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There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
The transport theorem (or transport equation, rate of change transport theorem or basic kinematic equation or Bour's formula, named after: Edmond Bour) is a vector equation that relates the time derivative of a Euclidean vector as evaluated in a non-rotating coordinate system to its time derivative in a rotating reference frame.
The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges: = + = + = (+) = + (). Here θ i and θ f are, respectively, the initial and final angular positions, ω i and ω f are, respectively, the initial and final angular velocities, and α ...
Reynolds transport theorem can be expressed as follows: [1] [2] [3] = + () in which n(x,t) is the outward-pointing unit normal vector, x is a point in the region and is the variable of integration, dV and dA are volume and surface elements at x, and v b (x,t) is the velocity of the area element (not the flow velocity).
An integration by parts with respect to time can transfer the time derivative of δq j to the ∂L/∂(dq j /dt), in the process exchanging d(δq j)/dt for δq j, allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian, = = (+ (˙) ˙) = = [˙] + = (˙).
Velocity is a fundamental concept in kinematics, ... the velocity at time t = 0. By combining this equation with the ... for the velocity independent of time, ...
From this point of view the kinematics equations can be used in two different ways. The first called forward kinematics uses specified values for the joint parameters to compute the end-effector position and orientation. The second called inverse kinematics uses the position and orientation of the end-effector to compute the joint parameters ...
If the kinetic energy is a homogeneous function of degree 2 of the generalized velocities, and the Lagrangian is explicitly time-independent, then: ((˙), (˙ ˙),) = ((˙), ˙ ˙,), (, ˙), where λ is a constant, then the Hamiltonian will be the total conserved energy, equal to the total kinetic and potential energies of the system: = + =.