1. Establish relation (1.18)
\nu =\frac{\partial E(P,L,X_0,X_1,.....,X_n) }{\partial P }in which the velocity v is the intensive quantity conjugated to the momentum P.
2. Establish relation (1.21)
ω =\frac{\partial E(P,L,X_0,X_1,.....,X_n) }{\partial L }in which that the angular velocity ω is the intensive quantity conjugated to the angular momentum L.
Using the momentum definition (1.16),
P = Mν,
the expression (1.22)
E (P, L, X_0, X_1, . . . , X_n) =\frac{1 }{2 }ν.P+\frac{1 }{2 }ω · L + U(X_0, X_1, . . . , X_n).
for the energy E of the system is written as,
E (P, L, X_0, X_1, . . . , X_n)=\frac{p^2 }{2M } +\frac{1 }{2 } ω· L + U(X_0, X_1, . . . , X_n).
From it, we derive relation (1.18), i.e.
\nu =\frac{\partial E(P,L,X_0,X_1,…..,X_n) }{\partial P }.
ν=\frac{P }{M }=\frac{\partial E(P,L,X_0,X_1,…..,X_n) }{\partial P }.
2. Using the angular momentum definition (1.19),
L = Iω
the expression (1.22)
E (P, L, X_0, X_1, . . . , X_n) =\frac{1 }{2 }ν.P+\frac{1 }{2 }ω · L + U(X_0, X_1, . . . , X_n).
of the energy E of the system is written as,
E (P, L, X_0, X_1, . . . , X_n) = \frac{1 }{2 }ν.P+\frac{L^2 }{2I } + U(X_0, X_1, . . . , X_n).
From it, we derive relation (1.21), i.e.
ω =\frac{\partial E(P,L,X_0,X_1,…..,X_n) }{\partial L }.
ω =\frac{L }{I }=\frac{\partial E(P,L,X_0,X_1,…..,X_n) }{\partial L }.