Editing Phase Space
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If there is no force acting on the particle<m>F=0</m>, by Newton's law <m> F = dp/dt right dp/dt = 0</m> , it's linear momentum(<m>p</m>) will not change, and therefore, if we know the position(<m>r</m>) of the particle and it's momentum (<m>p</m>) at any point in time, we can determine the momentum and position of the particle at any point in the future. If we take the momentum as one coordinate, and the position as another, we have a 2-dimensional space. This space is what we call the '''Phase Space''' of the system. As we have just seen here, the Phase Space for mechanical system that can move in a ''single dimension'' is a ''two dimensional '' one. We have the position of the particle along the line, and it's momentum at that position, and this determines a unique point in the phase space<m> r * p </m>. This notion generalizes, and for any ''n-dimensional'' mechanical system, we have a ''2n-dimensional'' phase space. This turns out to be quite important. | If there is no force acting on the particle<m>F=0</m>, by Newton's law <m> F = dp/dt right dp/dt = 0</m> , it's linear momentum(<m>p</m>) will not change, and therefore, if we know the position(<m>r</m>) of the particle and it's momentum (<m>p</m>) at any point in time, we can determine the momentum and position of the particle at any point in the future. If we take the momentum as one coordinate, and the position as another, we have a 2-dimensional space. This space is what we call the '''Phase Space''' of the system. As we have just seen here, the Phase Space for mechanical system that can move in a ''single dimension'' is a ''two dimensional '' one. We have the position of the particle along the line, and it's momentum at that position, and this determines a unique point in the phase space<m> r * p </m>. This notion generalizes, and for any ''n-dimensional'' mechanical system, we have a ''2n-dimensional'' phase space. This turns out to be quite important. | ||
| - | Over the course of the semester, we have developed the idea of a | + | Over the course of the semester, we have developed the idea of a <m> Manifold </m>. Casually, a manifold can be viewed as an n-dimensional object which locally looks like < |
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==2 examples:== | ==2 examples:== | ||
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1) particle on line:2D plane w/ area form (pg.20) | 1) particle on line:2D plane w/ area form (pg.20) | ||
| - | 2) n particles on line: R^(2n) w/ symplectic form (pg. 39) | + | 2) n particles on line: R^(2n) w/ symplectic form (pg. 39) |
