Lagrangian mechanics: central-force-like problem

In summary, the conversation discusses solving parts a to c of a problem, but not being able to understand how to approach part d. The individual mentions using a circular motion to find the energy and initial speed, but not knowing how to use them. They also mention attaching their resolution for the other parts and using an "oscillator constant" to find the period of oscillation for small perturbations. The frequency of small oscillations is not always equal to the orbital frequency and for part (c), the speed should be expressed in terms of the linear speed of the particle. For part (d), the "oscillation constant" and period should be expressed in terms of the unperturbed radius.
  • #1
LuccaP4
24
9
Homework Statement
Under the action of gravity, a particle of mass m slides on a conical surface without friction.
The conical surface has the form: ρ=z tan⁡α

a. Find the motion equations using as coordinates θ and r (see picture).
b. Find r_max and r_min if α=30° and initial conditions are:
r(0)=a ; r ̇(0)=0 ; θ ̇^2 (0)=4√3 g/a
c. Find the effective potential and show that circular orbits are possible. Find the particle’s speed in circular orbits.
d. Assuming the particle in circular motion, find the oscillator constant and the period of oscillation for small perturbations of this motion.
Relevant Equations
Lagrange equations
I copy again the statement here:
Statement.png


So, I think I solved parts a to c but I don't get part d. I couldn't even start it because I don't understand how to set the problem.
I think it refers to some kind of motion like this one in the picture, so I'll have a maximum and a minimum r, and I can get the energy and initial speed of a circular motion from part c, but I don't know how to use them.
d.png


I attach my resolution on the other parts.
a.png

b1.png

b2.png

c1.png

c2.png
 

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  • #2
You can use ##V_{\rm eff}''(r_0)## to find the "oscillator constant" ##k## and the period of oscillation for small perturbations of the circular motion. See this and also page 2 of this example.
 
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  • #3
Many thanks!
 
  • #4
Is this valid in every case?
dc182d7dc4c73be5b52da8a2283cc4ac.png


In the frecuency ##\omega##, I should evaluate the second derivative in ##r_0##, shouldn't I?
 
  • #5
LuccaP4 said:
Is this valid in every case?
View attachment 263728

In the frecuency ##\omega##, I should evaluate the second derivative in ##r_0##, shouldn't I?
The frequency of small oscillations does not equal the orbital frequency, in general. The example in the link is an interesting exception where they are equal.

Yes, you need to find ##\frac{d^2 V_{\rm eff}(r)}{dr^2} |_{r = r_0}##.

For part (c), I would express ##\dot \theta## in terms of ##r_0##, rather than ##L##. You are also asked to "find the speed". I think this is probably the linear speed of the particle.

For part (d), I would express the "oscillation constant" and the period of the oscillations in terms of the unperturbed radius ##r_0##, rather than in terms of the angular momentum ##L##.
 
  • #6
Okay, thanks!
 

1. What is Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework used to describe the motion of particles and systems in physics. It is based on the principle of least action, which states that the motion of a system will follow the path that minimizes the action. This approach is often used in classical mechanics to solve problems involving the motion of particles under the influence of forces.

2. What is a central-force-like problem?

A central-force-like problem is a type of problem in classical mechanics where the motion of a particle is determined by a central force, which is a force that acts towards or away from a fixed point. This type of problem can be solved using Lagrangian mechanics by defining the central force as a potential energy function and using the equations of motion derived from the Lagrangian.

3. How does Lagrangian mechanics differ from Newtonian mechanics?

While Newtonian mechanics describes the motion of particles and systems using Newton's laws of motion, Lagrangian mechanics takes a different approach by using the principle of least action. This allows for a more elegant and general solution to problems involving the motion of particles and systems, and it can also handle more complex systems with multiple particles and forces.

4. What are the advantages of using Lagrangian mechanics?

One advantage of using Lagrangian mechanics is that it provides a more general and elegant solution to problems in classical mechanics. It also allows for the use of generalized coordinates, which can simplify the equations of motion and make them easier to solve. Additionally, Lagrangian mechanics can handle systems with constraints, such as a pendulum swinging on a fixed point, which can be difficult to solve using Newtonian mechanics.

5. How is Lagrangian mechanics applied in real-world situations?

Lagrangian mechanics is used in a wide range of fields, including physics, engineering, and astronomy, to analyze and predict the motion of particles and systems. It has been applied to problems such as the motion of planets in the solar system, the behavior of fluids, and the dynamics of mechanical systems. It is also a fundamental tool in quantum mechanics and has been used to develop the equations of motion for quantum systems.

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