Principle of least action used to predict particle movement?

In summary: Principle_of_least_action) for more info.In summary, the principle of least action is a mathematical technique used in classical mechanics to formulate theories and solve dynamics problems. It involves minimizing the action of a system as it evolves from an initial state to a final state. This technique can also be applied in quantum theories.
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epislon58
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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?
 
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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.
 
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Thanks for the help.
 
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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
 

1. What is the principle of least action?

The principle of least action is a fundamental principle in physics which states that the path taken by a moving particle will be the one that minimizes the action, or the integral of the Lagrangian over time. In other words, a particle will follow a path that minimizes the difference between its kinetic and potential energy.

2. How is the principle of least action used to predict particle movement?

The principle of least action is used to predict particle movement by formulating an equation called the Euler-Lagrange equation, which describes the particle's motion. This equation takes into account the Lagrangian, which is a function that describes the particle's potential and kinetic energy.

3. What is the Lagrangian and how is it related to the principle of least action?

The Lagrangian is a function that describes the potential and kinetic energy of a particle. It is related to the principle of least action because the path taken by a particle will be the one that minimizes the difference between its kinetic and potential energy, which is described by the Lagrangian.

4. Can the principle of least action be applied to all types of particles?

Yes, the principle of least action can be applied to all types of particles, including classical particles and quantum particles. It is a fundamental principle in physics and is used in various fields such as mechanics, electromagnetism, and quantum mechanics.

5. What are some real-life applications of the principle of least action?

The principle of least action has many real-life applications, such as in the prediction of the motion of planets and satellites, the behavior of particles in quantum mechanics, and in the optimization of mechanical systems. It is also used in the development of control systems and in the study of optimization and decision-making processes.

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