General mechanical engineering - General dynamics. Analytical form

Analytical mechanics, whose central body is made up of Lagrange's equations, is a tool that covers the same field of application as general theorems. It's a question of finding the motion caused by a mechanical action or, as a reciprocal problem, of finding the mechanical actions, the motion being known. Formally, there's a big difference: the formulation of analytical mechanics is essentially scalar, whereas that of general theorems is vectorial. Kinetics is based on kinetic energy, while mechanical actions are expressed in terms of virtual power. In addition, equation setting is automatic, whereas with general theorems, you have to select both the systems to which you apply them and the most appropriate theorems to avoid a series of useless equations. With Lagrange or Hamilton, the system considered is always the global system.

So why have two tools to solve the same problem? The choice is based on a simple criterion: economy of thought. For one problem, Lagrange provides a remarkably simple formulation. For another, the general theorems allow direct control and step-by-step verification of the equation, whereas Lagrange's equations appear as a black box with input-output, and you have to trust the calculations blindly. On the other hand, the equations are particularly useful when mechanical actions can be represented by a priori known functions (potential function, dissipation function).

Hamilton's equations deduced from Lagrange's equations have the advantage of directly providing the canonical system (first-order differential equations) and allow a highly unified writing of optimization problems.

Lagrange's and Hamilton's equations, which are not just used in engineering mechanics, are part of our basic knowledge.