/sci/ - Science & Math

>>7586380
>>7586549

The Lagrangian mechanics formalism is all about finding a Lagrangian function <span class="math"> L(q,\dot{q},t) [/spoiler] of some generalized coordinates q and velocities <span class="math"> \dot{q} [/spoiler] that describes the physical system you're interested in. The main idea considers the overall action S of your physical system, which is given by a time-integral over the Lagrangian:
<div class="math"> S = \int_{t_1}^{t_2} L(q(t),\dot{q}(t),t)\ dt . </div>
Generally, out of every potential path q(t) that *could* describe the time evolution of your system, you want to find that *particular* q(t) which minimizes the overall action S. It turns out that the actual path that Nature chooses for physical systems to follow is generally the one that minimizes this action. This is sometimes called the Principle of Least Action for unknown reasons. You use techniques from the calculus of variations to find the minimizing path q(t). Actually, you usually end up finding the Euler-Lagrange equations of motion, which frequently look like this
<div class="math"> \frac{\partial L}{\partial q} - \frac{d}{dt} \Big( \frac{\partial L}{\partial \dot{q}}\Big) = 0 . </div>
So you're often back to solving differential equations, as in the Newtonian framework.

However, the Lagrangian viewpoint is extremely helpful and deep, especially if there are symmetries in your system. There is an important theorem by Emmy Noether that helps you to easily enumerate all of the conserved quantities from the symmetries in your Lagrangian. For example, conservation of energy, momentum, angular momentum, charge, etc are all explained in this formulation in terms of underlying symmetries in the physics of the system or the universe.

>>7414880
If you don't know this by heart you're a popsci faggot.

>>6310495
This, for reference, is the equations of the whole Standard Model, sans gravity.

It's very likely that the real laws of physics are considerably simpler - the Standard Model has many explicit, pre-defined parameters, which are quite inelegant, and any attempt to fit it into gravity is a kludge that doesn't work too well.

The real laws of physics are probably quite simple.

Lagrangian is minimized.

Or quantum variant: System is most probable around the minimal Lagrangian solution.

Ur being lazy

file was removed :(

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