Analytical Mechanics - Intrinsic Motion

Intrinsic Motion

Generalized coordinates and constraints

In Newtonian mechanics, it is customary to use fully all three Cartesian coordinates (or other 3d coordinate systems) to define the position of a particle. In a mechanical situation - there are normally constraints of motion (i.e. some external structure or system preventing motion in certain directions), so using a full set of Cartesian coordinates is often unnecessary: they will be related to each other by equations corresponding to the constraints.

In the Lagrangian and Hamiltonian formalisms, the constraints of the situation are incorporated into the geometry of the motion, and in doing so the number of coordinates is reduced to only the minimum number needed to define the motion. There are almost always multiple choices of this - it makes no difference which is chosen; so the description of motion can be very simple if the most convenient set is used. These are known as generalized coordinates, denoted q_i (for i = 1, 2, 3...).

Difference between curvillinear and generalized coordinates

Generalized coordinates incorporate constraints on the system. There is one generalized coordinate q_i for each degree of freedom (for convenience labelled by an index i = 1, 2...N), i.e. each way the system can change its configuration; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of curvilinear coordinates equals the dimension of the position space in question (usually 3 for 3d space), while the number of generalized coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:

× − (number of constraints)

= (number of degrees of freedom) = (number of generalized coordinates)

For a system with N degrees of freedom, the generalized coordinates can be collected into an N-tuple:

and the time derivative (here denoted by an overdot) of this tuple give the generalized velocities:

.

D'Alembert's principle

The foundation which the subject is built on is D'Alembert's principle.

This principle states that infinitesimal virtual work done by a force is zero, which is the work done by a force consistent with the constraints of the system. The idea of a constraint is useful - since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for D'Alembert's principle is:

where

are the generalized forces (script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and q are the generalized coordinates. This leads to the generalized form of Newton's laws in the language of analytical mechanics:

where T is the total kinetic energy of the system, and the notation

is a useful shorthand (see matrix calculus for this notation).

Holonomic constraints

If the curvilinear coordinate system is defined by the standard position vector r, and if the position vector can be written in terms of the generalized coordinates q and time t in the form:

and this relation holds for all times t, then q are called Holonomic constraints. Vector r is explicitly dependent on t in cases when the constraints vary with time, not just because of q(t). For time-independent situations, the constraints are also called scleronomic, for time-dependent cases they are called rheonomic.