7.6: Kinetic Energy in Generalized Coordinates

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Application of Noether’s theorem to the conservation of energy requires the kinetic energy to be expressed in generalized coordinates. In terms of fixed rectangular coordinates, the kinetic energy for $$N$$ bodies, each having three degrees of freedom, is expressed as

$T=\frac{1}{2}\sum_{\alpha =1}^{N}\sum_{i=1}^{3}m_{\alpha }\dot{x}_{\alpha ,i}^{2}\label{7.17}$

These can be expressed in terms of generalized coordinates as $$x_{\alpha ,i}=x_{\alpha ,i}(q_{j},t)$$ and in terms of generalized velocities

$\dot{x}_{\alpha ,i}=\sum_{j=1}^{s}\frac{\partial x_{\alpha ,i}}{\partial q_{j}}\dot{q}_{j}+\frac{\partial x_{\alpha ,i}}{\partial t}$ Taking the square of $$\dot{x}_{\alpha ,i}$$ and inserting into the kinetic energy relation gives

$T(\mathbf{q},\mathbf{\dot{q}},t)=\sum_{\alpha }\sum_{i,j,k}\frac{1}{2} m_{\alpha }\frac{\partial x_{\alpha ,i}}{\partial q_{j}}\frac{\partial x_{\alpha ,i}}{\partial q_{k}}\dot{q}_{j}\dot{q}_{k}+\sum_{\alpha }\sum_{i,j}m_{\alpha }\frac{\partial x_{\alpha ,i}}{\partial q_{j}}\frac{ \partial x_{\alpha ,i}}{\partial t}\dot{q}_{j}+\sum_{\alpha }\sum_{i}\frac{1 }{2}m_{\alpha }\left( \frac{\partial x_{\alpha ,i}}{\partial t}\right) ^{2}$ This can be abbreviated as

$T(\mathbf{q},\mathbf{\dot{q}},t)=T_{2}(\mathbf{q},\mathbf{\dot{q}},t)+T_{1}( \mathbf{q},\mathbf{\dot{q}},t)+T_{0}(\mathbf{q},t)$

where

\begin{align} \label{7.21} T_{2}(\mathbf{q},\mathbf{\dot{q}},t) &=&\sum_{\alpha }\sum_{i,j,k}\frac{1}{2} m_{\alpha }\frac{\partial x_{\alpha ,i}}{\partial q_{j}}\frac{\partial x_{\alpha ,i}}{\partial q_{k}}\dot{q}_{j}\dot{q}_{k}=\sum_{j,k}a_{jk}\dot{q} _{j}\dot{q}_{k} \\ T_{1}(\mathbf{q},\mathbf{\dot{q}},t) &=&\sum_{\alpha }\sum_{i,j}m_{\alpha } \frac{\partial x_{\alpha ,i}}{\partial q_{j}}\frac{\partial x_{\alpha ,i}}{ \partial t}\dot{q}_{j}=\sum_{j,k}b_{j}\dot{q}_{j} \\ T_{0}(\mathbf{q},t) &=&\sum_{\alpha }\sum_{i}\frac{1}{2}m_{\alpha }\left( \frac{\partial x_{\alpha ,i}}{\partial t}\right) ^{2}\end{align}

where $a_{jk}\equiv \sum_{\alpha =1}^{n}\sum_{i,=1}^{3}\frac{1}{2}m_{\alpha }\frac{ \partial x_{\alpha ,i}}{\partial q_{j}}\frac{\partial x_{\alpha ,i}}{ \partial q_{k}}$

When the transformed system is scleronomic, time does not appear explicitly in the transformation equations to generalized coordinates since $$\frac{\partial x_{\alpha ,i}}{\partial t}=0$$. Then $$T_{1}=T_{0}=0$$, and the kinetic energy reduces to a homogeneous quadratic function of the generalized velocities $T(\mathbf{q},\mathbf{ \dot{q}},t)=T_{2}(\mathbf{q},\mathbf{\dot{q}},t) \label{7.25}$

A useful relation can be derived by taking the differential of Equation \ref{7.21} with respect to $$\dot{q}_{l}$$. That is

$\frac{\partial T_{2}(\mathbf{q},\mathbf{\dot{q}},t)}{\partial \dot{q}_{l}} =\sum_{k}a_{lk}\dot{q}_{k}+\sum_{j}a_{jl}\dot{q}_{j}$

Multiply this by $$\dot{q}_{l}$$ and sum over $$l$$ gives

$\sum_{l}\dot{q}_{l}\frac{\partial T_{2}(\mathbf{q},\mathbf{\dot{q}},t)}{ \partial \dot{q}_{l}}=\sum_{k,l}a_{lk}\dot{q}_{k}\dot{q}_{l}+\sum_{j,l}a_{jl} \dot{q}_{j}\dot{q}_{l}=2\sum_{j,k}a_{lk}\dot{q}_{k}\dot{q}_{l}=2T_{2}$

Similarly, the products of the generalized velocities $$\dot{q},$$ with the corresponding derivatives of $$T_{1}$$ and $$T_{0}$$ give \begin{align} \label{7.27} \sum_{l}\dot{q}_{l}\frac{\partial T_{2}}{\partial \dot{q}_{l}} &=&2T_{2} \\ \sum_{l}\dot{q}_{l}\frac{\partial T_{1}(\mathbf{q},\mathbf{\dot{q}},t)}{ \partial \dot{q}_{l}} &=&T_{1}(\mathbf{q},\mathbf{\dot{q}},t) \\ \sum_{l}\dot{q}_{l}\frac{\partial T_{0}(\mathbf{q},t)}{\partial \dot{q}_{l}} &=&0\end{align}

Equation \ref{7.25} gives that $$T=T_{2}$$ when the transformed system is scleronomic, i.e. $$\frac{\partial x_{\alpha ,i}}{\partial t}=0,$$ and then the kinetic energy is a quadratic function of the generalized velocities $$\dot{q}_{j}$$. Using the definition of the generalized momentum equation $$(7.2.3)$$, assuming $$T=T_{2}$$, and that the potential $$U$$ is velocity independent, gives that

$p_{l}\equiv \frac{\partial L}{\partial \dot{q}_{l}}=\frac{\partial T}{\partial \dot{q} _{l}}-\frac{\partial U}{\partial \dot{q}_{l}}=\frac{\partial T_{2}}{\partial \dot{q}_{l}}$

Then Equation \ref{7.27} reduces to the useful relation that

$T_{2}=\frac{1}{2}\sum_{l}\dot{q}_{l}p_{l}=\frac{1}{2}\mathbf{\dot{q}\cdot p}$

where, for compactness, the summation is abbreviated as a scalar product.

This page titled 7.6: Kinetic Energy in Generalized Coordinates is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.