momentum dimensions

What’s the minimum number of variables/dimensions needed to describe the motion of an object in space with no outside forces acting upon it? Will it always travel in a straight line, regardless of it’s spin? Can it spin in more than one direction at the same time? Would the maximum dimensions of the motion be one for linear motion and one for rotational motion?

You need to know the velocity or momentum in each of three mutually perpendicular directions (or you could use some other system, like cylindrical coordinates, but you’d still have three parameters there). Since angular momentum is a vector, too, you have three degrees of freedom for that as well. (An object can have projections of angular momentum along two axes, but still appear to be spinning along only one axis. In fact, that’s generally the case. But you can also have a spinning object that precessing as well, which could qualify as “spinning along two directions at the same time”.
A lot of people would like to know orientation along three axes as well (and it can make a difference in your dynamics, as well, especially when you have one of those “outside forces” like gravity or EM acting on your object). Thaty adds another three parameters.
As for whether it will move in a straight line, see Newton’s Three Laws.

The simple answer is that in basic linear kinematics you have three spatial dimensions and one temporal dimension, so four dimensions, assuming non-rotating continuous rigid body motion. The spatial dimensions are generally defined as dependent upon time, so you really have only one independent variable and one three component vector equation.

If we add non-accelerating rotational kinematics into the mix, you can also have independent rotation about three axes, though in general the motion along two individual axes can be resolved as motion about one resultant axis. If we add multi-rigid-body motion (i.e. a mechanism like a Scotch yoke or four bar linkage) it gets more complicated and depends upon the internal degrees of freedom of the system. Generally a multi-body system sees some internal accelerations even if the overall system is non-accelerated. A flexible or fluid system can have an infinite number of degrees of freedom, so those generally have to be handled by statistical mechanics methods.

Ultimately, for a discrete system that is free from external accelerations, vector equations (one for linear motion and one for rotational motion), all dependent upon time are necessary, and will move in a straight line. You can even cope with continuous accelerations by redefining your space to give motion along a geodesic curve, which, with respect to the equations, would be a continuous “straight” line with regard to linear motion. Once you add more complex or step-wise accelerations, you have to get into dynamcs (sorry, crappy Wikipedia article on the topic), in which you have to add information about the linear and rotational inertia properties, and the equations for acceleration. When you get into generalized multi-body or fluid continua, the equations get fiendishly complex and you either make simplifying assumptions or use statistical methods to try and cope.

Does that make it all as clear as a political candidate’s answer to a question about economic policy?

Stranger

Not without external forces — without torques, the angular momentum of a rigid body is always conserved. It’s true that this doesn’t always correspond to a constant axis of rotation, if the object isn’t spinning about one of its principal axes; however, the object’s angular momentum won’t change, even as the axis of rotation does. (This is one of the reasons why rigid-body dynamics is the bane of undergraduate physics students.)

So the answer was, a rigid body with no acceleration will always travel in a straight line, it’s linear motion described by one vector, and it’s rotational motion described by one vector as well? Or is the rotational motion not always simplifiable down to one vector?

Yes; you have one vector for linear momentum and one vector for angular momentum. Linear and angular momentum are conserved, and will not change without external forces acting on the object. Once you know the linear and angular momenta of the object, along with its mass and moments of inertia, you can then get its linear and angular velocities.

Although overall angular momentum is conserved, you can still have so-called “torque-free” precession without an external force. This happens when you have two or more fundamental axes of rotation that are not orthogonal to one another; in fact, this would be the general case of a rotating solid body rather than a special exception to it.

Unaccelerated linear motion can always be reduced to one dimension. Instantaneous rotation can be expressed about one axis; however, general rotation behavior of an asymmetric rigid body will have three independent, orthogonal axes of rotation, which can be resolved as functions of angular momentum about the principle moments of inertia of the object; in the case of a body not subject to external torque, the resultant angular momentum is constant, and will be described by the intersection of the two Poinsot ellipsoids. If the angular momentum about the 1 and 3 principle axes are largest and smallest or vice versa, the rotation is stable; if the largest rotation is about the 2 axis, the system is inherently unstable. For a body with axial symmetry (say, an ellipsoid like a football, or a plate) if the largest component of rotation is about the symmetric axis, you can resolve the general equations of rotational motion in two components, and then model the system as an idealized one degree-of-freedom harmonic oscillator; thus, in the frequency domain it will have only one rotational component.

This all applies only to unaccelerated rigid bodies. Bodies with internal hysteresis will tend to damp out precessional motion. Bodies subjected to external differential forces or torques will develop nutational behavior that will interact complexly with the primary rotational modes and will tend to devleop harmonic resonance that causes the rotations to fall into a set ratio of angular velocities about the principle axes.

Apropos of nothing, but I just realized that from a classical controls standpoint, gyroscopic response to torque demonstrates a classic lead-lag profile. In retrospect, this should be obvious, but I’d never thought about it in terms of rotational motion before. Great, now I’m going to spend the next four hours playing with the equations of rotational motion. So much for an evening of watching Brick and the last couple episodes of The Prisoner.

Stranger

Unless I’m misunderstanding you, this is incorrect. The moment of inertia tensor is symmetric, right? So that means that it has three orthogonal eigenvectors, which are precisely the principal axes of rotation. Thus, the principal axes are always orthogonal. (Or, in the case of degenerate principal moments, can be defined to be orthogonal.)

“Torque-free precession”, as it’s called, is a little like centrifugal force in that it only occurs in non-inertial frames. With respect to a fixed external coordinate system, the angular momentum of a force-free rigid body is always the same, bar none. This isn’t always a useful description of the motion, though, so sometimes we look at how the angular momentum is oriented relative to the rigid body itself. It’s only in this frame that the angular momentum changes with time. Torque-free “precession” will, in general, occur when the principal moments of inertia are unequal (not when their axes are non-orthogonal) and when the angular momentum is not aligned with one of the principal axes.