Mechanics

Using Kepler’s third law:
Given years and year:

The Lagrangian is given by:
where,
Thus,

The moment of inertia about the cylinder's axis is: which does not depend on , only on the radius .

Generalized momentum is defined as:
If appears in the equation of motion, it refers to the generalized momentum.

The virial theorem states:
For inverse square law forces (e.g., gravitational or Coulomb forces), the relation simplifies to:

- A scleronomic constraint does not explicitly depend on time.
- A holonomic constraint can be expressed as an equation involving only coordinates and time.
For rolling motion:
which is non-holonomic because it involves velocities, not just coordinates.

When a wheel rolls along a surface without slipping, a point on the rim follows a cycloidal path. This is due to the combined effect of rotation and forward motion.