$$ \begin{aligned} \vec{A}^{\prime} &= \vec{A} + \nabla \psi \\ \phi^{\prime} &= \phi - \frac{\partial \psi}{\partial t} \\ \text{Where } \psi &= \psi(\vec{r}, t) \end{aligned} $$

The Lagrangian for a charge particle is: $$ \begin{aligned} L=T-V & =\frac{1}{2} m v^{2}-\left(V_{\text {electric }}+V_{\text {magnetic }}\right) \\ & =\frac{1}{2} m v^{2}-(q \phi+-q(\vec{v} \cdot \vec{A})) \\ & =\frac{1}{2} m v^{2}-q \phi+q(\vec{v} \cdot \vec{A}) \end{aligned} $$

Under Gauge transformations: $$ (\vec{A}, \phi) \rightarrow (\vec{A'}, \phi') $$

we get a new Lagrangian as $$ \begin{aligned} L'&= \frac{1}{2} m v^2 - q \phi' + q(\vec{v} \cdot \vec{A'})\\ & =\frac{1}{2} m v^{2}-q\left(\phi-\frac{\partial \psi}{\partial t}\right)+q(\vec{v} \cdot(\vec{A}+\nabla \psi)) \\ & =\frac{1}{2} m v^{2}-q \phi+q(\vec{v} \cdot \vec{A})+q \frac{\partial \psi}{\partial t}+q(\vec{v} \cdot \nabla \psi) \\ & =L+q\left(\frac{\partial \psi}{\partial t}+\vec{v} \cdot \nabla \psi\right) \\ & =L+q\left(\vec{v} \cdot \nabla \psi+\frac{\partial \psi}{\partial t}\right) \\ & =L+q\left( \nabla\psi\cdot\vec{v} +\frac{\partial \psi}{\partial t}\right) \\ & =L+q\left(\frac{\partial \psi}{\partial r} \cdot \vec{r}+\frac{\partial \psi}{\partial t}\right) \\ & =L+q\left(\frac{\partial \psi}{\partial r} \cdot \frac{d \vec{r}}{d t}+\frac{\partial \psi}{\partial t} \frac{d t}{d t}\right) \\ & =L+q\left(\frac{d \psi}{d t}\right)=L+ \frac{d}{d t}(q \psi)\\ \text{Hence }\,\,L' &=L+\frac{d F}{d t}\\ \end{aligned} $$

\[\begin{align} & L=\frac{1}{2}m{{v}^{2}}-q\phi +q\left( \vec{v}\cdot \vec{A} \right) \\ & =\frac{1}{2}m\left( v_{x}^{2}+v_{y}^{2}+v_{z}^{2} \right)-q\phi +q\left( {{v}_{x}}{{A}_{x}}+{{v}_{y}}{{A}_{y}}+{{v}_{z}}{{A}_{z}} \right) \\ & \text{The x-component of Lagrange equation of motion gives:} \\ & \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}} \right)-\frac{\partial L}{\partial x}=0 \\ & \Rightarrow \frac{d}{dt}\left( \frac{\partial L}{\partial {{v}_{x}}} \right)-\frac{\partial L}{\partial x}=0 \\ & \Rightarrow \frac{d}{dt}\left( m{{v}_{x}}+q{{A}_{x}} \right)-\left( -q\frac{\partial \phi }{\partial x}+q\left( {{v}_{x}}\frac{\partial {{A}_{x}}}{\partial x}+{{v}_{y}}\frac{\partial {{A}_{y}}}{\partial x}+{{v}_{z}}\frac{\partial {{A}_{z}}}{\partial x} \right) \right)=0 \\ & \Rightarrow m{{a}_{x}}+q\frac{d{{A}_{x}}}{dt}=-q\frac{\partial \phi }{\partial x}+q\left( {{v}_{x}}\frac{\partial {{A}_{x}}}{\partial x}+{{v}_{y}}\frac{\partial {{A}_{y}}}{\partial x}+{{v}_{z}}\frac{\partial {{A}_{z}}}{\partial x} \right) \\ & \Rightarrow m{{a}_{x}}=q\left( -\frac{\partial \phi }{\partial x}-\frac{d{{A}_{x}}}{dt} \right)+q\left( {{v}_{x}}\frac{\partial {{A}_{x}}}{\partial x}+{{v}_{y}}\frac{\partial {{A}_{y}}}{\partial x}+{{v}_{z}}\frac{\partial {{A}_{z}}}{\partial x} \right) \\ & =q\left( -\frac{\partial \phi }{\partial x}-\left( \frac{\partial {{A}_{x}}}{\partial x}\frac{dx}{dt}+\frac{\partial {{A}_{x}}}{\partial y}\frac{dy}{dt}+\frac{\partial {{A}_{x}}}{\partial z}\frac{dz}{dt}+\frac{\partial {{A}_{x}}}{\partial t}\frac{dt}{dt} \right) \right)+q\left( {{v}_{x}}\frac{\partial {{A}_{x}}}{\partial x}+{{v}_{y}}\frac{\partial {{A}_{y}}}{\partial x}+{{v}_{z}}\frac{\partial {{A}_{z}}}{\partial x} \right) \\ & =q\left( -\frac{\partial \phi }{\partial x}-\left( \frac{\partial {{A}_{x}}}{\partial x}\dot{x}+\frac{\partial {{A}_{x}}}{\partial y}\dot{y}+\frac{\partial {{A}_{x}}}{\partial z}\dot{z}+\frac{\partial {{A}_{x}}}{\partial t} \right) \right)+q\left( {{v}_{x}}\frac{\partial {{A}_{x}}}{\partial x}+{{v}_{y}}\frac{\partial {{A}_{y}}}{\partial x}+{{v}_{z}}\frac{\partial {{A}_{z}}}{\partial x} \right) \\ & =q\left( -\frac{\partial \phi }{\partial x}-\left( \frac{\partial {{A}_{x}}}{\partial x}{{v}_{x}}+\frac{\partial {{A}_{x}}}{\partial y}{{v}_{y}}+\frac{\partial {{A}_{x}}}{\partial z}{{v}_{z}}+\frac{\partial {{A}_{x}}}{\partial t} \right) \right)+q\left( {{v}_{x}}\frac{\partial {{A}_{x}}}{\partial x}+{{v}_{y}}\frac{\partial {{A}_{y}}}{\partial x}+{{v}_{z}}\frac{\partial {{A}_{z}}}{\partial x} \right) \\ & =q\left( -\frac{\partial \phi }{\partial x}-\frac{\partial {{A}_{x}}}{\partial t} \right)-q\left( \frac{\partial {{A}_{x}}}{\partial x}{{v}_{x}}+\frac{\partial {{A}_{x}}}{\partial y}{{v}_{y}}+\frac{\partial {{A}_{x}}}{\partial z}{{v}_{z}} \right)+q\left( {{v}_{x}}\frac{\partial {{A}_{x}}}{\partial x}+{{v}_{y}}\frac{\partial {{A}_{y}}}{\partial x}+{{v}_{z}}\frac{\partial {{A}_{z}}}{\partial x} \right) \\ & =q{{E}_{x}}-q\left( \frac{\partial {{A}_{x}}}{\partial y}{{v}_{y}}+\frac{\partial {{A}_{x}}}{\partial z}{{v}_{z}}-{{v}_{y}}\frac{\partial {{A}_{y}}}{\partial x}-{{v}_{z}}\frac{\partial {{A}_{z}}}{\partial x} \right) \\ & =q{{E}_{x}}-q\left( {{v}_{y}}\left( \frac{\partial {{A}_{x}}}{\partial y}-\frac{\partial {{A}_{y}}}{\partial x} \right)+{{v}_{z}}\left( \frac{\partial {{A}_{x}}}{\partial z}-\frac{\partial {{A}_{z}}}{\partial x} \right) \right) \\ & =q{{E}_{x}}+q\left( {{v}_{y}}\left( \frac{\partial {{A}_{y}}}{\partial x}-\frac{\partial {{A}_{x}}}{\partial y} \right)+{{v}_{z}}\left( \frac{\partial {{A}_{z}}}{\partial x}-\frac{\partial {{A}_{x}}}{\partial z} \right) \right) \\ & =q{{E}_{x}}+q\left( {{v}_{y}}{{\left( \nabla \times \vec{A} \right)}_{z}}-{{v}_{z}}{{\left( \nabla \times \vec{A} \right)}_{y}} \right) \\ & =q{{E}_{x}}+q{{\left( \vec{v}\times \left( \nabla \times \vec{A} \right) \right)}_{x}} \\ & =q{{E}_{x}}+q{{\left( \vec{v}\times \vec{B} \right)}_{x}} \\ & \because \\ & \left[ \nabla \times \vec{A}=\left( \begin{matrix} i & j & k \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ {{A}_{x}} & {{A}_{y}} & {{A}_{z}} \\ \end{matrix} \right)=\left( \frac{\partial {{A}_{z}}}{\partial y}-\frac{\partial {{A}_{y}}}{\partial z} \right)\hat{i}+\left( \frac{\partial {{A}_{x}}}{\partial z}-\frac{\partial {{A}_{z}}}{\partial x} \right)\hat{j}+\left( \frac{\partial {{A}_{y}}}{\partial x}-\frac{\partial {{A}_{x}}}{\partial y} \right)\hat{k} \right] \\ & \And \\ & \vec{v}\times \left( \nabla \times \vec{A} \right)=\left( \begin{matrix} i & j & k \\ {{v}_{x}} & {{v}_{y}} & {{v}_{z}} \\ {{\left( \nabla \times \vec{A} \right)}_{x}} & {{\left( \nabla \times \vec{A} \right)}_{y}} & {{\left( \nabla \times \vec{A} \right)}_{z}} \\ \end{matrix} \right) \\ & \Rightarrow {{\left[ \vec{v}\times \left( \nabla \times \vec{A} \right) \right]}_{x}}={{v}_{y}}{{\left( \nabla \times \vec{A} \right)}_{z}}-{{v}_{z}}{{\left( \nabla \times \vec{A} \right)}_{y}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\left( \nabla \times \vec{A} \right)}_{x}}\hat{i}+{{\left( \nabla \times \vec{A} \right)}_{y}}\hat{j}+{{\left( \nabla \times \vec{A} \right)}_{z}}\hat{k} \\ \end{align}\]

\[\begin{align} {{f}_{xi}} &=-\frac{\partial R}{\partial {{v}_{xi}}},{{f}_{yi}}=-\frac{\partial R}{\partial {{v}_{yi}}},{{f}_{zi}}=-\frac{\partial R}{\partial {{v}_{zi}}} \\ \Rightarrow \vec{f} &={{f}_{xi}}\hat{i}+{{f}_{yi}}\hat{j}+{{f}_{zi}}\hat{k} \\ & =-\left( \frac{\partial R}{\partial {{v}_{xi}}}\hat{i}+\frac{\partial R}{\partial {{v}_{yi}}}\hat{j}+\frac{\partial R}{\partial {{v}_{zi}}}\hat{k} \right) \\ \Rightarrow G_{k}^{'} &=\vec{f}\cdot \frac{\partial {{{\bar{r}}}_{i}}}{\partial {{q}_{k}}}=\left( {{f}_{xi}}\hat{i}+{{f}_{yi}}\hat{j}+{{f}_{zi}}\hat{k} \right)\cdot \frac{\partial {{{\dot{\vec{r}}}}_{i}}}{\partial {{{\dot{q}}}_{k}}} \\ & =-\left( \frac{\partial R}{\partial {{v}_{xi}}}\hat{i}+\frac{\partial R}{\partial {{v}_{yi}}}\hat{j}+\frac{\partial R}{\partial {{v}_{zi}}}\hat{k} \right)\cdot \frac{\partial {{{\dot{\vec{r}}}}_{i}}}{\partial {{{\dot{q}}}_{k}}} \\ & =-\left( \frac{\partial R}{\partial {{v}_{xi}}}\frac{\partial {{{\dot{x}}}_{i}}}{\partial {{{\dot{q}}}_{k}}}+\frac{\partial R}{\partial {{v}_{yi}}}\frac{\partial {{{\dot{y}}}_{i}}}{\partial {{{\dot{q}}}_{k}}}+\frac{\partial R}{\partial {{v}_{zi}}}\frac{\partial {{{\dot{z}}}_{i}}}{\partial {{{\dot{q}}}_{k}}} \right) \\ & =-\left( \frac{\partial R}{\partial {{v}_{xi}}}\frac{\partial {{v}_{xi}}}{\partial {{{\dot{q}}}_{k}}}+\frac{\partial R}{\partial {{v}_{yi}}}\frac{\partial {{v}_{yi}}}{\partial {{{\dot{q}}}_{k}}}+\frac{\partial R}{\partial {{v}_{zi}}}\frac{\partial {{v}_{zi}}}{\partial {{{\dot{q}}}_{k}}} \right) \\ & =-\frac{\partial R}{\partial {{{\dot{q}}}_{k}}} \\ & \Rightarrow \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{k}}} \right)-\left( \frac{\partial L}{\partial {{q}_{k}}} \right)=G_{k}^{'} \\ & \Rightarrow \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{k}}} \right)-\left( \frac{\partial L}{\partial {{q}_{k}}} \right)=-\frac{\partial R}{\partial {{{\dot{q}}}_{k}}} \\ & \Rightarrow \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{k}}} \right)-\left( \frac{\partial L}{\partial {{q}_{k}}} \right)+\frac{\partial R}{\partial {{{\dot{q}}}_{k}}}=0 \\ \end{align}\]

\begin{equation} \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{k}}} \right)-\left( \frac{\partial L}{\partial {{q}_{k}}} \right)+\frac{\partial R}{\partial {{{\dot{q}}}_{k}}}=0 \label{eq:routhian2} \end{equation}

\[\begin{align} & \frac{d}{dt}\left( \frac{\partial T}{\partial {{{\dot{q}}}_{k}}} \right)-\frac{\partial T}{\partial {{q}_{k}}}={{G}_{k}}=\left( {{{\vec{F}}}_{\text{Cons}}}+{{{\vec{F}}}_{\text{Non-Cons}}} \right)\cdot \left( \frac{\partial \vec{r}}{\partial {{q}_{k}}} \right)=-\frac{\partial V}{\partial {{q}_{k}}}+{{F}_{\text{Non-Cons}}} \\ & ={{{\vec{F}}}_{\text{Cons}}}\cdot \left( \frac{\partial \vec{r}}{\partial {{q}_{k}}} \right)+{{{\vec{F}}}_{\text{Non-Cons}}}\cdot \left( \frac{\partial \vec{r}}{\partial {{q}_{k}}} \right) \\ & =-\frac{\partial V}{\partial {{q}_{k}}}+G_{k}^{'} \\ & \Rightarrow \frac{d}{dt}\left( \frac{\partial \left( T-V \right)}{\partial {{{\dot{q}}}_{k}}} \right)-\frac{\partial \left( T-V \right)}{\partial {{q}_{k}}}=G_{k}^{'} \\ & \Rightarrow \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{k}}} \right)-\frac{\partial L}{\partial {{q}_{k}}}=G_{k}^{'} \\ \end{align}\]

\begin{align*} dW &=-{{{\vec{f}}}_{i}}\cdot d\vec{r} \\ & =-{{{\vec{f}}}_{i}}\cdot {{{\vec{v}}}_{i}}dt \\ & ={{k}_{i}}{{{\vec{v}}}_{i}}\cdot {{{\vec{v}}}_{i}}dt \\ & =2\sum{{{k}_{i}}v_{i}^{2}}dt \\ \Rightarrow\qquad R&= \sum{{{k}_{i}}v_{i}^{2}}=\frac{1}{2}\frac{dW}{dt} \\ \end{align*}

\[\begin{align} & L'=L+\frac{dF}{dt} \\ & \Rightarrow \dfrac{d}{dt}\left( \dfrac{\partial L'}{\partial {{{\dot{q}}}_{k}}} \right)-\frac{\partial L'}{\partial {{q}_{k}}} \\ & =\frac{d}{dt}\left( \frac{\partial }{\partial {{{\dot{q}}}_{k}}}\left( L+\frac{dF}{dt} \right) \right)-\frac{\partial }{\partial {{q}_{k}}}\left( L+\frac{dF}{dt} \right) \\ & =\frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{k}}} \right)+\frac{d}{dt}\left( \frac{\partial }{\partial {{{\dot{q}}}_{k}}}\frac{dF}{dt} \right)-\frac{\partial L}{\partial {{q}_{k}}}-\frac{\partial }{\partial {{q}_{k}}}\frac{dF}{dt} \\ & =\frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{k}}} \right)-\frac{\partial L}{\partial {{q}_{k}}}+\frac{d}{dt}\left( \frac{\partial }{\partial {{{\dot{q}}}_{k}}}\left( \frac{dF}{dt} \right) \right)-\frac{\partial }{\partial {{q}_{k}}}\left( \frac{dF}{dt} \right) \\ & =0+\frac{d}{dt}\left( \frac{\partial }{\partial {{{\dot{q}}}_{k}}}\left( \frac{\partial F}{\partial {{q}_{k}}}{{{\dot{q}}}_{k}}+\frac{\partial F}{\partial t} \right) \right)-\frac{\partial }{\partial {{q}_{k}}}\left( \frac{dF}{dt} \right) \\ & =\frac{d}{dt}\left( \frac{\partial }{\partial {{{\dot{q}}}_{k}}}\left( \frac{\partial F}{\partial {{q}_{k}}}{{{\dot{q}}}_{k}} \right)+\frac{\partial }{\partial {{{\dot{q}}}_{k}}}\left( \frac{\partial F}{\partial t} \right) \right)-\frac{\partial }{\partial {{q}_{k}}}\left( \frac{dF}{dt} \right) \\ & =\frac{d}{dt}\left( \frac{\partial }{\partial {{{\dot{q}}}_{k}}}\left( \frac{\partial F}{\partial {{q}_{k}}} \right){{{\dot{q}}}_{k}}+\frac{\partial F}{\partial {{q}_{k}}}\frac{\partial {{{\dot{q}}}_{k}}}{\partial {{{\dot{q}}}_{k}}}+\frac{\partial }{\partial {{{\dot{q}}}_{k}}}\left( \frac{\partial F}{\partial t} \right) \right)-\frac{\partial }{\partial {{q}_{k}}}\left( \frac{dF}{dt} \right) \\ & =\frac{d}{dt}\left( \frac{\partial }{\partial {{q}_{k}}}\left( \frac{\partial F}{\partial {{{\dot{q}}}_{k}}} \right){{{\dot{q}}}_{k}}+\frac{\partial F}{\partial {{q}_{k}}}+\frac{\partial }{\partial t}\left( \frac{\partial F}{\partial {{{\dot{q}}}_{k}}} \right) \right)-\frac{\partial }{\partial {{q}_{k}}}\left( \frac{dF}{dt} \right) \\ & =\frac{d}{dt}\left( \frac{\partial }{\partial {{q}_{k}}}\left( 0 \right){{{\dot{q}}}_{k}}+\frac{\partial F}{\partial {{q}_{k}}}+\frac{\partial }{\partial t}\left( 0 \right) \right)-\frac{\partial }{\partial {{q}_{k}}}\left( \frac{dF}{dt} \right) \\ & =\frac{d}{dt}\left( \frac{\partial F}{\partial {{q}_{k}}} \right)-\frac{\partial }{\partial {{q}_{k}}}\left( \frac{dF}{dt} \right) \\ & =\frac{d}{dt}\left( \frac{\partial F}{\partial {{q}_{k}}} \right)-\frac{d}{dt}\left( \frac{\partial F}{\partial {{q}_{k}}} \right) \\ & =0 \end{align}\]

\[\begin{align} L'&=\dfrac{1}{2}mv{{'}^{2}}-U \\ & =\dfrac{1}{2}m{{\left( v-V \right)}^{2}}-U \\ & =\dfrac{1}{2}m\left( {{v}^{2}}+{{V}^{2}}-2vV \right)-U \\ & =\dfrac{1}{2}m{{v}^{2}}-U+\frac{1}{2}m{{V}^{2}}-\dfrac{1}{2}mvV \\ & \dfrac{d}{dt}\left( \dfrac{\partial L'}{\partial v} \right)-\dfrac{\partial L'}{\partial x}=\dfrac{d}{dt}\left( mv-\dfrac{1}{2}mV \right)-\dfrac{\partial L'}{\partial x} \\ & =\dfrac{d}{dt}\left( mv-\dfrac{1}{2}mV \right)-\left( -\dfrac{\partial U}{\partial x} \right) \\ & = \\ & L=\dfrac{1}{2}m{{v}^{2}}-U \\ & =\dfrac{d}{dt}\left( \dfrac{\partial L}{\partial v} \right)-\dfrac{\partial L}{\partial x} \\ \end{align}\]