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actual forces actual motion angular velocity axis Bertrand's Theorem calculus of variations center of gravity components configuration conservative forces constant constraints coordinates q cos2 cyclic coordinates degrees of freedom differential equations dt cqk dt dt dx dy dz effective forces electromotive force equal equations of motion expressed in terms expression Mq force function forming the Lagrangian geometrical equations given Hamilton's principle Hamiltonian equations Hamiltonian function hanging particle Hence homogeneous quadratic hypothetical motion ignored coordinates impressed forces impulsive forces initial motion integral kinetic energy Lagrangian equation liquid modified expression momenta momentum moving system moving under conservative number of degrees potential energy problem of Art qv q2 ratic rectangular coordinates remaining coordinates rigid body sin2 solve sphere starts from rest system starts Thomson's Theorem throughout the motion tonian tractrix velocities q virtual wedge Whence zero