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There are two big formalisms that are used almost everywhere in modern physics:
The following table lists the names of the approaches where a given theory is formulated in terms of a specific formalism:
Classical Mechanics | Quantum Mechanics | Quantum Field Theory | |
Lagrangian Framework | Lagrangian mechanics | Path Integral Quantum Mechanics | Path Integral Quantum Field Theory |
Hamiltonian Framework | Hamiltonian Mechanics | Canonical Quantum Mechanics | Canonical Quantum Field Theory |
The connection between a Lagrangian and the corresponding Hamiltonian is given by the Legendre transformation. This can be understood with the help of Noether's theorem.
In quantum mechanics one usually uses the Hamiltonian framework and in quantum field theory the Lagrangian framework. However, again it is also possible to describe all physics equally with the other frameworks.
Let's consider an object attached to a mechanical spring.
In Newtonian mechanics, the movement of the object is described by the equation
$$ m \frac{d^2}{dt^2} x=-kx , $$ wher $x$ denotes the position of the object and $k$ the spring constant that characterises the mechanical spring. (This is known as Hooke's law.)
The corresponding Hamiltonian, that is used in the Hamiltonian framework to characterise the system is
$$ H = \frac{1}{2m}p^2 + \frac{k}{2}x^2 \, .$$
Starting from this Hamiltonian one can derive the equation of motion $m \frac{d^2}{dt^2} x=-kx$ that is used in the Newtonian framework. The Hamiltonian represents the total energy of the object. This means $H= T+V$ where $T=\frac{1}{2m}p^2 $ is the kinetic energy and $V= \frac{k}{2}x^2$ is the potential energy due to the compression and elongation of the spring.
In the Lagrangian framework the same system is characterised by the action
$$ S = \int dt \left( \frac{m}{2} \left(\frac{dx}{dt}\right) - \frac{k}{2}x^2 \right) ,$$ where $L= \frac{m}{2} \left(\frac{dx}{dt}\right) - \frac{k}{2}x^2$ is called the Lagrangian.
Starting from this action, one can derive the equation of motion $m \frac{d^2}{dt^2} x=-kx$ which is used in the Newtonian framework. Moreover, starting from this Lagrangian we can derive the corresponding Hamiltonian through a Legendre transformation. Moreover, we can also derive the Lagrangian from the corresponding Hamiltonian by making use of the Legendre transformation.
To summarize:
Hamiltonian: $ \frac{1}{2m}p^2 + \frac{k}{2}x^2$ | Lagrangian: $\int dt \left( \frac{m}{2} \left(\frac{dx}{dt}\right) - \frac{k}{2}x^2 \right)$ | |||||||||||||
Newtonian: $m \frac{d^2}{dt^2} x = -kx$ | ||||||||||||||
by Jakob Schwichenberg