Mathematical Principles of Theoretical Physics

2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 87 which implies that ∫ QT δLψ ̃∗dxdt= ∫ QT [ ih ̄ ∂ ψ ∂t + ̄h^2 2 m ∆ψ−Vψ ] ψ ̃∗dxd ...

88 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS whereψ= (ψ 1 ,ψ 2 ,ψ 3 ,ψ 4 )Tis the Dirac spinor, andγμ= (γ^0 ,γ^1 ,γ^2 ,γ^3 )i ...

2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 89 2.5.5 Symmetries and conservation laws The importance of PLD lies in the followin ...

90 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Now, we discuss the symmetry from the viewpoint of action functional and group a ...

2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 91 Remark 2.39.The correspondencebetween symmetries and conservation laws in the Noe ...

92 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Hence we derive from the Noether Theorem that I= 〈δL δx ̇ , d dλ ∣ ∣ ∣ λ= 0 Aλx ...

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 93 whereH=H(q,p,t)is the total energy,Lis the Lagrangian, andq,psatisfy the Hamilto ...

94 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS The state variablesqk( 1 ≤k≤N)represent positions, andpk( 1 ≤k≤N)represent momen ...

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 95 In this example, the two relations (2.6.10) and (2.6.11) are obtained directly b ...

96 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Conversely, if we know the PHD system (2.6.14) and (2.6.15), then it follows fro ...

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 97 introduced in the last subsection can be generalized to all physical fields. The ...

98 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS Theorem 2.46(Energy Conservation).Let(u,v)be the solutions of (2.6.33), andHdoes ...

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 99 which means that the rate of energy change inΩequals to the difference of the in ...

100 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS 2.6.3 PHD for Maxwell electromagnetic fields PHD is also valid in the Maxwell e ...

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 101 and the fields of PHD are A 1 ,A 2 ,A 3 and E 1 ,E 2 ,E 3. It is clear that 1 4 ...

102 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS In view of PHD for a quantum system, the conjugate fields are taken as real and ...

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 103 whereψ= (ψ 1 ,ψ 2 )T, andψk=ψk^1 +iψ^2 k( 1 ≤k≤ 2 ). The Hamilton energyHof (2. ...

104 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS whereHis given by (2.6.55). 4.Klein-Gordon equation: (2.6.56) 1 c^2 ∂^2 ψ ∂t^2 ...

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 105 whereHreads H= ∫ R^3 H(ψ,Dψ)dx. When taking the variation, the functional ̃L= ∫ ...

106 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS ...