formalisms:lagrangian_formalism
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formalisms:lagrangian_formalism [2018/12/30 00:49] thomas_abshier [Concrete] typos |
formalisms:lagrangian_formalism [2020/04/02 20:19] 184.147.122.3 |
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| Thus, the principle of minimal action does not simply mean that potential energy gets maximized, but that we have a trade off. Nature tries to make the potential energy as large as possible, while keeping the kinetic energy at a reasonable value. This explains why the ball almost stops at the top, and is the fastest close to the ground. | Thus, the principle of minimal action does not simply mean that potential energy gets maximized, but that we have a trade off. Nature tries to make the potential energy as large as possible, while keeping the kinetic energy at a reasonable value. This explains why the ball almost stops at the top, and is the fastest close to the ground. | ||
| - | Now with all this in mind, let's revisit the example discussed in the "Laymen" section. | + | Now with all this in mind, let's revisit the example discussed in the "Intuitive" section. |
| ** Light Again ** | ** Light Again ** | ||
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| In the Lagrangian approach we focus on the position and velocity of a particle, and | In the Lagrangian approach we focus on the position and velocity of a particle, and | ||
| - | compute what the particle does starting from the Lagrangian $L(q, q˙)$, which is a function | + | compute what the particle does starting from the Lagrangian $L(q, \dot{q})$, which is a function |
| - | $$ L: TQ \to \mathbb{R} $$ | + | $$ L\colon TQ \to \mathbb{R} $$ |
| where the tangent bundle is the space of position-velocity pairs. But we're led to consider momentum | where the tangent bundle is the space of position-velocity pairs. But we're led to consider momentum | ||
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| * See: http://astro1.panet.utoledo.edu/~ljc/bader1.html | * See: http://astro1.panet.utoledo.edu/~ljc/bader1.html | ||
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| </tabbox> | </tabbox> | ||
formalisms/lagrangian_formalism.txt · Last modified: 2023/03/08 17:35 by 62.4.55.178
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