Ah, sorry. Legendre transformation. That was what I meant. But doesn't it always exist?Orodruin said:Can you be more specific and give a specific example?
In general, you can get to the Hamiltonian formulation of the same theory by applying the Legendre transform (which is what I assume that you mean) to your Lagrangian. In order for this to work, the Legendre transform needs to exist, which in turn requires the Lagrangian to be a convex function of the time derivative of your generalised coordinates.
No. See the wikipedia page.Narasoma said:Ah, sorry. Legendre transformation. That was what I meant. But doesn't it always exist?
The Hamiltonian of a physical theory is a mathematical function that describes the total energy of a system. It is derived from the Lagrangian, which is the mathematical function that describes the dynamics of the system.
The Lagrangian and the Hamiltonian are two different mathematical functions that describe different aspects of a physical system. The Lagrangian describes the dynamics of the system in terms of its position and velocity, while the Hamiltonian describes the total energy of the system.
The Hamiltonian is derived from the Lagrangian through a mathematical process called Legendre transformation. This transformation allows us to express the dynamics of a system in terms of its energy instead of its position and velocity.
No, the Hamiltonian and Lagrangian cannot be used interchangeably. They describe different aspects of a physical system and have different mathematical forms. However, they are closely related and can be used together to fully describe the dynamics of a system.
The Hamiltonian is important in physics because it allows us to understand and predict the behavior of physical systems. It is a fundamental concept in classical mechanics and is also used in other branches of physics such as quantum mechanics and statistical mechanics.