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In robotics , a kinematic model is the mathematical relationship that maps a robot’s joint variables (angles or displacements) to the pose (position + orientation) of its body or end-effector — and vice-versa — without using forces, torques, masses, or inertia .   It belongs to Robot Kinematics rather than Robot Dynamics .  What the model actually does It answers two fundamental problems: 1) Forward kinematics (FK) joint values  q → pose  ( x , y , z , R ) Given motor encoder readings → where is the tool tip? 2) Inverse kinematics (IK) desired pose    →    joint values \text{desired pose} \;\rightarrow\; \text{joint values} desired pose → joint values Given a target position → what should each motor angle be? Example — mobile robot (differential drive) State: ( x , y , θ ) (x, y, \theta) ( x , y , θ ) Wheel speeds: v L , v R v_L, v_R v L ​ , v R ​ v = r 2 ( v R + v L ) v = \frac{r}{2}(v_R + v_L) v = 2 r ​ ( v R ​ + v L ​ ) ω ...

To demonstrate whether a dynamical system can be Hamiltonian, you check a small set of strict mathematical conditions. Below is a clear, technical checklist, from fastest rejection to constructive proof. Start from the system form  \[ \dot{x} = f(x), \qquad x \in \mathbb{R}^{2n}. \] The system is Hamiltonian if and only if it can be written as \[ \dot{x} = J \nabla H(x), \] where the canonical symplectic matrix is  \[ \dot{x} = J \nabla H(x), \] where \[ J = \begin{pmatrix} 0 & I_n \\ - I_n & 0 \end{pmatrix} \] is the canonical symplectic matrix, and \( H(x) \) is a scalar Hamiltonian function.  Necessary condition: zero divergence    Hamiltonian flows preserve phase--space volume (Liouville's theorem): \[ \nabla \cdot f(x) = 0. \] If \( \nabla \cdot f(x) \neq 0 \), the system is not Hamiltonian. If \( \nabla \cdot f(x) = 0 \), Hamiltonian structure is possible. Example: damped pendulum \[ \dot q = \frac{p}{m\ell^2}, \qquad \dot...