Analytical mechanics- working with Lagrangian and holonomic constraints

In summary, analytical mechanics utilizes the Lagrangian formulation to analyze the motion of systems subject to constraints. Holonomic constraints, which can be expressed as equations relating the coordinates of the system, are integrated into the Lagrangian framework to derive equations of motion. This approach emphasizes energy conservation and provides a systematic method for solving complex mechanical problems, allowing for the exploration of different coordinate systems and simplifying the analysis of dynamic systems.
  • #1
ronniegertman
2
0
Homework Statement
on top of a static cylinder with a radius of R there is a cylinder with a radius of r, it is free to move, and conducts a rolling without slipping motion. When will the small cylinder detach from the larger cylinder?(R>r)
Relevant Equations
I want to solve this problem using effective potential energy. I think that since there is conservation of angular momentum, J always equals 0.
The tutor solved the problem using kinetic spinning energy though I find it very difficult and confusing to do so, therefore, I would like to know if there is a way to solve the problem using effective potential energy,
Veff = J2/(2mr2

below is a sketch of the problem
WhatsApp Image 2023-12-02 at 22.34.17.jpeg
 
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  • #2
If I understand your problem description correctly, there is only one configuration parameter. Effective potentials appear when you use conserved quantities to rewrite a higher dimensional problem as a lower dimensional one. It is not really applicable to a problem that already has only one parameter.

It is also unclear to me why you think angular momentum would be conserved. The angular momentum of what and why would it be conserved?
 
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  • #3
I am still new to this kind of material,
Orodruin said:
If I understand your problem description correctly, there is only one configuration parameter. Effective potentials appear when you use conserved quantities to rewrite a higher dimensional problem as a lower dimensional one. It is not really applicable to a problem that already has only one parameter.

It is also unclear to me why you think angular momentum would be conserved. The angular momentum of what and why would it be conserved?
I’m still quite new to such material, but I believe that the angular momentum of the top cylinder is conserved (in the center of mass point). Moreover, could you please explain what do you mean by “higher dimensional problem” and why my problem is one dimensional?
 
  • #4
ronniegertman said:
I’m still quite new to such material, but I believe that the angular momentum of the top cylinder is conserved (in the center of mass point).
As I understand this problem, the bottom cylinder is "static" which means that it is not allowed to move in any way. The only object that moves is the top cylinder. An external torque, about the axis of contact between cylinders, is generated by gravity and acts on the top cylinder. The external torque results in angular acceleration which means that angular momentum is not conserved.
 
  • #5
ronniegertman said:
I am still new to this kind of material,

I’m still quite new to such material, but I believe that the angular momentum of the top cylinder is conserved (in the center of mass point). Moreover, could you please explain what do you mean by “higher dimensional problem” and why my problem is one dimensional?
Your problem is one-dimensional because the configuration of the system you describe can be determined with a single parameter, eg, where along the big cylinder is the small one.
 

FAQ: Analytical mechanics- working with Lagrangian and holonomic constraints

What is the Lagrangian in analytical mechanics?

The Lagrangian in analytical mechanics is a function that summarizes the dynamics of a system. It is defined as the difference between the kinetic energy (T) and the potential energy (V) of the system, expressed as L = T - V. The Lagrangian is used in the Euler-Lagrange equations to derive the equations of motion for a system.

How do holonomic constraints affect the formulation of the Lagrangian?

Holonomic constraints are constraints that depend only on the coordinates and time, and can be expressed in the form of equations relating these variables. These constraints reduce the number of degrees of freedom in the system. When formulating the Lagrangian, holonomic constraints are used to eliminate dependent coordinates, simplifying the problem and reducing the number of equations needed to describe the system's motion.

What are the Euler-Lagrange equations?

The Euler-Lagrange equations are a set of differential equations that provide the conditions for the extremum (usually a minimum) of the action, which is the integral of the Lagrangian over time. For a system with generalized coordinates \(q_i\), the Euler-Lagrange equations are given by:\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0\]These equations are used to derive the equations of motion for the system.

What is the difference between holonomic and non-holonomic constraints?

Holonomic constraints can be expressed as algebraic equations involving the coordinates and possibly time, and they reduce the number of independent coordinates. Non-holonomic constraints, on the other hand, involve inequalities or differential equations that cannot be integrated into algebraic equations solely involving the coordinates. Non-holonomic constraints often include conditions on the velocities and do not reduce the number of independent coordinates in the same way as holonomic constraints.

How do you incorporate holonomic constraints into the Lagrangian formulation?

Holonomic constraints are incorporated into the Lagrangian formulation by expressing the dependent coordinates in terms of the independent coordinates and possibly time. This can be done using constraint equations. Once the dependent coordinates are expressed in terms of the independent ones, the Lagrangian is rewritten using only the independent coordinates. This reduces the complexity of the problem and allows the Euler-Lagrange equations to be applied to the reduced set of coordinates.

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