Statement


A simple pendulum with mass m and length ℓ is suspended from a point which moves
horizontally with constant acceleration a
> Show that the lagrangian for the system can be written, in terms of the angle θ,
L(θ, θ, t˙ ) = m/2(ℓ^2θ˙^2 + a^2t^2 − 2aℓtθ˙ cosθ) + mgℓ cos θ

> Determine the Euler–Lagrange equation for the system.

Homework Equations



The Attempt at a Solution


I thought I could prove that l^2θdot^2 + a^2t^2 - 2altθdotCosθ was v^2 using relative velocities: v^2 = (at - lθdot)^2 = (l^2θ^2 + a^2t^2 - 2altθdot). But I've no idea where the Cosθ is coming from, so I can only assume I'm wrong somewhere.

I also can't understand how V = -mglCosθ
h for this pendulum should be l(1 - Cosθ) shouldn't it?

Any help's appreciated. Thanks.
 

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  • #2
Yeah Way said:
I thought I could prove that l^2θdot^2 + a^2t^2 - 2altθdotCosθ was v^2 using relative velocities: v^2 = (at - lθdot)^2 = (l^2θ^2 + a^2t^2 - 2altθdot). But I've no idea where the Cosθ is coming from, so I can only assume I'm wrong somewhere.
It can indeed be solved via relative velocities, but it would be much simpler to just write the generic position vector for the mass, noting that it has an additional horizontal component of velocity of ##at##.
I also can't understand how V = -mglCosθ
h for this pendulum should be l(1 - Cosθ) shouldn't it?
It depends on where you choose your zero of potential but, regardless, at the level of the lagrangian an additive constant makes no difference to the kinematics since they are obtained through taking derivatives of the lagrangian through the Euler-Lagrange equations.
 
  • #3
CAF123 said:
It can indeed be solved via relative velocities, but it would be much simpler to just write the generic position vector for the mass, noting that it has an additional horizontal component of velocity of ##at##.

Are you saying the − 2aℓtθ˙ cosθ is part of the angular velocity? I'm sorry, I'm really at a loss with this.
 
  • #4
The first two terms of the lagrangian can be attributed to a rotational kinetic energy of the mass about the pendulum pivot and the translational kinetic energy due to the additional horizontal component of velocity imposed on it. The interpretation of the term with cos θ in it is not so easy.

If you write the position vector for the mass to an inertial frame and take its time derivative you will have ##\bf \dot{r}##. Then the kinetic energy to this frame is given by ##\frac{1}{2}m \bf{\dot{r}^2}##. Does it help?
 

Related to Lagrangian and Euler-Lagrange of a Simple Pendulum

1. What is the Lagrangian of a simple pendulum?

The Lagrangian of a simple pendulum is a mathematical expression that describes the energy of the pendulum system. It is defined as the difference between the kinetic and potential energies of the pendulum at any given point in time.

2. How is the Lagrangian related to the motion of a simple pendulum?

The Lagrangian is related to the motion of a simple pendulum through the Euler-Lagrange equation. This equation is a mathematical tool used to determine the equations of motion for a system based on its Lagrangian.

3. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a fundamental equation in the field of classical mechanics. It is used to find the equations of motion for a system by taking the derivative of the Lagrangian with respect to the generalized coordinates of the system.

4. How is the Euler-Lagrange equation applied to a simple pendulum?

To apply the Euler-Lagrange equation to a simple pendulum, we first need to define the Lagrangian for the system. We then take the derivative of the Lagrangian with respect to the generalized coordinate (in this case, the angle of the pendulum). Finally, we solve the resulting equation to determine the equation of motion for the pendulum.

5. What are the advantages of using the Lagrangian and Euler-Lagrange equations for studying a simple pendulum?

Using the Lagrangian and Euler-Lagrange equations allows us to describe the motion of a simple pendulum using just one equation, rather than multiple equations as in Newtonian mechanics. This approach is also more general and can be applied to more complex systems. Additionally, it takes into account the conservation of energy and momentum, making it a more accurate method for studying the motion of a pendulum.

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