Noether's theorem for Lagrangian systems #
Let E be a real normed space, the configuration space of a mechanical system, and let
L : ℝ → E × E → ℝ be a possibly time-dependent Lagrangian, a function L t (q, v) of time,
positions q and velocities v. A curve q with velocity v solves the Euler–Lagrange
equations when d/dt ∂_v L(t, q, v) = ∂_q L(t, q, v).
An infinitesimal symmetry of L is a possibly time-dependent vector field ξ(t, q) on the
configuration space. It moves a point (q, v) of the velocity phase space along its first
prolongation (ξ(t, q), ∂_t ξ(t, q) + D_q ξ(t, q) v); the second component is
Dξ_(t, q)(1, v), the derivative of ξ in the direction of the velocity (1, v) of the curve
t ↦ (t, q(t)). The derivative of L along this prolongation is the infinitesimal variation
δ_ξ L(t, q, v) = ∂_q L(t, q, v)(ξ(t, q)) + ∂_v L(t, q, v)(Dξ_(t, q)(1, v))
(TauCeti.lagrangianVariation). The vector field ξ is a symmetry up to a gauge term
B(t, q) when this variation is the total time derivative DB_(t, q)(1, v) of B, that is,
δ_ξ L = dB/dt along curves. Noether's theorem then says that the Noether charge
J(t, q, v) = ∂_v L(t, q, v)(ξ(t, q)) - B(t, q)
(TauCeti.noetherCharge) is conserved along every solution of the Euler–Lagrange equations
(TauCeti.hasDerivAt_noetherCharge, TauCeti.noetherCharge_eq_of_eulerLagrange). The symmetry
condition is only used along the solution, and only at the times considered.
Two classical sources of symmetries with B = 0 are invariance of L under translation in a
direction e, which makes the momentum ∂_v L(e) conserved
(TauCeti.lagrangianVariation_const_eq_zero), and invariance under a one-parameter family of
linear maps g s with g 0 = 1 and generator A = g'(0)
(TauCeti.lagrangianVariation_comp_snd_eq_zero), such as the rotations generated by a
skew-adjoint A. For a linear generator ξ(t, q) = A q and B = 0, the Noether charge is the
cotangent-lift moment map ⟨μ(q, p), A⟩ = p(A q) of the linear action
(TauCeti.cotangentMomentMap) evaluated at the Legendre image (q, ∂_v L(t, q, v))
(TauCeti.noetherCharge_eq_cotangentMomentMap). The Galilean boosts ξ(t, q) = t e of a
particle L = (m / 2) ‖v‖² - U(q) in a potential invariant in the direction e need the gauge
term B(t, q) = m ⟪q, e⟫. Their charge m (t ⟪v, e⟫ - ⟪q, e⟫) is conserved, which says that
the e-component of the motion is uniform (TauCeti.mul_inner_sub_inner_eq_of_eulerLagrange).
The charge, the variation and the symmetry condition are independent of the configuration
coordinates. Let φ : E → F be a change of coordinates and L' a Lagrangian on F with
L(t, q, v) = L'(t, φ(q), Dφ_q v), and let ξ' be φ-related to ξ, that is,
ξ'(t, φ(q)) = Dφ_q ξ(t, q). Then the charges agree at (q, v) and its image
(φ(q), Dφ_q v) when the gauge terms do (TauCeti.noetherCharge_eq_of_comp). For C²
coordinate changes, the variations agree as well (TauCeti.lagrangianVariation_eq_of_comp), as do
the total time derivatives of gauge terms related by B(t, q) = B'(t, φ(q)), so the symmetry
condition holds in one set of coordinates exactly when it holds in the other
(TauCeti.lagrangianVariation_eq_fderiv_iff_of_comp). No invertibility of φ is needed.
All statements are pointwise in time and in the velocity phase space, so they apply equally to local solutions and to local coordinate changes.
Main definitions #
TauCeti.noetherCharge L ξ B: the Noether charge∂_v L(t, q, v)(ξ(t, q)) - B(t, q).TauCeti.lagrangianVariation L ξ: the infinitesimal variation ofLalong the first prolongation ofξ.
Main results #
TauCeti.hasDerivAt_noetherCharge,TauCeti.noetherCharge_eq_of_eulerLagrange: Noether's theorem.TauCeti.noetherCharge_eq_cotangentMomentMap: comparison with the cotangent-lift moment map.TauCeti.noetherCharge_eq_of_comp,TauCeti.lagrangianVariation_eq_of_comp,TauCeti.lagrangianVariation_eq_fderiv_iff_of_comp: the transformation law under changes of configuration coordinates.
References #
- E. Noether, Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1918), 235–257.
- V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Graduate Texts in Mathematics 60, Springer, 1989, Section 20.
- J. E. Marsden, T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Texts in Applied Mathematics 17, Springer, 1999, Sections 7.3 and 11.4.
- P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Graduate Texts in Mathematics 107, Springer, 1993, Section 4.4 (divergence symmetries).
The Noether charge and the infinitesimal variation #
The Noether charge of a time-dependent Lagrangian L, an infinitesimal symmetry ξ and a
gauge term B: at time t and at (q, v), the momentum ∂_v L(t, q, v) paired with ξ(t, q),
minus B(t, q).
Instances For
The infinitesimal variation of a time-dependent Lagrangian L under a time-dependent vector
field ξ: at time t and at (q, v), the derivative of L t in the direction of the first
prolongation (ξ(t, q), Dξ_(t, q)(1, v)) of ξ, where
Dξ_(t, q)(1, v) = ∂_t ξ(t, q) + D_q ξ(t, q) v.
Instances For
Where L t is differentiable, the infinitesimal variation is
∂_q L(t, q, v)(ξ(t, q)) + ∂_v L(t, q, v)(Dξ_(t, q)(1, v)).
Noether's theorem #
Noether's theorem, pointwise in time. Let q have velocity v t at t and satisfy the
Euler–Lagrange equation d/dt ∂_v L(t, q, v) = ∂_q L(t, q, v) there. If at t the infinitesimal
variation of L under ξ is the total time derivative DB_(t, q)(1, v) of the gauge term B
along the curve, then the Noether charge has derivative zero at t.
Noether's theorem. Let q solve the Euler–Lagrange equations with velocity v on an open
preconnected set of times s, and suppose that, at every time of s, the infinitesimal variation
of L under ξ along the solution is the total time derivative of the gauge term B. Then the
Noether charge takes the same value at all times of s.
Symmetries from invariance of the Lagrangian #
For a constant vector field e, the infinitesimal variation is the derivative ∂_q L(e) of
L in the position direction e.
Translation invariance. If L t is differentiable at x and invariant under translation
of the position in the direction e near x, then the constant vector field e is an
infinitesimal symmetry of L at t and x, with zero variation. Its Noether charge with zero
gauge term is the momentum ∂_v L(t, q, v)(e).
For a linear vector field ξ(t, q) = A q, the infinitesimal variation is the derivative of L
in the direction (A q, A v) of the tangent lift of A.
Invariance under a linear family. Let g s be continuous linear maps of E with g 0 = 1
and derivative A at 0. If L t is differentiable at x and L t (g s q, g s v) = L t (q, v)
for s near 0, then the linear vector field A is an infinitesimal symmetry of L at t and
x, with zero variation. For the rotations g s = exp(s A) generated by a skew-adjoint A, the
Noether charge is the angular momentum ∂_v L(A q).
The Noether charge of a linear symmetry is the cotangent-lift moment map. For a linear
vector field ξ(t, q) = A q and zero gauge term, the Noether charge at (q, v) is the moment map
⟨μ(q, p), A⟩ = p(A q) of the cotangent-lifted linear action, at the momentum
p = ∂_v L(t, q, v).
Changes of configuration coordinates #
The Noether charge is independent of coordinates. Let φ : E → F be a map of
configuration spaces, and let L' be a Lagrangian on F with L(t, q, u) = L'(t, φ q, Dφ_q u)
for velocities u near v, where the velocity map u ↦ L'(t, φ q, u) is differentiable at
Dφ_q v. If ξ'(t, φ q) = Dφ_q ξ(t, q) and B(t, q) = B'(t, φ q), then the Noether charges agree
at (q, v) and at (φ q, Dφ_q v).
The infinitesimal variation is independent of coordinates. Let φ : E → F be C² at q,
and let L' be a Lagrangian on F with L(t, y, u) = L'(t, φ y, Dφ_y u) near (q, v), with
L' t differentiable at (φ q, Dφ_q v). If ξ'(s, φ y) = Dφ_y ξ(s, y) near (t, q), with ξ
and ξ' differentiable at (t, q) and (t, φ q), then the infinitesimal variations agree at
(q, v) and at (φ q, Dφ_q v).
The symmetry condition is independent of coordinates. Under the hypotheses of
TauCeti.lagrangianVariation_eq_of_comp, let the gauge terms be related by
B(s, y) = B'(s, φ y) near (t, q), with B' differentiable at (t, φ q). Then the infinitesimal
variation of L under ξ at (q, v) is the total time derivative of B exactly when the
infinitesimal variation of L' under ξ' at (φ q, Dφ_q v) is the total time derivative of
B'.
Galilean boosts #
Galilean boosts. Let a particle of nonzero mass m move in a potential U, with Lagrangian
L(q, v) = (m / 2) ‖v‖² - U(q), along a solution of the Euler–Lagrange equations on an open
preconnected set of times. If along the solution U is differentiable with DU(e) = 0, as when
U is invariant under translation in the direction e, then the boost ξ(t, q) = t e is a
symmetry with gauge term B(t, q) = m ⟪q, e⟫. Its conserved Noether charge shows that
t ⟪v t, e⟫ - ⟪q t, e⟫ is constant: the e-component of the position moves uniformly.