Symmetry and conservation
A generalized coordinate q_j is called cyclic if the system’s Lagrangian \mathcal{L}(q_j, \dot{q}_j, t) does not explicitly depend on q_j. This means:
\frac{\partial \mathcal{L}}{\partial q_j} = 0
and Lagrange’s equations reduce to:
\frac{\mathrm d}{\mathrm dt}\left( \frac{\partial \mathcal{L}}{\partial \dot{q}_j} \right) = 0
which immediately shows the corresponding momentum p_j is a constant of motion.
Translational symmetry: linear momentum conservation
If a system of N particles is such that its potential energy V(\mathbf{x}_1, \ldots, \mathbf{x}_N) is invariant under translation along a direction \mathbf{n}, then:
V(\mathbf{x}_1 + s\mathbf{n}, \ldots, \mathbf{x}_N + s\mathbf{n}) = V(\mathbf{x}_1, \ldots, \mathbf{x}_N)
Taking the derivative of the Lagrangian \mathcal{L} with respect to the translation parameter s and applying Lagrange’s equations gives:
\frac{\mathrm d}{\mathrm dt} \left( \mathbf{P} \cdot \mathbf{n} \right) = 0
where \mathbf{P} is the total linear momentum. This shows that invariance of the potential under translations directly implies conservation of linear momentum along the direction \mathbf{n}.
Rotational symmetry: angular momentum conservation
Now, assume the potential is invariant under rotations by angle \theta about an axis defined by unit vector \mathbf{n}:
V(\mathcal{R}_\mathbf{n}(\theta)\mathbf{x}_1, \ldots, \mathcal{R}_\mathbf{n}(\theta)\mathbf{x}_N) = V(\mathbf{x}_1, \ldots, \mathbf{x}_N)
For an infinitesimal rotation, the position transforms as:
\delta\mathbf{x}_i = \delta\theta\, (\mathbf{n} \times \mathbf{x}_i)
and the chain rule and the invariance of \mathcal{L} yield:
\frac{\mathrm d}{\mathrm dt} \left( \mathbf{L}_{tot} \cdot \mathbf{n} \right) = 0
where \mathbf{L}_{tot} is the total angular momentum. Hence, rotational symmetry implies the conservation of the angular momentum component along \mathbf{n}.
Conclusion
Translational and rotational invariance immediately lead to conservation of linear and angular momentum components respectively, through the structure of the Lagrangian formalism. This relationship between symmetry and conservation is one of the most direct and far-reaching results in classical mechanics.
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