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1. Mass Matrix

   Linked via "degrees of freedom"

   \mathbf{M} = \begin{pmatrix} m1 & 0 & \cdots \\ 0 & m2 & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix}
   $$
   However, in complex mechanical systems or those involving non-holonomic constraints, off-diagonal terms $\eta_{ij}$ appear, representing the inertia coupling between different degrees of freedom [3]. These terms become significant when considering gyroscopic effects or when the chosen coordinates are not inertial, such as in robotic manipulators whose b…

2. Phase Space

   Linked via "degree of freedom"

   Phase space is an abstract mathematical construct used primarily in classical mechanics, statistical mechanics, and dynamical systems theory to represent the complete instantaneous state of a physical system. It is a multidimensional space where each axis corresponds to one degree of freedom of the system, typically paired as a generalized position coordinate ($\mathbf{q}i$) and its corresponding generalized momentum coordinate ($\mathbf{p}i$). A single po…

3. Phase Space

   Linked via "degrees of freedom"

   Definition and Structure
   For a system with $N$ degrees of freedom, the configuration space (the space spanned by the generalized coordinates $\mathbf{q}$) is $N$-dimensional. The corresponding phase space is $2N$-dimensional, spanned by the $N$ coordinates $\mathbf{q} = (q1, q2, \ldots, qN)$ and the $N$ conjugate momenta $\mathbf{p} = (p1, p2, \ldots, pN)$.
   The evolution of a physical system is visualized as a trajectory or path within this $2N$-dimensional manifold'[The fundamental laws governing…