An Etymological Dictionary of Astronomy and Astrophysics
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Made general. → generalized coordinates; → generalized velocities.

P.p. of → generalize

In a material system, the independent parameters which completely specify the configuration of the system, i.e. the position of its particles with respect to the frame of reference. Usually each coordinate is designated by the letter q with a numerical subscript. A set of generalized coordinates would be written as q₁, q₂, ..., q_n. Thus a particle moving in a plane may be described by two coordinates q₁, q₂, which may in special cases be the → Cartesian coordinates x, y, or the → polar coordinates r, θ, or any other suitable pair of coordinates. A particle moving in a space is located by three coordinates, which may be Cartesian coordinates x, y, z, or → spherical coordinates r, θ, φ, or in general q₁, q₂, q₃. The generalized coordinates are normally a "minimal set" of coordinates. For example, in Cartesian coordinates the simple pendulum requires two coordinates (x and y), but in polar coordinates only one coordinate (θ) is required. So θ is the appropriate generalized coordinate for the pendulum problem.

→ generalized; → coordinate.

In → Lagrangian dynamics, forces related to → generalized coordinates. For any system with n generalized coordinates q_i (i = 1, ..., n), generalized forces are expressed by F_i = ∂L/∂q_i, where L is the → Lagrangian function.

→ generalized; → force.

In → Lagrangian dynamics, momenta related to → generalized coordinates. For any system with n generalized coordinates q_i (i = 1, ..., n), generalized momenta are expressed by p_i = ∂L/∂q^._i, where L is the → Lagrangian function.

→ generalized; → momentum.

The time → derivatives of the → generalized coordinates of a system.

→ generalized; → velocity.