Some words about the principle of least action

What is the most important object physicists manipulate everyday?

My answer with no hesitation is : the Lagrangian !

What is it? simply the object that contain everything one need to know about a given physical system.

As many should know, the Lagrangian is simply the kinetic energy minus the potential energy of the system (it's all about moving and interacting objects). Before that, there was the newtonian formulation of mechanics, a vectorial equation, meaning three scalars equations to solve. It is natural to wonder whether it's possible or not to summarize these equations into a single one (Lagrange showed that this was possible, though it's Hamilton who introduced the Lagrangian as it is now teached to every physics student). That's one approach to the problem, the modern one.

As usual, the historical approach is more cumbersome. My curiosity was aroused recently when I discovered that it was possible to read some of Maupertuis works in wikipedia :" Derivation of the laws of motion and equilibrium from a metaphysical principle" and " Accord between different laws of Nature that seemed incompatible". Maupertuis is known for having introduced the principle of least action in mechanics. In the first work, it is pretty clear that it was inspired by the work of Fermat on laws of optics. The latter has brilliantly demonstrated them by simply finding the path minimizing the time it takes to a ray to cover it. If my memory is good, the argument is clearly detailed in the second tome of Feynman lectures of physics. Anyway, there is a funny exercise that one can do to get the point. Imagine there's a girl drowning in the sea, a guy see her from the beach and want to reach her as fast as possible, but he doesn't swim as fast as he runs, therefore the straight line is not the fastest trajectory (assuming there are not aligned on the same line perpendicular to the waterfront).

Going back to Maupertuis, instead of time, he decided to consider the velocity, which mixes time and space. Of course, in the case of light, time and distance are related by a constant coefficient, explaining his success in deriving on his own the Snell relations. But his wish was clearly to introduce a general principle that could be used to deduce the trajectories of every mechanical system, mainly for metaphysical reasons, exchanging the newtonian approach, relating the causes (the applied forces) to the consequences (the acceleration), with a new one : the final cause, which assumes roughly that nature will tend to the simplest possible dynamics, which, rephrased in a modern way implies the minimization of a given quantity.

However it is not him but Lagrange who succeeded in that task, thanks to the mathematical apparatus developped by Euler. The latter has been inspired by the Maupertuis work, but also by the work of his close friends, the Bernoulli brothers (who were the two sons of Euler's teacher) on the equilibrium shape of different objects (see the catenary and the brachistrone). It is this static approach which led them to consider the potential energy as the quantity to work with and to minimize.

For interested people, it is easy to find the book of Lagrange on the internet (in french !). And for even more interested people, I recommand the reading of the first chapter of the first tome of Landau and Lifschitz lectures, about mechanics. There is a rigorous derivation of the Lagrangian by considering galilean relativity principle.

My answer with no hesitation is : the Lagrangian !

What is it? simply the object that contain everything one need to know about a given physical system.

As many should know, the Lagrangian is simply the kinetic energy minus the potential energy of the system (it's all about moving and interacting objects). Before that, there was the newtonian formulation of mechanics, a vectorial equation, meaning three scalars equations to solve. It is natural to wonder whether it's possible or not to summarize these equations into a single one (Lagrange showed that this was possible, though it's Hamilton who introduced the Lagrangian as it is now teached to every physics student). That's one approach to the problem, the modern one.

As usual, the historical approach is more cumbersome. My curiosity was aroused recently when I discovered that it was possible to read some of Maupertuis works in wikipedia :" Derivation of the laws of motion and equilibrium from a metaphysical principle" and " Accord between different laws of Nature that seemed incompatible". Maupertuis is known for having introduced the principle of least action in mechanics. In the first work, it is pretty clear that it was inspired by the work of Fermat on laws of optics. The latter has brilliantly demonstrated them by simply finding the path minimizing the time it takes to a ray to cover it. If my memory is good, the argument is clearly detailed in the second tome of Feynman lectures of physics. Anyway, there is a funny exercise that one can do to get the point. Imagine there's a girl drowning in the sea, a guy see her from the beach and want to reach her as fast as possible, but he doesn't swim as fast as he runs, therefore the straight line is not the fastest trajectory (assuming there are not aligned on the same line perpendicular to the waterfront).

Going back to Maupertuis, instead of time, he decided to consider the velocity, which mixes time and space. Of course, in the case of light, time and distance are related by a constant coefficient, explaining his success in deriving on his own the Snell relations. But his wish was clearly to introduce a general principle that could be used to deduce the trajectories of every mechanical system, mainly for metaphysical reasons, exchanging the newtonian approach, relating the causes (the applied forces) to the consequences (the acceleration), with a new one : the final cause, which assumes roughly that nature will tend to the simplest possible dynamics, which, rephrased in a modern way implies the minimization of a given quantity.

However it is not him but Lagrange who succeeded in that task, thanks to the mathematical apparatus developped by Euler. The latter has been inspired by the Maupertuis work, but also by the work of his close friends, the Bernoulli brothers (who were the two sons of Euler's teacher) on the equilibrium shape of different objects (see the catenary and the brachistrone). It is this static approach which led them to consider the potential energy as the quantity to work with and to minimize.

For interested people, it is easy to find the book of Lagrange on the internet (in french !). And for even more interested people, I recommand the reading of the first chapter of the first tome of Landau and Lifschitz lectures, about mechanics. There is a rigorous derivation of the Lagrangian by considering galilean relativity principle.

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