Abstract
Here we show that for systems of particles subject to holonomic constraints and forces derivable from a generalized potential, Lagrange’s equations can also be obtained by finding the stationary values of a functional, called the “action.” By doing so, we convert the problem of finding the equations of motion to a problem in the calculus of variations. We also introduce the Hamiltonian for a system of particles, which can be obtained from the Lagrangian by performing a Legendre transformation. This process leads to Hamilton’s equations of motion, which are 1st-order differential equations for the generalized coordinates and momenta. The Hamiltonian formulation is particularly well-suited for illustrating the intimate connection between continuous symmetries of the system and conserved quantities.
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Notes
- 1.
If you’ve had any exposure to quantum mechanics, you may recall another quantity with the same dimensions as the action. Planck’s constant \(h = 6.626\times 10^{-34}~\mathrm{J}\cdot \mathrm{s}\), which appears in quantum mechanics, is also called the “quantum of action”.
- 2.
The underbar on the index \(\underline{a}\) in conservation law I/II indicates that this is a particular (i.e., single) value of the index; it should not be thought of as a placeholder for all possible values, as the index a without an underbar usually represents.
- 3.
We are abusing notation slightly in (3.38), writing the determinant of the matrix of 2nd partial derivatives of F with respect to the \(v^i\) as the determinant of the matrix components. We will occasionally do this whenever the abstract matrix notation is more cumbersome or less informative than the matrix component notation.
- 4.
Basically F, G, u, v, w defined in the previous section are replaced by L, H, q, \(\dot{q}\), p; and the possible explicit dependence on the time t just goes along for the ride.
- 5.
Semi-holonomic constraints are described in more detail in Problem 3.2.
- 6.
By “mathematical structure” we simply mean an operation or rule acting on the elements of a set. For example, the dot product of two vectors is an additional mathematical structure on the space of vectors.
- 7.
We will not consider transformations that simply rescale the integrand, such as \(Q^a = q^a\), \(P_a = \lambda p_a\), where \(\lambda \) is a constant, which also preserve Hamilton’s equations with \(H'=\lambda H\). See Goldstein et al. (2002) for more information about such scaling transformations.
- 8.
In (3.105) and in all subsequent relevant equations, we ignore terms that are 2nd-order or higher in \(\lambda \). Also, we write the derivatives of q and p as ordinary derivatives with respect to \(\lambda \) (and not partial derivatives), as we are treating \(q_0\) and \(p_0\) as fixed parameters in the expressions \(q(q_0,p_0,\lambda )\) and \(p(q_0,p_0,\lambda )\).
- 9.
Recall that the commutator of two operators \(\hat{A}\) and \(\hat{B}\) is defined by \([\hat{A},\hat{B}]\equiv \hat{A}\hat{B}-\hat{B}\hat{A}\). In general, operators do not commute, i.e., \(\hat{A}\hat{B}\ne \hat{B}\hat{A}\).
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Benacquista, M.J., Romano, J.D. (2018). Hamilton’s Principle and Action Integrals. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68780-3_3
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DOI: https://doi.org/10.1007/978-3-319-68780-3_3
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