✎ Formalisms [The Physics Travel Guide]

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====== Formalisms ======

There are four big formalisms that are used almost everywhere in modern physics:

* The [[formalisms:hamiltonian_formalism|Hamiltonian Framework]], where we describe the evolution of our system as a trajectory in [[basic_tools:phase_space|phase space]].
  * The [[formalisms:lagrangian_formalism|Lagrangian Framework]], where we describe the evolution of our system as a trajectory in [[basic_tools:configuration_space|configuration space]].
  * The Newtonian formalism where we describe the system in terms of trajectories in real space.
  * The Schrödinger formalism, where we describe the system in terms of abstract vectors living in [[basic_tools:hilbert_space|Hilbert space]].

Each formalism has strengths and weaknesses. Which one is better depends on the system we wish to describe.

----

**Comparision of the Lagrangian and Hamiltonian Formalism**((//Table adapted from Principles of Quantum Mechanics by R. Shankar//))

^ 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 [[theories|theory]] is formulated in terms of a specific formalism:

The connection between a Lagrangian and the corresponding Hamiltonian is given by the [[advanced_tools:legendre_transformation|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 [[advanced_tools:legendre_transformation|Legendre transformation]]. Moreover, we can also derive the Lagrangian from the corresponding Hamiltonian by making use of the Legendre transformation.

**To summarize:
**
<diagram>
| Hamiltonian: $ \frac{1}{2m}p^2 + \frac{k}{2}x^2$ |-@a| Lagrangian: $\int dt \left( \frac{m}{2} \left(\frac{dx}{dt}\right) - \frac{k}{2}x^2 \right)$|
| |!| | | | !| |
| |`@2| AAA|  '@8| |AAA=Newtonian: $m \frac{d^2}{dt^2} x =  -kx$
</diagram>

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