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Well, I studied that a physical law is an equation between tensors that are function of position and time because when the frame is changed tensors change in order to leave the equation true: $$T_1(\vec x, t)=T_2(\vec x, t)$$

But there are some equations like Lagrange's equations that apparently aren't physical laws by this definition: $$\frac d {dt} \frac {\partial L} {\partial \dot q}-\frac {\partial L} {\partial q} =0 $$

Does it mean that Lagrange's equation are just mathematical stuff or do I miss something?

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    $\begingroup$ "a physical law is an equation between tensors that are function of position and time" - where does this definition come from? $\endgroup$ – probably_someone Jul 17 at 12:49
  • $\begingroup$ My professor told me, and he said it's because tensors change in order to maintain the equation true for every frame $\endgroup$ – SimoBartz Jul 17 at 12:51
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    $\begingroup$ It seems like your professor is using the principle of covariance (en.wikipedia.org/wiki/Principle_of_covariance) to determine whether something is a physical law or not. This is one possible definition of many. Also, how do you know that the Euler-Lagrange equation doesn't follow this format? Remember that scalars, vectors, and 1-forms are types of tensors as well. $\endgroup$ – probably_someone Jul 17 at 12:55
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    $\begingroup$ The EL equations are covariant under general coordinate transformations, cf. this Phys.SE post, if that's what you're asking. $\endgroup$ – Qmechanic Jul 17 at 13:15
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    $\begingroup$ Your professor is incorrect. Physics is defined by position, time, and velocity - according to the Lagrange equations or equivalently to the Least Action Principle, which is the primary physical law valid in every branch of physics, including Classical Mechanics, Quantum Mechanics, and General Relativity. $\endgroup$ – safesphere Jul 17 at 15:10
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This is one of those slightly ambiguous things that people may start arguing about without reaching any definite conclusions, so I may be opening a Pandora’s box here. But for what it’s worth: no, the Euler-Lagrange equation is not a physical law. I have never seen it referred to as a physical law (explicitly or in an implied way) in any textbook or other authoritative source, and this is consistent with my own understanding of what a physical law is.

As a side remark, your own definition of a physical law is wrong. Physical laws are simply statements that describe some aspect of how the universe behaves and are found through experiment to be correct (within some possibly limited regime of parameters or circumstances). Examples include Newton’s laws of motion, Schrodinger’s equation, the ideal gas law, Coulomb’s law, Maxwell’s equations, etc. Physical laws need not necessarily mention time, position, or involve tensors. But they have to say something specific about the physical universe! For example, $2+2=4$ is not a physical law, and neither is the formula $$ (fg)’ = f’g+fg’ $$ for the derivative of a product of two functions.

As for the Euler-Lagrange equation, what it is is a general mathematical device that enables us to encode a qualitative statement about a function $q(x,t)$ - the fact that it is a stationary element for a least action principle - in an explicit way as a differential relationship satisfied by the function. It is of course very important in physics, since it turns out that many physical laws can in fact be formulated as least action principles, but note that the Euler-Lagrange equation holds true for action functional associated with arbitrary Lagrangians, including those that have no physical meaning. So, at the end of the day the EL equation has no more claim to be called a physical law than the chain rule, the fundamental theorem of calculus, or other general mathematical results of this sort that get used all the time in math and physics.

Perhaps the most that one can say is that there are specific Lagrangians such that when one writes down the EL equation for that Lagrangian one gets a differential equation that is some well-known physical law - that I can certainly agree with. But to say that the EL equation itself is a physical law would be quite misleading, and, well, simply not true.

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Yes, Lagrangian equations are physical laws. The Lagrangian formulation is the same mathematically as the Hamiltonian formulation (the kind most people are more familiar with), there's just been a few tricks (the calculus of variations) to shift the work of finding solutions from a zero of change of energy to minimizing an integral.

Besides, arrange the generic Lagrangian and get $$ \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}}} = \frac{\partial{L}}{\partial{q}}$$ which as $q$s are a set of arbitrary coordinates in configurations space, which is a mapping from position and time, it's really a tensorial law after all.

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