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There are four big formalisms that are used almost everywhere in modern physics:
Each formalism has strengths and weaknesses and which one is better depends on the system we wish to describe.
Comparision of the Lagrangian and Hamiltonian Formalism^{1)}
Lagrangian formalism | Hamiltonian formalism |
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We describe the state of a system with $n$ degrees of freedom with the $n$ coordinates $(q_1,\ldots, q_n)$ and the $n$ velocities $(\dot{q}_1,\ldots , \dot{q}_n)$ | We describe the state the a system with $n$ degrees of freedom by the $n$ coordinates $(q_1,\ldots, q_n)$ and the $n$ momenta $(p_1,\ldots , p_n)$ |
We represent the state of the system by a point moving with a definite velocity in an $n$-dimensional configuration space | We represent the state of the system by a point moving with a definite velocity in an $2n$-dimensional phase space with coordinates $(q_1,\ldots, q_n; p_1,\ldots , p_n)$ |
The $n$ configuration space coordinates evolve according to $n$ second-order equations | The $2n$ phase space coordinates evolve according to $2n$ first-order equations |
For a given Lagrangian $\mathcal{L}$ different trajectories can pass through the same given point in our configuration space, depending on $\dot q$. | For a given Hamiltonian $\mathcal{H}$ only one trajectory passes through a given point in phase space. |
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 | |
Newtonian Formalism | Newtonian Mechanics | Bohmian mechanics |
Lagrangian Formalism | Lagrangian mechanics | Path Integral Quantum Mechanics |
Hamiltonian Formalism | Hamiltonian Mechanics | Phase space quantum mechanics |
Schrödinger Formalism | Koopman-von-Neumann Mechanics | Canonical quantum mechanics |
The connection between a Lagrangian and the corresponding Hamiltonian is given by the Legendre transformation.
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: $m \frac{d^2}{dt^2} x = -kx$ | ||||||||||||||
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