Hamiltonian Function: Conservation Laws In general:

Closed System: No interaction with anything outside • If the Lagrangian of a system, closed or the system. otherwise, is invariant with respect to a translation in a certain direction, then the Any closed system has: seven constants or integrals of linear momentum of the system in that motion: direction is constant in time.

• three components of the linear momentum

• three components of angular momentum " Conservation of angular momentum: • the total energy

• Property of inertial frame:

o Space is isotropic in an inertial frame (a

closed system is unaffected by orientation or " Conservation of linear momentum: rotation of the entire system).

• Property of inertial frame: • The Lagrangian of a closed system remains Space is homogeneous in an inertial system o invariant if the system is rotated through an or frame (a closed system is unaffected by a infinitesimal angle. translation of the system in space).

G G G J = r × p = const. • The Lagrangian of a closed system in an inertial frame is invariant. G pk ≡ p ∂L G p = mq = = const. qk ≡ r k k  ∂qk

1 2 In general: d L d ∂L ∴ = [ q ]= 0 ∑ k dt k dt ∂qk • If the Lagrangian remains invariant under rotation about an axis, then the angular d  ∂L   q − L = 0 momentum of the system about this axis will ∑ k  dt  k ∂qk  remain constant in time.

∴ the quantity in parentheses must be constant in time. " Conservation of Energy:

 ∂L  • Property of inertial frame: H =  q − L = p q − L =const. ∑ k   ∑ k k o The time is homogeneous within an inertial  k ∂qk  k reference frame. H is called the Hamiltonian. • The Lagrangian of a closed system can not be an H = H ( pk ,qk ) explicit function of time.

• H is a constant of motion if L is not an explicit ∂L ∴ = o function of time ∂t d L ∂L dq ∂L dq dq T = k + k = 0 k = q • If the kinetic energy is a homogeneous dt ∑ ∂q dt ∑ ∂q dt k k k k k dt quadratic function of the qk and the potential

energy V is a function of the qk alone, we will from Lagrange equation: have:

d ∂L ∂L d L d ∂L ∂L ( ) = = q + q = 0  ⇒ ∑  k ∑  k dt ∂qk ∂qk dt k dt ∂qk k ∂qk H = 2T − L = 2T − (T −V ) = T +V = E = const.

3 4 Andδ the variation of H : Hamiltonian Dynamics: Hamilton’s Equations of Motion δ  ∂H ∂H  Consider: H =  pk + δ qk  ∑ ∂ p ∂q k  k k 

L = L(qk ,qk ) ∂H ∂L    = qk pk = ∂ p   k ∂qk  ∂H ∂L ∴  = − p p = k k ∂qk ∂qk

⇒ qk = qk ( pk , qk ) These are Hamiltonian canonical equation of motion.

p q ∴ H can be expressed as a function of k and k .

H ( p ,q ) = p q ( p ,q )− L k k ∑ k k k k δ k

δ H δ p δq The variation of correspondingδ to k and k :

δ  ∂L ∂L  H = p q + q p − q − δ q ∑  k k k k k k  k  ∂qk ∂qk  δ δ H = []q p − p δ q ∑ k k k k k

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