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Scalar angular momentum from Lagrangian mechanics

Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate ϕ {\displaystyle \phi } expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass m {\displaystyle m} constrained to move in a circle of radius r {\displaystyle r} in the absence of any external force field. The kinetic energy of the system is T = 1 2 m r 2 ω 2 = 1 2 m r 2 ϕ ˙ 2. {\displaystyle T={\tfrac {1}{2}}mr^{2}\omega ^{2}={\tfrac {1}{2}}mr^{2}{\dot {\phi }}^{2}.}