Studying the dynamics of some Lagrangian systems by nonlocal - PowerPoint PPT Presentation

Gaetano Zampieri Università di Verona, Italy gaetano.zampieri@univr.it Studying the dynamics of some Lagrangian systems by nonlocal constants of motion This is a joint work with Gianluca Gorni Università di Udine, Italy Castro Urdiales, June 2019

Theorem in the variational case 2

Theorem in the variational case 2 � For smooth scalar valued Lagrangian function L ( t, q, ˙ q ) , q ∈ R n , Euler-Lagrange equation t ∈ R , q, ˙ d � � � � dt∂ ˙ q L t, q ( t ) , ˙ q ( t ) − ∂ q L t, q ( t ) , ˙ q ( t ) = 0

Theorem in the variational case 2 � For smooth scalar valued Lagrangian function L ( t, q, ˙ q ) , q ∈ R n , Euler-Lagrange equation t ∈ R , q, ˙ d � � � � dt∂ ˙ q L t, q ( t ) , ˙ q ( t ) − ∂ q L t, q ( t ) , ˙ q ( t ) = 0 � Theorem. Let q ( t ) be a sol. to the Euler-Lagrange eq. and let q λ ( t ) , λ ∈ R , be a smooth family of perturbed motions, such that q 0 ( t ) ≡ q ( t ) . Then the following function of t is constant � t ∂ � �� � � � ∂ ˙ q L t, q ( t ) , ˙ q ( t ) · ∂ λ q λ ( t ) λ =0 − ∂λL s, q λ ( s ) , ˙ q λ ( s ) λ =0 ds . � � t 0

Theorem in the variational case 3 We call it the constant of motion associated to the fam- ily q λ ( t ) .

Theorem in the variational case 3 We call it the constant of motion associated to the fam- ily q λ ( t ) . � The constant of motion can be unuseful or trivial. In general it is nonlocal, i.e. its value at a time t depends not only on the state ( q ( t ) , ˙ q ( t )) at time t , but also on the whole history between t 0 and t .

Theorem in the variational case 3 We call it the constant of motion associated to the fam- ily q λ ( t ) . � The constant of motion can be unuseful or trivial. In general it is nonlocal, i.e. its value at a time t depends not only on the state ( q ( t ) , ˙ q ( t )) at time t , but also on the whole history between t 0 and t . ◦ Proof. Taking the time derivative we have d − ∂ �� � � � � � � ∂ ˙ q L t, q ( t ) , ˙ q ( t ) · ∂ λ q λ ( t ) ∂λL t, q λ ( t ) , ˙ q λ ( t ) � λ =0 = � � λ =0 dt = d · d � � � � � � dt∂ ˙ q L t, q ( t ) , ˙ q ( t ) · ∂ λ q λ ( t ) � λ =0 + ∂ ˙ q L t, q ( t ) , ˙ q ( t ) dt∂ λ q λ ( t ) � λ =0 + � � � � � � − ∂ q L t, q ( t ) , ˙ q ( t ) · ∂ λ q λ ( t ) � λ =0 − ∂ ˙ q L t, q ( t ) , ˙ q ( t ) · ∂ λ ˙ q λ ( t ) � λ =0 = 0 since the sum of the red terms vanishes by means of the Euler-Lagrange equation and the blu terms are equal by reversing the derivation order. q.e.d.

Angular momentum 4

Angular momentum 4 � The perturbed motions q λ ( t ) were originally inspired by the mechanism that Noether’s theorem uses to deduce conservation laws whenever the Lagrangian function L enjoys certain invariance properties. A simple classical example, particle of mass m in the plane that is driven by a central force field q ) := 1 q | 2 − U q = ( q 1 , q 2 ) ∈ R 2 . � � L ( t, q, ˙ 2 m | ˙ t, | q | ,

Angular momentum 4 � The perturbed motions q λ ( t ) were originally inspired by the mechanism that Noether’s theorem uses to deduce conservation laws whenever the Lagrangian function L enjoys certain invariance properties. A simple classical example, particle of mass m in the plane that is driven by a central force field q ) := 1 q | 2 − U q = ( q 1 , q 2 ) ∈ R 2 . � � L ( t, q, ˙ 2 m | ˙ t, | q | , To exploit the rotational symmetry of L it is natural to take the rotated family � cos λ − sin λ � � q 1 ( t ) � � � � q λ ( t ) := , ∂ λ q λ ( t ) λ =0 = − q 2 ( t ) , q 1 ( t ) . � sin λ cos λ q 2 ( t )

Angular momentum 4 � The perturbed motions q λ ( t ) were originally inspired by the mechanism that Noether’s theorem uses to deduce conservation laws whenever the Lagrangian function L enjoys certain invariance properties. A simple classical example, particle of mass m in the plane that is driven by a central force field q ) := 1 q | 2 − U q = ( q 1 , q 2 ) ∈ R 2 . � � L ( t, q, ˙ 2 m | ˙ t, | q | , To exploit the rotational symmetry of L it is natural to take the rotated family � cos λ − sin λ � � q 1 ( t ) � � � � q λ ( t ) := , ∂ λ q λ ( t ) λ =0 = − q 2 ( t ) , q 1 ( t ) . � sin λ cos λ q 2 ( t ) It is clear that L ( t, q λ ( t ) , ˙ q λ ( t )) does not depend on λ . The constant of motion associated to the rotation family reduces to Noether’s theorem and gives the angular momentum as constant of motion: � ∂ ˙ q L · ∂ λ q λ λ =0 = m ˙ q · ( − q 2 , q 1 ) = m ( q 1 ˙ q 2 − q 2 ˙ q 1 ) . �

Energy conservation 5

Energy conservation 5 � Next, we revisit another classical example, from our point of view. For time q ), q ∈ R n , and the time-shift family q λ ( t ) = q ( t + λ ): indep. L ( t, q, ˙ q ) = L ( q, ˙ q ( t ) = d �� � ∂ λ L t, q λ ( t ) , ˙ q λ ( t ) λ =0 = ∂ q L · ˙ q ( t ) + ∂ ˙ q L · ¨ dt L ( q ( t ) , ˙ q ( t )) . �

Energy conservation 5 � Next, we revisit another classical example, from our point of view. For time q ), q ∈ R n , and the time-shift family q λ ( t ) = q ( t + λ ): indep. L ( t, q, ˙ q ) = L ( q, ˙ q ( t ) = d �� � ∂ λ L t, q λ ( t ) , ˙ q λ ( t ) λ =0 = ∂ q L · ˙ q ( t ) + ∂ ˙ q L · ¨ dt L ( q ( t ) , ˙ q ( t )) . � The constant of motion is � t d ∂ ˙ q L · ˙ q ( t ) − ds L ( q ( s ) , ˙ q ( s )) ds = t 0 = ∂ ˙ q L ( q ( t ) , ˙ q ( t )) · ˙ q ( t ) − L ( q ( t ) , ˙ q ( t )) + L ( q ( t 0 ) , ˙ q ( t 0 )) = = E ( q ( t ) , ˙ q ( t )) + L ( q ( t 0 ) , ˙ q ( t 0 )) . Energy E ( q, ˙ q ) = ∂ ˙ q L ( q, ˙ q ) · ˙ q − L ( q, ˙ q ) up to a trivial additive const.

Energy conservation 5 � Next, we revisit another classical example, from our point of view. For time q ), q ∈ R n , and the time-shift family q λ ( t ) = q ( t + λ ): indep. L ( t, q, ˙ q ) = L ( q, ˙ q ( t ) = d �� � ∂ λ L t, q λ ( t ) , ˙ q λ ( t ) λ =0 = ∂ q L · ˙ q ( t ) + ∂ ˙ q L · ¨ dt L ( q ( t ) , ˙ q ( t )) . � The constant of motion is � t d ∂ ˙ q L · ˙ q ( t ) − ds L ( q ( s ) , ˙ q ( s )) ds = t 0 = ∂ ˙ q L ( q ( t ) , ˙ q ( t )) · ˙ q ( t ) − L ( q ( t ) , ˙ q ( t )) + L ( q ( t 0 ) , ˙ q ( t 0 )) = = E ( q ( t ) , ˙ q ( t )) + L ( q ( t 0 ) , ˙ q ( t 0 )) . Energy E ( q, ˙ q ) = ∂ ˙ q L ( q, ˙ q ) · ˙ q − L ( q, ˙ q ) up to a trivial additive const. q | 2 − U ( q ) gives E ( q, ˙ q | 2 + U ( q ) q ) = 1 q ) = 1 For instance L ( q, ˙ 2 m | ˙ 2 m | ˙

Homogeneous potentials 6

Homogeneous potentials 6 � From now on, our results. Lagrangian q | 2 − U ( q ) , q ) := 1 q ∈ R n , L ( t, q, ˙ q ) = L ( q, ˙ 2 m | ˙ potential U positively homogeneous of degree α U ( sq ) = s α U ( q ) , s > 0 .

Homogeneous potentials 6 � From now on, our results. Lagrangian q | 2 − U ( q ) , q ) := 1 q ∈ R n , L ( t, q, ˙ q ) = L ( q, ˙ 2 m | ˙ potential U positively homogeneous of degree α U ( sq ) = s α U ( q ) , s > 0 . Well known that: if q ( t ) is solution to Euler-Lagrange eq. ¨ q = −∇ U ( q ) then q λ ( t ) = e λ q e λ ( α/ 2 − 1) t � � , λ ∈ R , solution too. Our theorem with this family, and some computations, give the constant of motion

Homogeneous potentials 6 � From now on, our results. Lagrangian q | 2 − U ( q ) , q ) := 1 q ∈ R n , L ( t, q, ˙ q ) = L ( q, ˙ 2 m | ˙ potential U positively homogeneous of degree α U ( sq ) = s α U ( q ) , s > 0 . Well known that: if q ( t ) is solution to Euler-Lagrange eq. ¨ q = −∇ U ( q ) then q λ ( t ) = e λ q e λ ( α/ 2 − 1) t � � , λ ∈ R , solution too. Our theorem with this family, and some computations, give the constant of motion � � t � α � α � m ˙ q ( t ) · q ( t ) + t 2 − 1 E ( q ( t ) , ˙ q ( t )) − 2 + 1 t 0 L ( q ( s ) , ˙ q ( s )) ds . q | 2 + U ( q ), the energy conserved too. with E := 1 2 m | ˙

Homogeneous potentials 7 In the special case α = − 2 we get a time-dependent first integral in the usual sense F ( q, ˙ q ) := m ˙ q · q − 2 t E ( q, ˙ q )

Homogeneous potentials 7 In the special case α = − 2 we get a time-dependent first integral in the usual sense F ( q, ˙ q ) := m ˙ q · q − 2 t E ( q, ˙ q ) � Examples central U ( q ) = − k/ | q | 2 and Calogero’s � U ( q 1 , . . . , q n ) = g 2 ( q j − q k ) − 2 , 1 ≤ j<k ≤ n for q j ∈ R , q j � = q k when j � = k .

Homogeneous potentials 7 In the special case α = − 2 we get a time-dependent first integral in the usual sense F ( q, ˙ q ) := m ˙ q · q − 2 t E ( q, ˙ q ) � Examples central U ( q ) = − k/ | q | 2 and Calogero’s � U ( q 1 , . . . , q n ) = g 2 ( q j − q k ) − 2 , 1 ≤ j<k ≤ n for q j ∈ R , q j � = q k when j � = k . � Notice that for α = − 2 the integrand in the formula of the theorem does not vanish.

Homogeneous potentials 8 � Take the antiderivative in time of 0 = mq ( t ) · ˙ q ( t ) − 2 tE − F and obtain one more time-dependent constant of motion F 1 = 1 2 m | q ( t ) | 2 − t 2 E − tF . We can also solve for | q ( t ) | : √ t 2 E + tF + F 1 . | q ( t ) | = 2 m This formula gives the time-dependence of distance from the origin even though we don’t know the shape of the orbit. So we generalized a formula known in the central case.