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formalisms
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Table of Contents
Formalisms
There are two big formalisms that are used almost everywhere in modern physics:
Both have strength and weaknesses and which one is better depends on the system we wish to describe.
In addition to these formalisms, we also can use the Newtonian formalism where we describe the system in terms of trajectories in real space. A fourth possibility is the Schrödinger formalism, where we describe the system in terms of states living in Hilbert space.
===== Comparision of the Lagrangian and Hamiltonian Formalism =====1)
| Lagrangian formalism | Hamiltonian formalism |
|---|---|
| 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 Formalisms in Practice
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 | |
| Lagrangian Formalism | Lagrangian mechanics | Path Integral Quantum Mechanics |
| Hamiltonian Formalism | Hamiltonian Mechanics | Phase space quantum mechanics |
The connection between a Lagrangian and the corresponding Hamiltonian is given by the Legendre transformation.
A concrete example
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: $m \frac{d^2}{dt^2} x = -kx$ | ||||||||||||||
1)
Table adapted from Principles of Quantum Mechanics by R. Shankar
formalisms.1525445832.txt.gz · Last modified: 2018/05/04 14:57 (external edit)
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