# Principle of least action used to predict particle movement?

1. Aug 21, 2013

### epislon58

I am having a tough time understanding the principle of least action and I would really appreciate it if someone would clear it up for me. And from what I understand from it, it can be used to predict particle motion?

Last edited: Aug 22, 2013
2. Aug 22, 2013

### Jolb

The principle of least action is not really that intuitive. It is highly useful in physics because it allows one to formulate theories from scratch pretty readily, but it is more mathematically abstract than, say, a newtonian picture of forces and accelerations. But the two pictures are equivalent and certain problems are only possible to solve using the "least action" formulation of Newtonian mechanics--the famous example is the Brachistochrone problem: http://en.wikipedia.org/wiki/Brachistochrone_problem . Furthermore, theories that go beyond Newtonian mechanics, like quantum field theory, are formulated using the "least action" formulation. In classical mechanics you run into two (equivalent) versions of the "least action" formulation: the Lagrangian formulation and the Hamiltonian formulation.

Let me attempt to explain the principle of least action and the technique of using it to derive the Newtonian equations of motion. First, the definition of action is S = ∫ T-V dt, where T is the kinetic energy of the system and V is the potential energy. We call the quantity T-V=:L, the Lagrangian.

The principle of least action says that if a system evolves from a given initial state to a given final state, the trajectory it will take will be the one which minimizes (or in some cases maximizes) the value of S. In classical mechanics, one can show that this reproduces all the Newtonian equations of motion if you "vary" the integral using a mathematical technique called calculus of variations. Calculus of variations basically says--"let me determine a path for the particle to take between the initial state to the final state such that any small deviation from this path increases S". If you formulate this all correctly you get a description equivalent to Newton's laws. A similar but somewhat more sophisticated procedure applies to quantum theories as well.

3. Aug 22, 2013

### epislon58

Thanks for the help.

4. Aug 23, 2013

### chris gray

The Principle of Least Action can indeed be used in place of the equations of motion to solve dynamics problems in classical (and quantum) mechanics. See one of the sections in "Principle of Least Action" on Scholarpedia