#calculusofvariations — Public Fediverse posts

Noether’s Theorem: Symmetry→Conservation Law.

Symmetries of the action imply conserved quantities per Noether's theorem.

A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.

Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.

#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation

Noether’s Theorem: Symmetry→Conservation Law.

Symmetries of the action imply conserved quantities per Noether's theorem.

A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.

Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.

#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation

Noether’s Theorem: Symmetry→Conservation Law.

Symmetries of the action imply conserved quantities per Noether's theorem.

A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.

Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.

#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation

Noether’s Theorem: Symmetry→Conservation Law.

Symmetries of the action imply conserved quantities per Noether's theorem.

A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.

Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.

#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation

Noether’s Theorem: Symmetry→Conservation Law.

Symmetries of the action imply conserved quantities per Noether's theorem.

A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.

Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.

#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation

FUNDAMENTAL LEMMA OF THE CALCULUS OF VARIATIONS:
If a continuous multivariable function \(f\) on an open set \({\Omega\subset \mathbb {R} ^{d}}\) satisfies the equality
\[\displaystyle\int_\Omega\mathcal{f}(x)\mathcal{g}(x)\ \mathrm{d}x=0\]
for all compactly supported smooth functions \(g\) on \(\Omega\), then \(f\) is identically zero.
#lemma #calculusofvariations #fundamentallemma #multivariablefunctions #calculus #variations #functional #function #smoothfunction #analysis #mathematics

FUNDAMENTAL LEMMA OF THE CALCULUS OF VARIATIONS:
If a continuous multivariable function \(f\) on an open set \({\Omega\subset \mathbb {R} ^{d}}\) satisfies the equality
\[\displaystyle\int_\Omega\mathcal{f}(x)\mathcal{g}(x)\ \mathrm{d}x=0\]
for all compactly supported smooth functions \(g\) on \(\Omega\), then \(f\) is identically zero.
#lemma #calculusofvariations #fundamentallemma #multivariablefunctions #calculus #variations #functional #function #smoothfunction #analysis #mathematics

FUNDAMENTAL LEMMA OF THE CALCULUS OF VARIATIONS:
If a continuous multivariable function \(f\) on an open set \({\Omega\subset \mathbb {R} ^{d}}\) satisfies the equality
\[\displaystyle\int_\Omega\mathcal{f}(x)\mathcal{g}(x)\ \mathrm{d}x=0\]
for all compactly supported smooth functions \(g\) on \(\Omega\), then \(f\) is identically zero.
#lemma #calculusofvariations #fundamentallemma #multivariablefunctions #calculus #variations #functional #function #smoothfunction #analysis #mathematics

FUNDAMENTAL LEMMA OF THE CALCULUS OF VARIATIONS:
If a continuous multivariable function \(f\) on an open set \({\Omega\subset \mathbb {R} ^{d}}\) satisfies the equality
\[\displaystyle\int_\Omega\mathcal{f}(x)\mathcal{g}(x)\ \mathrm{d}x=0\]
for all compactly supported smooth functions \(g\) on \(\Omega\), then \(f\) is identically zero.
#lemma #calculusofvariations #fundamentallemma #multivariablefunctions #calculus #variations #functional #function #smoothfunction #analysis #mathematics