Try to use this code,
<form action="http://www.mydomain.com/wp-admin/admin.php" method="get">
<input type="hidden" name="page" value="my-plugin.php" />
<input type="text" class="mydatepicker" name="start_date" value=""/>
<input type="submit" value="submit">
</form>
This will show the url like this…
http://www.mydomain.com/wp-admin/admin.php?page=my-plugin.php&start_date=2014-04-03
Check and let me know your comments, if it works