Question

1. Lagrange bracket. The Lagrange bracket of u and v with respect to the (q,p) variables is defined as \begin{equation} \{u, v\}_{q,p} = \frac{\partial q_i}{\partial u}\frac{\partial p_i}{\partial v} - \frac{\partial p_i}{\partial u}\frac{\partial q_i}{\partial v}, \end{equation} or in matrix notation, \begin{equation} \{u, v\}_{\eta} = \frac{\partial \eta^T}{\partial u}\frac{\partial \eta}{\partial v}, \end{equation} where $\eta = (q_i, p_i)$ and $\gamma_{ij} = \delta_{i,j+n} - \delta_{j,i-n}$. Imagine $\eta$ as a column vector and $\eta^T$ as a row vector when using matrix notation. (a) Show the canonical invariance of the Lagrange bracket by proving $\{u, v\}_{\eta} = \{u, v\}_{\zeta}$ for $M_{ij} = \frac{\partial \zeta_i}{\partial \eta_j}$ and $M\gamma M^T = \gamma$. (1 point) (b) Prove the following fundamental Lagrange brackets: (1 point) \begin{equation} \{q_i, q_j\} = 0 = \{p_i, p_j\} \quad \text{and} \quad \{q_i, p_j\} = \delta_{ij}, \end{equation} or in matrix notation, \begin{equation} \{\eta, \eta\} = \gamma. \end{equation} (c) Show that the Lagrange bracket and Poisson bracket satisfy the following relation: (1 point) \begin{equation} \{u, u\}[u, u] = -1 \quad \text{or} \quad \sum_{k=1}^{2n} \{u_k, u\}[u_k, u_j] = \delta_{ij}. \end{equation}

          1. Lagrange bracket. The Lagrange bracket of u and v with respect to the (q,p) variables
is defined as
\begin{equation}
\{u, v\}_{q,p} = \frac{\partial q_i}{\partial u}\frac{\partial p_i}{\partial v} - \frac{\partial p_i}{\partial u}\frac{\partial q_i}{\partial v},
\end{equation}
or in matrix notation,
\begin{equation}
\{u, v\}_{\eta} = \frac{\partial \eta^T}{\partial u}\frac{\partial \eta}{\partial v},
\end{equation}
where $\eta = (q_i, p_i)$ and $\gamma_{ij} = \delta_{i,j+n} - \delta_{j,i-n}$. Imagine $\eta$ as a column vector and $\eta^T$ as a row
vector when using matrix notation.
(a) Show the canonical invariance of the Lagrange bracket by proving $\{u, v\}_{\eta} = \{u, v\}_{\zeta}$
for $M_{ij} = \frac{\partial \zeta_i}{\partial \eta_j}$ and $M\gamma M^T = \gamma$. (1 point)
(b) Prove the following fundamental Lagrange brackets: (1 point)
\begin{equation}
\{q_i, q_j\} = 0 = \{p_i, p_j\} \quad \text{and} \quad \{q_i, p_j\} = \delta_{ij},
\end{equation}
or in matrix notation,
\begin{equation}
\{\eta, \eta\} = \gamma.
\end{equation}
(c) Show that the Lagrange bracket and Poisson bracket satisfy the following relation:
(1 point)
\begin{equation}
\{u, u\}[u, u] = -1 \quad \text{or} \quad \sum_{k=1}^{2n} \{u_k, u\}[u_k, u_j] = \delta_{ij}.
\end{equation}

$1. Lagrange bracket. The Lagrange bracket of u and v with respect to the (q,p) variables is defined as {u, v}q,p = (∂ qi)/(∂ u)(∂ pi)/(∂ v) - (∂ pi)/(∂ u)(∂ qi)/(∂ v), or in matrix notation, {u, v}η = (∂η^T)/(∂ u)(∂η)/(∂ v), where η = (qi, pi) and γij = δi,j+n - δj,i-n. Imagine η as a column vector and η^T as a row vector when using matrix notation. (a) Show the canonical invariance of the Lagrange bracket by proving {u, v}η = {u, v}ζ for Mij = (∂)/(∂) and Mγ M^T = γ. (1 point) (b) Prove the following fundamental Lagrange brackets: (1 point) {qi, qj} = 0 = {pi, pj} and {qi, pj} = δij, or in matrix notation, {η, η} = γ. (c) Show that the Lagrange bracket and Poisson bracket satisfy the following relation: (1 point) {u, u}[u, u] = -1 or ∑k=1^2n{uk, u}[uk, uj] = δij.$

Added by Victoria C.

Question

Please give Ace some feedback