Lagrange Function and Energy Conservation

Extract

[...] Taking y as the generalised coordinate, we must express the positions of the masses in terms of y. On the x-axis, the mass m1 is located at the position While on the y-axis, the mass m2 is located at the position . It is known that at any instant:" so and the speed of the mass m1 is . On the other hand, the speed of mass m2 is therefore Therefore, the total kinetic energy of the system is: The potential energy of gravity of the system is: Thus the Lagrange function is: 3. [...]

[...] The total energy of the system is: As then : The total energy of the system is constant. Figure 1 : Phase portrait in the space with l = 1 in blue the trajectories for small E and in red for large E Code MatLAB for plotting (solution by Euler's method): Figure 2 : Angular velocity at an instant t as a function of the value of phi with l = 1 m Code MatLAB for plotting (integration by the rectangle method): Sources: https://python-prepa.github.io/systemes_dynamiques.html http://res-nlp.univ-lemans.fr/NLP_E_M01_G03_01/co/NLP_E_M01_G03_01.html Rax, Jean-Marcel. Analytical Mechanics: Adiabaticity, Resonances, Chaos. Malakoff: Dunod Print. [...]