https://www.superphysics.org/research/lagrange/analytical/part-1/ Recent content in Analytical Mechanics on Superphysics Hugo en Wed, 25 Mar 2026 00:00:00 +0000 https://www.superphysics.org/research/lagrange/analytical/part-1/preface/ Wed, 25 Mar 2026 00:00:00 +0000 https://www.superphysics.org/research/lagrange/analytical/part-1/preface/ <p>We already have several Treatises on Mechanics.</p> <p>But this one is entirely new.</p> <p>I have proposed to reduce the theory of this Science, and the art of solving the problems related to it, to general formulas that give all the equations necessary for the solution of each problem.</p> https://www.superphysics.org/research/lagrange/analytical/part-1/section-01/ Wed, 18 Mar 2026 00:00:00 +0000 https://www.superphysics.org/research/lagrange/analytical/part-1/section-01/ <p>Statics is the science of the equilibrium of forces.</p> <p>By force or power is the cause which impresses or tends to impress motion on the body to which it is supposed to be applied.</p> https://www.superphysics.org/research/lagrange/analytical/part-1/section-01b/ Wed, 18 Mar 2026 00:00:00 +0000 https://www.superphysics.org/research/lagrange/analytical/part-1/section-01b/ <ol start="5"> <li>The ratio of power to weight on an inclined plane was for a long time a problem among modern mechanicians.</li> </ol> <p>Stevin was the first to solve it.</p> <p>But his solution is founded on an indirect consideration independent of the theory of the lever.</p> https://www.superphysics.org/research/lagrange/analytical/part-1/section-01c/ Wed, 18 Mar 2026 00:00:00 +0000 https://www.superphysics.org/research/lagrange/analytical/part-1/section-01c/ <ol start="9"> <li>The second fundamental principle of Statics is that of the composition of forces.</li> </ol> <p>If 2 forces act at the same time on a body according to different directions, these forces are then equivalent to a single force, capable of imparting to the body the same motion as would be given by the two forces acting separately.</p> https://www.superphysics.org/research/lagrange/analytical/part-1/section-01d/ Wed, 18 Mar 2026 00:00:00 +0000 https://www.superphysics.org/research/lagrange/analytical/part-1/section-01d/ <ol start="16"> <li>The third principle is that of virtual velocities.</li> </ol> <p>Virtual velocity is the velocity that a body in equilibrium is disposed to receive in case the equilibrium comes to be broken, i.e. the velocity that this body would really take in the first instant of its motion; and the principle in question consists in this: that powers are in equilibrium when they are in inverse ratio to their virtual velocities, estimated according to the directions of these powers.</p> https://www.superphysics.org/research/lagrange/analytical/part-1/section-02/ Wed, 18 Mar 2026 00:00:00 +0000 https://www.superphysics.org/research/lagrange/analytical/part-1/section-02/ <ol> <li>The general law of equilibrium in machines is that forces or powers are reciprocally proportional to the velocities of the points where they are applied, estimated according to the direction of these powers:</li> </ol> <p>This law constitutes what is commonly called the principle of virtual velocities, a principle long recognized as the fundamental principle of equilibrium, as we have shown in the previous Section, and which can, consequently, be regarded as a kind of axiom of Mechanics.</p>