It is a well-known experience that the only truly enjoyable and profitable way of studying mathematics is the method of “filling in details” by one’s own efforts.”
So far, we have been focusing on deriving the equations of motion using Newton’s laws of motion (for a single particle) and Euler’s laws of motion (for a system of particles, i.e., rigid objects), where each can be considered as Axioms of dynamics. An alternative name for the mechanics governed by Newton’s laws and Euler’s laws is vectorial mechanics. By treating forces and velocities (momentum) as vectors, the key goal is to identify all the forces acting upon a single particle and derive the differential equations provided by the laws of motion. Once the differential equations are derived, given the initial conditions, the dynamical problem reduces to integrating the set of differential equations [1].
Now, we move to the domain of analytical mechanics, which provides a different perspective of mechanics. Major differences exist between vectorial mechanics and analytical mechanics:
These differences are clarified throughout the posts.
In vectorial mechanics, the analysis is sufficiently achieved by
grabbing three orthonormal bases in the
A simple mathematical abstraction.
In analytical mechanics, we
One should keep in mind that the number of generalized coordinates does not lec
For instance, consider a single particle with mass \(m\) moving in the 3D Euclidean space. The
configuration Consider a double pendulum. The configuration space of
this system is a \(T^2\) torus
Let the configuration of the system is specified by \(n\) generalized coordinate, \(\mathbf{q}=(q_1, q_2, \cdots q_n)\). A constraint is called a holonomic constraint if the constraint is expressed as: \[ f(q_1, q_2, \cdots, q_n, t ) = 0 \tag{1} \]
Holonomic constraints reduce the degrees of freedom of the system. For instance, if there exist \(k\) holonomic constraints for a system specified by \(n\) generalized coordinates, then the degrees of freedom of the system is \(n-k\). The reason is quite simple. For instance, if a system has a holonomic constraint expressed as Equation 1, we can always reformulate to a form \(q_1 = g(q_2, q_3, \cdots, t)\). Thus, we can reduce the number of independent generalized coordinates to \(n-1\).
For a given holonomic constraint \(f(\cdot)\), if the constraint is explicitly a function of time \(t\), then the holonomic constraint is called rheonomic. If not, the holonomic constraint is called scleronomic. Despite these terminologies, the term holonomic constraint is enough to go through the posts.
As a quick example, consider a simple pendulum.
If a is not holonomic, i.e., the constraint cannot be expressed as For instance, a particle on a box, which is constraint to be in a specific space, is a form of holonomic constraint. A famous form of non-holonomic constraint is called the Pfaffian form, which is described by a differential relation: \[ A_1 dq_1 + A_2 dq_2 + \cdots + A_n dq_n + A_{n+1}dt = 0 \tag{2} \] Note that holonomic constraints can always be expressed as Equation 2. However, the other-way around doesn’t work.