formalisms
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formalisms [2018/05/04 16:56] jakobadmin [Comparision of the Lagrangian and Hamiltonian Formalism] |
formalisms [2020/04/02 20:08] (current) 184.147.122.3 |
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| - | There are two big formalisms that are used almost everywhere in modern physics: | + | There are four big formalisms that are used almost everywhere in modern physics: |
| - | * The [[formalisms:hamiltonian_formalism|Hamiltonian Framework]] | + | * The [[formalisms:hamiltonian_formalism|Hamiltonian formalism]], where we describe the evolution of our system as a trajectory in [[basic_tools:phase_space|phase space]]. |
| - | * The [[formalisms:lagrangian_formalism|Lagrangian Framework]] | + | * The [[formalisms:lagrangian_formalism|Lagrangian formalism]], 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 everyday space. | ||
| + | * The Schrödinger formalism, where we describe the system in terms of abstract vectors living in [[basic_tools:hilbert_space|Hilbert space]]. | ||
| - | Both have strength and weaknesses and which one is better depends on the system we wish to describe. | + | Each formalism has strengths 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 ===== | + | **Comparision of the Lagrangian and Hamiltonian Formalism**((//Table adapted from Principles of Quantum Mechanics by R. Shankar//)) |
| ^ Lagrangian formalism ^ Hamiltonian formalism ^ | ^ 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 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 of 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 $(q_1,\ldots, q_n; p_1,\ldots , p_n)$ | | | 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 | | | 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.| | | 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.| | ||
| - | //Table adapted from Principles of Quantum Mechanics by R. Shankar// | + | |
| ===== The Formalisms in Practice===== | ===== 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 following table lists the names of the approaches where a given [[theories|theory]] is formulated in terms of a specific formalism: | ||
| - | ^ | **Classical Mechanics ** | **Quantum Mechanics** | | + | ^ | **Classical Mechanics ** | **Quantum Mechanics** | |
| - | | ** Lagrangian Formalism** | [[theories:classical_mechanics:lagrangian_mechanics|Lagrangian mechanics]] | [[theories:quantum_mechanics:path_integral|Path Integral]] Quantum Mechanics | | + | | **Newtonian Formalism** | [[theories:classical_mechanics:newtonian|Newtonian Mechanics]] | [[theories:quantum_mechanics:bohmian|Bohmian mechanics]] | |
| - | | **Hamiltonian Formalism** | [[theories:classical_mechanics:hamiltonian_mechanics|Hamiltonian Mechanics]] | [[theories:quantum_mechanics:phase_space_quantum_mechanics|Phase space quantum mechanics]] | | + | | ** Lagrangian Formalism** | [[theories:classical_mechanics:lagrangian|Lagrangian mechanics]] | [[theories:quantum_mechanics:path_integral|Path Integral Quantum Mechanics]] | |
| + | | **Hamiltonian Formalism** | [[theories:classical_mechanics:hamiltonian|Hamiltonian Mechanics]] | [[theories:quantum_mechanics:phase_space|Phase space quantum mechanics]] | | ||
| + | | **Schrödinger Formalism** | [[theories:classical_mechanics:koopman_von_neumann_mechanics|Koopman-von-Neumann Mechanics]] | [[theories:quantum_mechanics:canonical|Canonical quantum mechanics]] | | ||
| The connection between a Lagrangian and the corresponding Hamiltonian is given by the [[advanced_tools:legendre_transformation|Legendre transformation]]. | The connection between a Lagrangian and the corresponding Hamiltonian is given by the [[advanced_tools:legendre_transformation|Legendre transformation]]. | ||
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| $$ m \frac{d^2}{dt^2} x=-kx , $$ | $$ m \frac{d^2}{dt^2} x=-kx , $$ | ||
| - | 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.) | + | where $x$ denotes the position of the object and $k$ the spring constant that characterises the mechanical spring. (This is known as Hooke's law.) |
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