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formalisms [2018/05/06 09:23]
jakobadmin
formalisms [2020/04/02 20:08] (current)
184.147.122.3
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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.1525591397.txt.gz · Last modified: 2018/05/06 07:23 (external edit) · Currently locked by: 162.158.167.86,216.244.66.248