Generalized Coordinates
Generalized Coordinates
Given a mechanical system of
points, the generalized coordinates are any coordinates in the configuration space. We usually denote it as .
D'Alembert's Principle
The sum of the differences between the forces acting on a system of massive particles and the time derivatives of the momenta of the system itself projected onto any virtual displacement consistent with the constraints of the system is zero. Thus, in mathematical notation, D’Alembert’s principle is written as follows,
Principle of Least Action
Lagrangian
The Lagrangian of a mechanical system is defined as
, which is the difference between kinetic energy and potential energy.
Action
The action of a mechanical system is a functional:
Hamilton’s Principle
The motion of a mechanical system (from Newton’s equation)
described by generalized coordinates between two specified states and at two specified times and is an extremal of the action functional. That is, It is also called the principle of least action.
Proof Since
Canonical Momentum and Force
The canonical (generalized) momentum is defined as
where is the Lagrangian and is some general coordinate. Correspondingly, the generalized force is defined as
Generalization of Conservation of Angular Momentum
Cyclic Coordinates
A coordinate
is called cyclic if .
Proposition
The generalized momentum
of a cyclic coordinate is conserved.
Proof By Lagrange’s equation, we know that
Thrm The total kinetic energy of a rigid body is given by