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"quote": "位置-速度空间与相空间之间的变换称为[Legendre变换(Legendre transform)](../../../../../concepts/mechanics/legendre-transform)，在良好情形下（例如当 $L$ 对所有速度都是二次的时），它是一个同构。不过一般而言，这并不是同构，勒让德变换常常将位置-速度空间映到相空间的一个较低维子空间。这类情形并不罕见，并且即使作为经典力学系统，也需要复杂得多的形式体系（这一主题称为“[约束哈密顿动力学(constrained Hamiltonian dynamics)](../../../../../concepts/mechanics/constrained-hamiltonian-dynamics)”）。我们将在第 46 章研究的一个重要例子是自由电磁场，其运动方程为麦克斯韦方程。在这种情况下，位形空间坐标是矢势的分量 $\\left( A _ { 0 } , A _ { 1 } , A _ { 2 } , A _ { 3 } \\right)$，问题之所以出现，是因为拉格朗日量不依赖于 $\\dot { A } _ { 0 }$",
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"quote": "between position-velocity and phase space is known as the [Legendre transform](../../../../../concepts/mechanics/legendre-transform), and in good cases (for instance when $L$ is quadratic in all the velocities) it is an isomorphism. In general though, this is not an isomorphism, with the Legendre transform often taking position-velocity space to a lower dimensional subspace of phase space. Such cases are not unusual and require a much more complicated formalism, even as classical mechanical systems (this subject is known as “[constrained Hamiltonian dynamics](../../../../../concepts/mechanics/constrained-hamiltonian-dynamics)”). One important example we will study in chapter 46 is that of the free electromagnetic field, with equations of motion the Maxwell equations. In that case the configuration space coordinates are the components $\\left( A _ { 0 } , A _ { 1 } , A _ { 2 } , A _ { 3 } \\right)$ of the vector potential, with the problem arising because the Lagrangian does not depend on $\\dot { A } _ { 0 }$",
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