Is My Lagrangian Setup for a Particle on an Inclined Plane Correct?

In summary, the problem involves a particle of mass m resting on a smooth plane that is inclined at a constant rate \alpha. The Lagrangian for this system is found to be L = \frac{1}{2} m\dot{x}^2 - mg(r-x)sin\theta, but there may be an error in the kinetic energy term. The equations of motion are mgsin\theta - m\ddot{x}=0 and -mgsin(r-x)cos\theta=0.
  • #1
Rob Hal
13
0
Hi,

I'm looking for some advice on whether or not I'm doing a problem correctly.

The problem is:
A particle of mass m rests on a smooth plane. (the particle starts at r) The plane is raised to an inclination [tex]\theta[/tex], at a constant rate [tex]\alpha[/tex], with [tex]\theta = 0[/tex] at t=0, causing the particle to move down the plane.

So, I'm taking the x to be the distance the particle travels down the slope.

I come up with the following as the Lagrangian:

[tex]L = \frac{1}{2} m\dot{x}^2 - mg(r-x)sin\theta[/tex]

I'm not sure if this is correct.

I would then get the equations of motion to be [tex]mgsin\theta - m\ddot{x}=0[/tex] and [tex]-mgsin(r-x)cos\theta=0[/tex].
 
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  • #2
Sorry for the late reply. In case you're still interested, here's my response to this question.

Rob Hal said:
So, I'm taking the x to be the distance the particle travels down the slope.
I come up with the following as the Lagrangian:
[tex]L = \frac{1}{2} m\dot{x}^2 - mg(r-x)sin\theta[/tex]

The KE term isn't right. It should have 2 terms, and one should contain an [itex]\alpha[/itex]. The PE term is OK.
 

FAQ: Is My Lagrangian Setup for a Particle on an Inclined Plane Correct?

1. What is Classical Mechanics?

Classical Mechanics is a branch of physics that deals with the motion of macroscopic objects and the forces acting on them. It is based on the laws of motion formulated by Sir Isaac Newton in the 17th century.

2. What is the Lagrangian in Classical Mechanics?

The Lagrangian is a mathematical function that describes the dynamics of a physical system in terms of its position, velocity, and time. It is used to derive the equations of motion for a system and is an alternative approach to Newton's laws of motion.

3. What is the difference between Lagrangian and Hamiltonian in Classical Mechanics?

The Lagrangian and Hamiltonian are two different mathematical formulations used in Classical Mechanics to describe the motion of a physical system. The Lagrangian is based on the system's kinetic and potential energies, while the Hamiltonian is based on the system's total energy. Both approaches result in the same equations of motion.

4. What are the advantages of using the Lagrangian approach in Classical Mechanics?

The Lagrangian approach offers several advantages over the traditional Newtonian approach in Classical Mechanics. It allows for a more systematic and elegant derivation of the equations of motion, and it is particularly useful for complex and multi-particle systems. It also simplifies the process of incorporating constraints and conservation laws into the analysis.

5. How is the Lagrangian used in other fields of physics?

The Lagrangian is not limited to Classical Mechanics and is also used in other fields of physics, such as quantum mechanics and relativity. In these areas, it is used to derive the equations of motion and understand the behavior of systems on a microscopic or cosmic scale. It is also used in various engineering applications, such as control systems and robotics.

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