Astronomy before computers!

Sect. 2.7: Energy Function & Energy Conservation One more conservation theorem which we would expect to get from the Lagrange formalism is: CONSERVATION OF ENERGY. Consider a general Lagrangian L, a function of the coords qj, velocities qj, & time t: L = L(qj,qj,t) (j = 1,…n) The total time derivative of L (chain rule): (dL/dt) = ∑j(∂L/∂qj)(dqj/dt) + ∑j(∂L/∂qj)(dqj/dt) + (∂L/∂t) Or: (dL/dt) = ∑j(∂L/∂qj)qj + ∑j(∂L/∂qj)qj + (∂L/∂t)

Total time derivative of L: (dL/dt) = ∑j(∂L/∂qj)qj + ∑j(∂L/∂qj)qj + (∂L/∂t) (1) Lagrange’s Eqtns: (d/dt)[(∂L/∂qj)] - (∂L/∂qj) = 0 Put into (1) (dL/dt) = ∑j(d/dt)[(∂L/∂qj)]qj + ∑j(∂L/∂qj)qj + (∂L/∂t) Identity: 1st 2 terms combine (dL/dt) = ∑j(d/dt)[qj(∂L/∂qj)] + (∂L/∂t) Or: (d/dt)[∑jqj(∂L/∂qj) - L] + (∂L/∂t) = 0 (2)

(d/dt)[∑jqj(∂L/∂qj) - L] + (∂L/∂t) = 0 (2) Define the Energy Function h: h  ∑jqj(∂L/∂qj) - L = h(q1,..qn;q1,..qn,t) (2)  (dh/dt) = - (∂L/∂t)  For a Lagrangian L which is not an explicit function of time (so that (∂L/∂t) = 0) (dh/dt) = 0 & h = constant (conserved) Energy Function h = h(q1,..qn;q1,..qn,t) Identical Physically to what we later will call the Hamiltonian H. However, here, h is a function of n indep coords qj & velocities qj. The Hamiltonian H is ALWAYS considered a function of 2n indep coords qj & momenta pj

Energy Function h  ∑jqj(∂L/∂qj) - L We had (dh/dt) = - (∂L/∂t)  For a Lagrangian for which (∂L/∂t) = 0 (dh/dt) = 0 & h = constant (conserved) For this to be useful, we need a Physical Interpretation of h. Will now show that, under certain circumstances, h = total mechanical energy of the system.

Physical Interpretation of h Energy Function h  ∑jqj(∂L/∂qj) - L Recall (Sect. 1.6) that we can always write KE as: T = M0 + ∑jMjqj + ∑jMjkqjqk M0  (½)∑imi(∂ri/∂t)2 , Mj ∑imi(∂ri/∂t)(∂ri/∂qj) Mjk  ∑i mi(∂ri/∂qj)(∂ri/∂qk) Or (schematically) T = T0(q) + T1(q,q) + T2(q,q) T0  M0 independent of generalized velocities T1  ∑jMjqj linear in generalized velocities T2  ∑jMjkqjqk quadratic in generalized velocities

L = L0(q,t) + L1(q,q,t) + L2(q,q,t) With almost complete generality, we can write (schematically) the Lagrangian for most problems of interest in mechanics as: L = L0(q,t) + L1(q,q,t) + L2(q,q,t) L0  independent of the generalized velocities L1  linear in generalized the velocities L2  quadratic in generalized the velocities For conservative forces, L has this form. Also does for some velocity dependent potentials, such as for EM fields.

Euler’s Theorem from mathematics: L = L0(q,t) + L1(q,q,t) + L2(q,q,t) (1) Euler’s Theorem from mathematics: If f = f(x1,x2,.. xN) = a homogeneous function of degree n of the variables xi, then ∑ixi(∂f/∂xi) = n f (2) Energy Function h  ∑jqj(∂L/∂qj) - L (3) For L of form (1): (2)  h = 0L0 + 1L1+2L2 - [L0 + L1 + L2] or h = L2 - L0

 Energy function h  ∑jqj(∂L/∂qj) - L = L2 - L0 L = L0(q,t) + L1(q,q,t) + L2(q,q,t)  Energy function h  ∑jqj(∂L/∂qj) - L = L2 - L0 Special case (both conditions!): a.) The transformation eqtns from Cartesian to Generalized Coords are time indep.  In the KE, T0 = T1 = 0  T = T2 b.) V is velocity indep.  L2 = T = T2 & L0 = -V  h = T + V = E  Total Mechanical Energy Under these conditions, if V does not depend on t, neither does L & thus (∂L/∂t) = 0 = (dh/dt) so h = E = constant (conserved)

Energy Conservation Summary: Different Conditions: Energy Function h  ∑jqj(∂L/∂qj) - L ALWAYS: (dh/dt) = - (∂L/∂t) SOMETIMES: L does not depend on t  (∂L/∂t) = 0, (dh/dt) = 0 & h = const. (conserved) USUALLY: L = L0(q,t) + L1(q,q,t) + L2(q,q,t)  h  ∑jqj(∂L/∂qj) - L = L2 - L0 SOMETIMES: T = T2 = L2 AND L0 = -V  h = T + V = E  Total Mechanical Energy  Conservation Theorem for Mechanical Energy: If h = E AND L does not depend on t, E is conserved!

 Can have conditions in which: 1. h is conserved & = E Clearly, the conditions for conservation of energy function h are DISTINCT from those which make it the total mechanical energy E.  Can have conditions in which: 1. h is conserved & = E 2. h is not conserved & = E 3. h is conserved &  E 4. h is not conserved &  E Most common case in classical (& quantum) mech. is case 1.

Stated another way: Two questions: 1. Does the energy function h = E for the system? 2. Is the mechanical energy E conserved for the system? Two aspects of the problem! DIFFERENT questions! May have cases where h  E, but E is conserved. For example: A conservative system, using generalized coords in motion with respect to fixed rectangular axes:  Transformation eqtns will contain the time  T will NOT be a homogeneous, quadratic function of the generalized velocities!  h  E, However, because the system is conservative, E is conserved! (This is a physical fact about the system, independent of coordinate choices!).

It is also worth noting: The Lagrangian L = T - U is independent of the choice of generalized coordinates. The Energy function h  ∑jqj(∂L/∂qj) - L depends on the choice of generalized coordinates. The most common case in classical (& quantum) mechanics is h = E and E is conserved.

Non-Conservative Forces Consider a non-conservative system: Frictional forces obtained from the dissipation function ₣ . Derivation with the energy function h becomes: (dh/dt) + (∂L/∂t) = - ∑jqj(∂₣ /∂qj) Ch. 1: The formulation of ₣ shows it is a homogeneous, quadratic function of the q’s.  Use Euler’s theorem again: ∑jqj(∂₣ /∂qj) = 2₣  (dh/dt) = - (∂L/∂t) - 2₣ If L is not an explicit function of time (∂L/∂t =0 ) AND h = E: (dE/dt) = - 2₣ That is, under these conditions, 2₣ = Energy dissipation rate.

Symmetry Properties & Conservation Laws (From Marion’s Book!) In general, in physical systems: A Symmetry Property of the System  Conservation of Some Physical Quantity Also: Conservation of Some Physical Quantity  A Symmetry Property of the System Not just valid in classical mechanics! Valid in quantum mechanics also! Forms the foundation of modern field theories (Quantum Field Theory, Elementary Particles,…)

We’ve seen in general that: Conservation Theorem: If the Generalized Coord qj is cyclic or ignorable, the corresponding Generalized (or Conjugate) Momentum, pj  (∂L/∂qj) is conserved. An underlying symmetry property of the system: If qj is cyclic, the system is unchanged (invariant) under a translation (or rotation) in the “qj direction”.  pj is conserved

Linear Momentum Conservation Conservation of linear momentum: If a component of the total force vanishes, nF = 0, the corresponding component of total linear momentum np = const (is conserved) Underlying symmetry property of the system: The system is unchanged (invariant) under a translation in the “n direction”.  np is conserved

Angular Momentum Conservation Conservation of angular momentum: If a component of total the torque vanishes, nN = 0, the corresponding component of total angular momentum nL = const (conserved) Underlying symmetry property of the system: The system is unchanged (invariant) under a rotation about the “n direction”.  nL is conserved

Energy Conservation Conservation of mechanical energy: If all forces in the system are conservative, the total mechanical energy E = const (conserved) Underlying symmetry property of the system: (More subtle than the others!) The system is unchanged (invariant) under a time reversal. (Changing t to -t in all eqtns of motion)  E is conserved

Summary: Conservation Laws Under the proper conditions, there can be up to 7 “Constants of the Motion”  “1st Integrals of the Motion”  Quantities which are Conserved (const in time): Total Mechanical Energy (E) 3 vector components of Linear Momentum (p) 3 vector components of Angular Momentum (L)