Standard Lagrangian

Lagrangian mechanics was based on the concepts of kinetic and protential energy.

$$ L=T-V $$

d’Almbert’s principle of virtual work → LM→ standard Lagrangian$L(\mathbf{q},\mathbf{\dot{q}},t)=T(\mathbf{\dot{q}},t)-U(\mathbf{q},t)$→ (defining the action functional) E-L variational equations

$$ \left\{\frac d{dt}\left(\frac{\partial L}{\partial\dot{q}j}\right)-\frac{\partial L}{\partial q_j}\right\}=Q_j^{EXC}+\sum{k=1}^m\lambda_k\frac{\partial g_k}{\partial q_j}(\mathbf{q},t) $$

the right terms extend the range of validity of using standard Legrangian in the E-L equations by introducing constraint 约束（标势约束→ 一般约束） and omitted forces（非完整约束→ 完整约束） explicitly显式地

Hamilton’s principle → LM/HM → Lagrangians statify Hamilton’s Action Principle.{standard/non-standard Lagrangian }→ E-L equation→ correct equations of motion

Guage invariance 规范不变性 of standard Lagrangian

there is a continuous specturm of equvalent standard Lagrangian that all lead to identical equatons of motion. That is beacuse $L$ is a scaler quantity that is invariant to coordiante transformations.

The following transformation change the standard Lagrangian, but leave the equation of motion unchanged:[equivalent Lagrangians]

1. addition of a constant to the scalar potetial.

2. addition of a constant kinetic energy

3. addition of a total time derivative $L_2\to L_1+\frac d{dt}[\Lambda(q_i,t)]$, for any differentiable function $\Lambda(q_i,t)$ of generalized coordinates plus time, that has continuous second derivatives.

   $$ \begin{aligned} L_2(\mathbf{q},\dot{q},t) &=L_1(\mathbf{q},\dot{q},t) +\frac{d\Lambda(\mathbf{q},t)}{dt} \\&=L_1(\mathbf{q},\dot{q},t) +\left(\frac{\partial\Lambda(\mathbf{q},t)}{\partial q_j}\dot{q}_j +\frac{\partial\Lambda(\mathbf{q},t)}{\partial t}\right) \\\frac d{dt}\left(\frac{\partial L_2}{\partial\dot{q}_j}\right) -\frac{\partial L_2}{\partial q_j} & =\frac d{dt}\left(\frac{\partial L_1}{\partial\dot{q}_j}\right) +\frac{\partial^2\Lambda(\mathbf{q},t)}{\partial q_j^2}\dot q_j +\frac{\partial^2\Lambda(\mathbf{q},t)}{\partial t\partial q_j}\\ &\quad -\frac{\partial L_1}{\partial q_j} -\frac{\partial^2\Lambda(\mathbf{q},t)}{\partial q_j^2}\dot q_j -\frac{\partial^2\Lambda(\mathbf{q},t)}{\partial t\partial q_j}
   \\ &=\frac d{dt}{\left(\frac{\partial L_1}{\partial\dot{q}_j}\right)} -\frac{\partial L_1}{\partial q_j} \end{aligned} $$

There is no unique choice amog the wide range of equivalent standard Lagrangians expressed in terms of generalized coorfinates. This discussion is an example of gauge invariance in physics.

Modern theories in physicd: potetial fields→ a property of field theory: gauge symmetry.

stactic electric potential field 静电场 gravitational potential field 重力场： $\varphi +C$

both classical and quantal 量子manifestations显现 of field theory, the basis of the Standard Model of eclectroweak and strong interactions.

Gauge theories constrain the laws of physics in that the impact of gauge transformations must cancel out when expressed in terms of the observables.