formalisms
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formalisms [2018/05/06 09:20] jakobadmin |
formalisms [2020/04/02 20:08] (current) 184.147.122.3 |
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| * 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. |
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| ^ 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 | | ||
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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.) |
formalisms.1525591246.txt.gz · Last modified: 2018/05/06 07:20 (external edit)
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