User Tools

Site Tools


formalisms

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
formalisms [2018/05/05 12:23]
jakobadmin ↷ Links adapted because of a move operation
formalisms [2020/04/02 20:08] (current)
184.147.122.3
Line 6: Line 6:
 There are four 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]], where we describe the evolution of our system as a trajectory in [[basic_tools:​phase_space|phase space]]. +  * 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]], where we describe the evolution of our system as a trajectory in [[basic_tools:​configuration_space|configuration space]]. +  * 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 real 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]]. ​   * 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. +Each formalism has strengths and weaknesses ​and which one is better depends on the system we wish to describe.
  
 ---- ----
Line 20: Line 19:
  
 ^ 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 |
Line 30: Line 29:
  
 ^                            | **Classical Mechanics **                                                                      | **Quantum Mechanics** ​                                                                      | ^                            | **Classical Mechanics **                                                                      | **Quantum Mechanics** ​                                                                      |
-| **Newtonian Formalism** ​   | [[theories:​classical_mechanics:​newtonian_mechanics|Newtonian Mechanics]] ​                     | [[theories:​quantum_mechanics:​bohmian_mechanics|Bohmian mechanics]] ​                         | +| **Newtonian Formalism** ​   | [[theories:​classical_mechanics:​newtonian|Newtonian Mechanics]] ​                     | [[theories:​quantum_mechanics:​bohmian|Bohmian mechanics]] ​                         | 
-| ** Lagrangian Formalism** ​ | [[theories:​classical_mechanics:​lagrangian_mechanics|Lagrangian mechanics]] ​                   | [[theories:​quantum_mechanics:​path_integral|Path Integral 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_mechanics|Hamiltonian Mechanics]] ​                 | [[theories:​quantum_mechanics:​phase_space|Phase space 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]] ​     | | **Schrödinger Formalism** ​ | [[theories:​classical_mechanics:​koopman_von_neumann_mechanics|Koopman-von-Neumann Mechanics]] ​ | [[theories:​quantum_mechanics:​canonical|Canonical quantum mechanics]] ​     |
  
Line 49: Line 48:
  
 $$ m \frac{d^2}{dt^2} x=-kx , $$ $$ m \frac{d^2}{dt^2} x=-kx , $$
-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.)
  
  
formalisms.1525515835.txt.gz · Last modified: 2018/05/05 10:23 (external edit)