Conservation Laws

One of the most eye-opening and intellectually satisfying aspects of this chapter is the way the conservation laws emerge directly from symmetry principles. It is quite elegant to see that conservation of energy follows from the homogeneity of time, that linear momentum follows as a conservation law from the homogeneity of space, and that angular momentum follows as a conservation law from the isotropy of space. In this way, Landau presents the basic structure of mechanics not as a list of separate facts, but as a unified consequence of the symmetries of nature.

Euler's theorem for homogeneous functions

Before discussing conservation laws, it is useful to recall Euler's theorem for homogeneous functions. Let \(f(x_1,\dots,x_n)\) be a differentiable function homogeneous of degree \(k\), meaning that

Define the auxiliary one-variable function \(g(\lambda)=f(\lambda x_1,\dots,\lambda x_n)\). By homogeneity, this can also be written as \(g(\lambda)=\lambda^k f(x_1,\dots,x_n)\). Differentiating with respect to \(\lambda\) gives \(g'(\lambda)=k\lambda^{k-1}f\). On the other hand, applying the chain rule directly to \(g(\lambda)=f(\lambda x_1,\dots,\lambda x_n)\) gives

Since both expressions represent the same derivative, they must be equal. Evaluating at \(\lambda=1\) one arrives at Euler's theorem,

Euler's theorem applied to the kinetic energy

In mechanics, the kinetic energy \(T=T(\dot{q}_1,\dots,\dot{q}_n)\) is a homogeneous function of degree \(2\) in the generalized velocities. Therefore, by Euler's theorem,

Momentum

Consider an infinitesimal translation of the whole system, \(\mathbf{r}_a \to \mathbf{r}_a+\boldsymbol{\epsilon}\), so that each position changes by \(\delta \mathbf{r}_a=\boldsymbol{\epsilon}\). Since \(\boldsymbol{\epsilon}\) is constant, the velocities do not change: \(\delta \dot{\mathbf{r}}_a=0\). The first-order Taylor expansion of the Lagrangian then reduces to

If space is homogeneous, a global translation cannot change the physics, so \(\delta L=0\). Because \(\boldsymbol{\epsilon}\) is arbitrary, \(\sum_a \partial L/\partial \mathbf{r}_a=0\). Using the Lagrange equations to replace \(\partial L/\partial \mathbf{r}_a\) by a time derivative, one obtains

Thus the sum of the conjugate momenta is conserved. This quantity is the total linear momentum,

Angular momentum

Consider an infinitesimal rotation by \(\delta \boldsymbol{\phi}\). The corresponding variations are \(\delta \mathbf{r}_a=\delta \boldsymbol{\phi}\times \mathbf{r}_a\) and \(\delta \mathbf{v}_a=\delta \boldsymbol{\phi}\times \mathbf{v}_a\). Using \(\mathbf{p}_a=\partial L/\partial \mathbf{v}_a\) and \(\partial L/\partial \mathbf{r}_a=\dot{\mathbf{p}}_a\), the first-order variation of the Lagrangian becomes

Recognizing the bracket as a total time derivative and imposing isotropy of space (\(\delta L=0\) for any rotation), one finds the conserved total angular momentum,

Mechanical similarity

The mechanical-similarity relations begin by scaling the trajectory and time as \(\mathbf{r}'=\alpha \mathbf{r}\) and \(t'=\beta t\). The velocity then scales as \(\mathbf{v}'=(\alpha/\beta)\,\mathbf{v}\), so the kinetic energy scales as \((\alpha/\beta)^2 T\). If the potential is homogeneous of degree \(k\), i.e. \(U(\alpha \mathbf{r})=\alpha^k U(\mathbf{r})\), then matching the scaling of the kinetic and potential terms requires \((\alpha/\beta)^2=\alpha^k\), hence

Writing \(\alpha=l'/l\), the time, velocity, energy, and angular momentum transform as

These relations show how time, velocity, energy, and angular momentum must transform when the coordinates are rescaled in a system whose potential is a homogeneous function of degree \(k\).