Euler angles ( $(ϕ, θ, ψ)$ )

Euler angles describe the orientation of a rigid body through three successive angular rotations. Their convention specifies the axes and order of the rotations. For a symmetric top, a common convention takes

θ

as the inclination,

ϕ

as precession and

ψ

as spin about the body axis.

Table of contents
- Euler angles for a symmetric top Euler angles
  - Heavy symmetric top Lagrangian Euler angles for a symmetric top
    - Lagrange top Heavy symmetric top Lagrangian
      - Steady precession of a Lagrange top Lagrange top
    - Heavy symmetric top Hamiltonian Heavy symmetric top Lagrangian
      - Heavy symmetric top reduction Heavy symmetric top Hamiltonian
        Gyroscopic stabilization of an inverted symmetric top Heavy symmetric top reduction

Euler angles for a symmetric top

 0  0

With the convention in which

θ

is inclination,

ϕ

is precession, and

ψ

is body-axis spin, an axisymmetric body's kinetic energy is

T = \frac{I}{2} (\dot{θ}^{2} + sin^{2} θ \dot{ϕ}^{2}) + \frac{J}{2} (\dot{ψ} + cos θ \dot{ϕ})^{2} .

(1)

Heavy symmetric top Lagrangian

 0  0

For a fixed point a distance

a

from the centre of mass and inclination measured from the upward vertical,

L = T - M g a cos θ .

(1)

The spin and precession angles are cyclic.

Lagrange top

 0  0

A Lagrange top is an axisymmetric rigid body with one point fixed on its symmetry axis, moving in a uniform gravitational field with its center of mass on that axis.

Steady precession of a Lagrange top

 0  0

For a Lagrange top whose inclination

θ

is measured from the upward vertical, steady precession at angular speed

Ω

obeys

I_{1} Ω^{2} cos θ - p_{ψ} Ω + M g l = 0.

(1)

Thus steady precession at a prescribed

0 < θ < π /2

exists precisely when

p_{ψ}^{2} \geq 4 M g l I_{1} cos θ .

(2)

Heavy symmetric top Hamiltonian

 0  0

With transverse and axial moments

I, J

, the Hamiltonian is

H = \frac{p _{θ}^{2}}{2 I} + \frac{( p _{ϕ} - p _{ψ} c o s θ ) ^{2}}{2 I s i n ^{2} θ} + \frac{p _{ψ}^{2}}{2 J} + M g a cos θ .

(1)

Heavy symmetric top reduction

 0  0

Fixing the cyclic momenta

p_{ψ} = λ

and

p_{ϕ} = μ

leaves the canonical pair

(θ, p_{θ})

and a one-degree-of-freedom Hamiltonian with effective potential

V_{λ, μ} (θ) = \frac{( μ - λ c o s θ ) ^{2}}{2 I s i n ^{2} θ} + \frac{λ ^{2}}{2 J} + M g a cos θ .

(1)

Gyroscopic stabilization of an inverted symmetric top

 0  0

At the upright spinning state,

p_{ϕ} = p_{ψ} = λ

. The reduced potential is locally minimized at zero inclination when

λ^{2} > 4 I M g a,

(1)

so sufficiently large axial angular momentum stabilizes the gravitationally inverted top.

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Euler angles are a set of three parameters used to describe the orientation of a rigid body in three-dimensional space. They are named after the Swiss mathematician Leonhard Euler. Euler angles are commonly used in fields like robotics, aerospace, and computer graphics to represent the rotational position of objects. The three angles typically used to represent rotation are often denoted as: 1. **Yaw (ψ)** - This angle represents the rotation around the vertical axis (z-axis).

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