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Wednesday, March 5, 2014

axioms of Lagrangian mechanics

imagine you'd want to formulate mechanics based on an extremal principle. what would you need to require such that all three Newton-axioms are fulfilled?

First, you would use the principle of least action for $S = \int (T - V) dt$ as usual in classical mechanics. This implies Newton's second law, i.e. $\vec F = m \vec a$. Newton's first law is already there since for $\vec F = 0$, we get $\vec a = 0$ and thus $\vec v = \text{const}$. As you know, Newton's first law is not really a dyncamical law but rather a definition of inertial frames.

However, you don't automatically get Newton's third law from this. One way to get it is under the restriction of a system of $n$ point masses and with a potential of the type $V(\vec r_1, \dots, \vec r_n) = \sum V(|\vec r_i - \vec r_j)$.

CQW would like to add a remark, that keeps puzzling one of the editors for a few years now and concerns Newton's first law... well, using mechanical similarity for motion in a potential of the form $\Phi(r)\propto r^n$ the time scale is related to the distance scale $t^2\propto r^{2-n}$, implying that $t\propto r$ and therefore $r/t=\upsilon = \mathrm{const}$ for a constant potential with $n=0$, i.e. the first law. for this to hold one does *not* need an equation of motion, only the relation that the potential energy scales $T\propto \upsilon^2$ and that in fact can be derived without an equation of motion either.

using Galilean invariance and considering the transformation of the kinetic energy $T(\upsilon^2)$ in a frame with relative velocity $\epsilon$ gives $T((\upsilon+\epsilon)^2) = T(\upsilon^2+2\epsilon\upsilon + \epsilon^2) \simeq T(\upsilon^2) + 2\epsilon\upsilon\partial T/\partial\upsilon^2$. for the second term not to matter it must be a total time derivative (that's the $\upsilon$-term), such that $T$ must be propto $\upsilon^2$ and the derivative $\partial T/\partial\upsilon^2$ is constant. for that reason mechanical similarity $t^2\propto r^{2-n}$ can be established with the idea that a Galilean-covariant Lagrange-formalism is valid, but without an actual equation of motion. in this sense, the law of inertia follows from the Galilean relativity.

and there's a second detail that keeps me awestruck: there's already a glimpse of general relativity in Newton's first law, because in the force-free case a body follows a *straight* line with uniform velocity, so one could almost think that Newton had an idea of geodesic motion.

First, you would use the principle of least action for $S = \int (T - V) dt$ as usual in classical mechanics. This implies Newton's second law, i.e. $\vec F = m \vec a$. Newton's first law is already there since for $\vec F = 0$, we get $\vec a = 0$ and thus $\vec v = \text{const}$. As you know, Newton's first law is not really a dyncamical law but rather a definition of inertial frames.

ReplyDeleteHowever, you don't automatically get Newton's third law from this. One way to get it is under the restriction of a system of $n$ point masses and with a potential of the type $V(\vec r_1, \dots, \vec r_n) = \sum V(|\vec r_i - \vec r_j)$.

thank you very much, Anonymous, for your answer!

DeleteCQW would like to add a remark, that keeps puzzling one of the editors for a few years now and concerns Newton's first law... well, using mechanical similarity for motion in a potential of the form $\Phi(r)\propto r^n$ the time scale is related to the distance scale $t^2\propto r^{2-n}$, implying that $t\propto r$ and therefore $r/t=\upsilon = \mathrm{const}$ for a constant potential with $n=0$, i.e. the first law. for this to hold one does *not* need an equation of motion, only the relation that the potential energy scales $T\propto \upsilon^2$ and that in fact can be derived without an equation of motion either.

ReplyDeleteusing Galilean invariance and considering the transformation of the kinetic energy $T(\upsilon^2)$ in a frame with relative velocity $\epsilon$ gives $T((\upsilon+\epsilon)^2) = T(\upsilon^2+2\epsilon\upsilon + \epsilon^2) \simeq T(\upsilon^2) + 2\epsilon\upsilon\partial T/\partial\upsilon^2$. for the second term not to matter it must be a total time derivative (that's the $\upsilon$-term), such that $T$ must be propto $\upsilon^2$ and the derivative $\partial T/\partial\upsilon^2$ is constant. for that reason mechanical similarity $t^2\propto r^{2-n}$ can be established with the idea that a Galilean-covariant Lagrange-formalism is valid, but without an actual equation of motion. in this sense, the law of inertia follows from the Galilean relativity.

and there's a second detail that keeps me awestruck: there's already a glimpse of general relativity in Newton's first law, because in the force-free case a body follows a *straight* line with uniform velocity, so one could almost think that Newton had an idea of geodesic motion.

ReplyDelete