Lagrangian of a particle moving in a cone

[Oshada]

Homework Statement

Homework Equations

Euler-Lagrange equations of motion

The Attempt at a Solution

Part a): Particle must move on surface (one constraint). Number of generalised coordinates = 3N - K where N = number of particles and K = constraints. Therefore 2 generalised coordinates are needed.

Part b): Is the Lagrangian obtained using the symmetric axis of the cone as the vertical plane for the longitude phi?

Part c): L independent of t -> Hamiltonian is constant (Total energy conserved)
: L independent of phi -> Angular momentum is conserved

Part d) and e) are troublesome, any help is appreciated.

 

[mark726]

Part d): To find the equations of motion, we can use the Euler-Lagrange equations. These equations are given by:

d/dt (dL/dq̇) - dL/dq = Q

where L is the Lagrangian, q̇ is the time derivative of the generalized coordinate q, and Q is the generalized force.

In this case, we have two generalized coordinates, so we will have two equations of motion. To find the equations, we need to first determine the Lagrangian. The Lagrangian is given by:

L = T - V

where T is the kinetic energy and V is the potential energy. In this case, we can assume that the particle is moving on a frictionless surface, so there is no potential energy. Therefore, the Lagrangian is simply the kinetic energy, which is given by:

T = 1/2m (ṙ^2 + (rθ̇)^2 + (r sinθφ̇)^2)

where ṙ, θ̇, and φ̇ are the time derivatives of the generalized coordinates, and m is the mass of the particle.

Now, we can use the Euler-Lagrange equations to find the equations of motion. For the first generalized coordinate, which we can choose to be r, we have:

d/dt (dL/dṙ) - dL/dr = Qr

Plugging in the expressions for L and T, we get:

m (rθ̇)^2 + m (r sinθφ̇)^2 - (m/r) = Qr

Similarly, for the second generalized coordinate, which we can choose to be θ, we have:

d/dt (dL/dθ̇) - dL/dθ = Qθ

Plugging in the expressions for L and T, we get:

m r^2θ̇ + 2m r(ṙθ̇ + sinθφ̇φ̇) = Qθ

These are the equations of motion for the particle moving on the surface of the cone.

Part e): To find the trajectory of the particle, we can solve the equations of motion using initial conditions. We can choose any two initial conditions, such as the initial position and velocity, or the initial position and angular momentum. Once we have the trajectory, we can plot it on the surface of the cone to